<![CDATA[ Advanced Mathematics - freeCodeCamp.org ]]> https://www.freecodecamp.org/news/ https://cdn.freecodecamp.org/universal/favicons/favicon.png <![CDATA[ Advanced Mathematics - freeCodeCamp.org ]]> https://www.freecodecamp.org/news/ Eleventy Fri, 22 May 2026 22:36:58 +0000 60 <![CDATA[ 120 Free Online Math Courses from the World’s Top Universities ]]> https://www.freecodecamp.org/news/math-online-courses-from-worlds-top-universities/ 66c376b59a8c3b7871c66a55 Sun, 12 Jan 2025 06:00:00 +0000 In this article, we’ve compiled over 300 online courses offered by the 60 best universities in the world for studying math.

We did so by combining popular university rankings to identify the best institutions, and then using the Class Central database to find all their math online courses.

Methodology

I built the list following the same data-driven approach used to build the list of computer science courses from the top CS universities.

First, I identified the leading world university rankings. Since I was specifically interested in math, I looked at their latest rankings of the best universities for studying math (or closest superset). Here are the rankings I ended up using:

Then, we crawled and scraped each ranking.

Now that I had some data, I used Jupyter with Python to process it. I combined the three rankings into one by averaging the position of each university in each ranking. Then, I filtered out the universities that didn’t offer online courses, and limited the list to the top 60 institutions.

For the latest update of this article, Suparn used the same sources and methodology to find the top 60 universities offering online math courses.

83ffe01a-ea37-4faf-9cde-264d7b9eab73

Combined ranking: top 10 universities for studying mathematics in 2024

As you can see in the image above, I found that the top three math institutions are:

  1. Harvard

  2. MIT

  3. Stanford

Finally, we used the Class Central database, with its 250K online courses, to find all the math courses offered by the universities in the ranking.

The end result is a list of more than 300 online courses offered by 60 best universities in the world for studying math in 2025.

While processing the data, I noticed something interesting: 59 of the top 60 universities offer online courses, a lot more than I would have guessed. The world’s top institutions are very prolific creators of online courses.

Stats

  • Enrollments range from 21 to over 13 million. There are 8 courses with over 1 million enrollments

  • Altogether, the courses in this list have over 52 million enrollments, with an average of over 296 thousand enrollments

  • 222 courses are free and 98 are paid

  • 291 courses are in English, 19 Chinese, 7 French, and 1 each in Spanish, German, Korean.

  • Together, they account for 2,118 reviews at Class Central, with an average of 28 reviews

  • Average Rating 4.51 out of 5.0

  • 80 courses are beginner level, 57 are intermediate level, and 11 are advanced level.

General Mathematics (45)

Linear Algebra (32)

Differential Equations (12)

Statistics & Probability (54)

Calculus (30)

Mathematical Modeling (9)

Convex Optimization (17)

Probability Theory (8)

Others (113)

More Courses

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With over 300 courses to pick from, we hope you find something you like. But if these aren’t enough, check out Class Central’s catalog of 250K online courses or our thematic collections:

]]> <![CDATA[ Practical Guide to Linear Algebra in Data Science and AI ]]> https://www.freecodecamp.org/news/linear-algebra-roadmap/ 66d4614e55db48792eed3fa5 Tue, 04 Jun 2024 20:22:06 +0000

"In God we trust; all others bring data." – W. Edwards Deming

This famous quote from Edwards Deming perfectly captures the essence of modern Data Science and AI.

Data is the lifeblood of Data Science and AI fields – Machine Learning, Deep Learning, Generative AI and much more. And understanding how to analyze and manipulate data it is key to unlocking its full potential.

The key to understanding all these concepts is linear algebra – the unsung hero behind many powerful algorithms and techniques.

If you've ever felt a disconnect between the linear algebra you learned in school and its practical use in your career, you're not alone. If you believe you should study and work your way through an entire book of Introduction to Linear Algebra, then you are again not alone.

Many aspiring data science and AI professionals struggle to bridge this gap and think they need to spend countless hours to master mathematics for Data Science and AI. But don't worry, this guide is here to help.

I'll show you how linear algebra isn't just a theoretical concept or old fashioned forgotten area of expertise. You'll learn how it's a practical tool that you can use to solve real-world problems in your field.

Linear Algebra combined with Mathematical Analysis (called Calculus I and II in many undergrad studies) form the backbone of Machine Learning, Deep Learning, Computer Vision, and Generative AI. From building recommendation systems and training Neural Networks to analyzing medical images, understanding linear algebra opens up a world of possibilities.

In this guide, you'll discover:

  • Real-World Applications: We'll explore how linear algebra is applied across various industries, from healthcare to finance, and everything in between (with a special and detailed focus on Data Science and AI).

  • Practical Tips: You'll learn how to translate theoretical concepts into actionable steps for your data science projects.

  • Linear Algebra RoadMap 2024: You will get a roadmap for Linear Algebra in 2024 – on paper and in a video tutorial.

  • Career Development Resources: I will provide you resources to help you learn linear algebra and accelerate your career in data science and AI.

Whether you're a student, a recent graduate, or an experienced professional aspiring to become technical professional, this guide will equip you with the knowledge and skills to learn and leverage linear algebra effectively in your work. And you won't have to spend all your time on endless browsing and searching.

"Mathematics is like the producer of the movies: you don't see them but they are actually running the show." – Tatev Aslanyan

Table of Contents

  1. Core Linear Algebra Concepts

  2. Linear Algebra Roadmap

  3. Real-World Applications of Linear Algebra

  4. Resources for Learning Linear Algebra

Core Concepts in Linear Algebra that You Will Actually Use

Let's dive into the heart of linear algebra and explore the core concepts that you will leverage daily in your Data Science, Machine Learning, or AI journey.

Vectors and Matrices: The Building Blocks of Your Data

Think of vectors as lists of numbers (like NumPy arrays), and matrices as tables of numbers (multiple arrays stacked next to each other). In the world of data science and AI, vectors and matrices are your bread and butter.

Vectors can represent anything from customer characteristics (salary, age, height, income, purchase history) to word embeddings (numerical representations of words, text, and strings in general in natural language processing [NLP]). These vectors in datasets are commonly referred to as features – or, if used as response variables, as labels, dependent variables, and so on.

Matrices are powerful data structures that store datasets, with each row representing a data point and each column representing a feature. When you load your data and store it in a dataframe, all the rows of your data are basically the rows of your matrix, while all the features and response variables combined are the columns in your matrix.

Simple vector or matrix operations like addition, subtraction, multiplication of vectors and matrices are tools for data manipulation and transformation. These tools are used to normalize or standardize features, scale the data, combine different datasets or even perform forward pass/backward pass when training neural networks.

Linear algebra operations all power these common and daily tasks in Data Science and Machine Learning.

Linear Transformations: Manipulating and Transforming Data

In the world of data, transformations are the key. You need transformations to rotate an image and resize it.

These are also common ways to perform data augmentation in Computer Vision. Maybe you want to adjust the colors or contrast. These tasks are all done through linear transformations, which are essentially functions that map one set of data points to another.

In the world of linear algebra, multiplying a matrix by a vector (or another matrix), transposing the matrix and inverting it, is like applying a specific transformation to your data. This is incredibly powerful for:

  • Image and signal processing: Enhancing images, removing noise, or transforming audio signals.

  • Data preprocessing: Scaling features, standardizing variables, and preparing data for machine learning models.

  • Feature engineering: Creating new features by combining or manipulating existing ones through linear combinations.

Eigenvalues and Eigenvectors: The Essence of Your Data

Think of eigenvalues and eigenvectors as the DNA of your data matrix. These sets of important values reveal the fundamental characteristics and directions, respectively, of largest variation (information).

Once you know the eigenvalues and eigenvectors, you can quickly figure out which features in your data contain the most variation (that is information). This is basically your golden ticket for feature selection.

Eigenvalues and eigenvectors are essential in linear algebra, as they offer insights into matrix properties. They are particularly useful across various disciplines such as engineering, physics, data science and AI.

  • Eigenvalues indicate the factor by which an eigenvector is scaled by a matrix, revealing key properties like system stability or oscillation.

  • Eigenvectors are vectors that remain directed along the same line under a matrix transformation, only scaled in magnitude. They help simplify complex systems and elucidate structural properties of transformations.

Eigenvalues and Eigenvectors are essential for:

  • Dimensionality Reduction (PCA): PCA uses eigenvectors to identify the directions of greatest variation (variance) in your data, allowing you to reduce the number of features while retaining the most important information.

  • PageRank Algorithm: Google's famous algorithm uses eigenvectors to determine the importance of web pages.

  • Understanding data clusters: Eigenvectors help us to identify groups or clusters within your data.

Don't be intimidated by the names – eigenvalues and eigenvectors are simply numbers and vectors that describe the inherent structure of your data. Understanding them gives you a powerful lens through which to analyze and interpret complex datasets.

Matrix Factorization: Uncover Hidden Patterns in Your Data

Imagine a massive table of article ratings from thousands of users. Hidden within this data are patterns that reveal user preferences and article similarities.

Matrix factorization, particularly a technique called Singular Value Decomposition (SVD), is the key to creating such a recommender system.

SVD breaks down large matrices into smaller, more manageable matrices that reveal what are called latent factors. These are the underlying characteristics that explain why users rate things (like movies) the way they do. This is the algorithm behind famous recommendation systems like Amazon or Netflix, which use these latent factors to suggest items and movies you'll love.

But matrix factorization isn't just for building powerful recommender systems. It's a versatile tool used for:

  • Dimensionality reduction: Simplify your data by identifying the most important features.

  • Topic modeling: Discover hidden topics in a collection of documents.

  • Image compression: Reduce the size of image files without sacrificing too much quality.

  • Recommendation systems: Predict user preferences and similarities to generate meaningful recommendations and suggest relevant items.

Linear Algebra RoadMap – Your Path to Success

Now let's look at a roadmap that'll help guide you as you master Linear Algebra for Data Science and AI. It's a structured journey that builds upon foundational concepts and progressively delves into advanced topics with real-world applications.

This roadmap, from LunarTechs 25+ hour Linear Algebra Course is aligned with resources such as the Linear Algebra and Its Applications by David C. Lay, Steven R. Lay,and Judi J. McDonald (Cambridge Linear Algebra Book) and the Interactive Linear Algebra by Dan Margalit and Joseph Rabinoff (UBC Linear Algebra Book). It provides you with a solid foundation to tackle real-world problems in data science and AI.

Image

Image Source: LunarTech - Fundamentals to Linear Algebra

Refresh your Memory of High School Algebra

  • Begin by refreshing your understanding of Real Numbers & Vector Spaces, ensuring you grasp the fundamental properties and operations of numbers and vectors.

  • Refresh your knowledge of Angles and Trigonometry, essential for understanding vector relationships and transformations.

  • Make sure you are clear on Norm vs. Euclidean Distance, as norms quantify vector magnitude, and Euclidean distance measures the distance between vectors. This is a very important concept for your future journey of implementing math in real world.

  • Refresh your knowledge on the Pythagorean Theorem and Orthogonality, crucial for concepts like projections and orthogonal transformations.

  • Make sure you are clear on Cartesian Coordinate System for visualizing vectors and understand the geometric side of vectors.

Foundations of Vectors

  • Dive into Vectors and Operations, including vector addition, subtraction, scalar multiplication, and their geometric interpretations.

  • Study Special Vectors and Operations, such as unit vectors, zero vectors, and linear combinations.

  • Explore Advanced Vector Concepts, including linear independence, span, basis, and dimension, crucial for understanding vector spaces.

  • Master the Dot Product and its Applications, understanding its role in calculating angles, projections, and vector similarity.

  • Understand the Cauchy-Schwarz inequality – related to dot product and trigonometric concepts, which provides bounds on the dot product and has applications in various fields.

Foundations of Linear Systems and Matrices

  • Master Matrices and Solving Linear Systems, as learning how to represent systems of equations in matrix form and solve them using techniques like Gaussian elimination will help you understand ML and AI for real.

  • Study Core Matrix Operations, including addition, subtraction, scalar multiplication, matrix multiplication, and transposition.

  • Practice Gaussian Reduction, REF, RREF, row echelon form (REF), and reduced row echelon form (RREF) for solving linear systems and finding inverses.

  • Explore the concepts of Null Space, Column Space, Basis, Rank, Full Rank, essential for understanding the solutions and properties of linear systems.

  • Learn the Algebraic Laws for Matrices with Proofs, solidifying your understanding of matrix algebra.

Linear Transformations and Matrices

  • Dive into Linear Transformations and Matrices, and make sure you understand how matrices can represent linear transformations in vector spaces.

  • Learn how to Transpose a Matrix and its properties.

  • Study Determinants and Their Properties, understanding their significance in determining invertibility and calculating areas/volumes.

  • Master Transpose and Inverses of Matrices (2x2) and (3x3), essential for solving linear systems and understanding matrix transformations.

  • Explore Vector Spaces and Projections, understanding subspaces, orthogonal projections, and their applications in data science.

  • Understand and pratice the Gram-Schmidt Process for orthogonalizing a set of vectors, crucial for QR decomposition (popular Matrix Factorization technique) and other applications.

Advanced Linear Algebra Topics

  • Delve into Matrix Factorization, understanding techniques like QR decomposition, eigenvalue decomposition, and singular value decomposition (SVD).

  • QR Decomposition: Learn how to decompose a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R), useful for solving linear systems and least squares problems.

  • Eigenvalues, Eigenvectors, and Eigen Decomposition: Understand how to find these fundamental characteristics of a matrix and their applications in dimensionality reduction (PCA) and other areas.

  • Singular Value Decomposition (SVD): Learn this powerful matrix factorization technique widely used in data science for dimensionality reduction, recommendation systems, and other applications.

Here is the YouTube tutorial, Linear Algebra Roadmap 2024, which explains in even more detail the Linear Algebra Roadmap topic by topic.

By following this roadmap, you'll gain a comprehensive understanding of linear algebra concepts, starting from the basics and gradually progressing to advanced topics, equipping you with the skills necessary to tackle real-world problems in data science and AI.

Linear Algebra in Action: Real-World Applications in Data Science, AI, and Beyond

Mathematics is like producer of the movies: you don't see them but they are actually running the show.

In this section, we'll delve into specific examples that showcase the practical power of linear algebra across various cutting edge fields. You'll see how seemingly abstract concepts translate into real-world solutions that drive innovation and impact our daily lives.

Let's explore how linear algebra is revolutionizing many different industries.

Linear Algebra in Data Science and Machine Learning

Linear Regression

Linear Regression, which is a fundamental ML algorithm, relies on linear algebra to find the best-fit line (or hyperplane) that minimizes the error between predicted and actual values.

Matrices and vectors are used to represent data and model parameters, while matrix operations like inversion and transpose are crucial for solving the regression equations.

Application - House Price Prediction: Predicting housing prices based on features like square footage, number of bedrooms, and location. You can check out a complete end-to-end case study here.

Imagine you're a real estate agent trying to predict the price of a house. You have data on various features of different houses: the square footage, the number of bedrooms, and so on.

These features are put into a table-like structure called a matrix, denoted as X. Each row of X represents a different house, and each column represents a specific feature – for instance, one column might be the square footage, another the number of bedrooms. The prices of the corresponding houses are stored in another matrix, Y.

Your goal is to predict the price (Y) of a new house based on its features (X). Linear regression uses linear algebra to find the relationship between these features and the price.

The "line of best fit" is defined by a set of coefficients called Beta (β). Each element in Beta corresponds to a particular feature in X and tells you how much that feature influences the final price. We also add an error term, epsilon (ε), to account for any random variation in house prices that can't be explained by the features we have.

Under the hood, linear regression uses matrix operations like transposes, inverses, and matrix multiplication to calculate the Beta values that give the best prediction. So, while you might not see the complex math directly, linear algebra is the engine that powers the price estimates you see on real estate websites!

Logistic Regression

This algorithm uses linear algebra to model the relationship between customer features (like tenure, usage patterns, and demographics) and the probability of churn. Coefficients learned through linear algebra determine the importance of each feature in predicting churn.

Application - Customer Churn Prediction*:* A telecommunications company might use logistic regression to identify customers at high risk of switching to a competitor. The model analyzes factors like call duration, data usage, customer service interactions, and billing issues.

Support Vector Machines (SVM)

SVM is a powerful classification algorithm that uses linear algebra to find the optimal hyperplane separating different classes of data. The concept of vector dot products is central to calculating distances and determining the margin between classes.

Application - Spam Email Identification: classifies emails as spam or not spam based on features like word frequency and email length.

Feature Extraction

Techniques like Principal Component Analysis (PCA) leverage linear algebra to extract the most important features from image data, reducing dimensionality and improving computational efficiency.

Application - Object Detection: Object detection algorithms often use PCA to reduce the complexity of image features before classification.

Principal Component Analysis (PCA)

PCA leverages linear algebra, specifically eigenvalues and eigenvectors, to identify the directions of greatest variance in high-dimensional data. By projecting data onto these principal components, PCA reduces dimensionality while preserving the most important information.

Application - Genomics: In genomics research, PCA is used to analyze gene expression data from thousands of genes. By reducing the dimensionality of the data, researchers can more easily visualize patterns and identify relationships between genes.

Linear Algebra in Deep Learning and Generative AI

Neural Networks

The foundation of deep learning, neural networks are essentially interconnected layers of nodes (neurons) that process information using linear algebra operations. Matrices represent weights and biases, while matrix multiplication and activation functions propagate signals through the network.

Application - Image Classification with CNNs: Image classification using convolutional neural networks (CNNs), where linear algebra is used for filtering operations and feature extraction.

Image Transformations

Linear algebra is used extensively for image manipulation, including rotation, scaling, translation, and shearing. Matrices are used to represent these transformations, and matrix multiplication is used to apply them to images.

Application in Facial Recognition: Facial recognition software uses linear transformations to align and normalize face images for comparison.

Generative Adversarial Networks (GANs)

GANs, a type of generative model, use linear algebra operations within their neural networks to learn and generate new data samples, such as images or text.

Application in Generating Images: Generating realistic images of human faces or creating artwork in the style of famous painters.

Variational Autoencoders (VAEs)

These generative models use linear algebra to encode high-dimensional data into a lower-dimensional latent space. This space is structured to follow a standard distribution (usually a Gaussian), making it easier to sample new data points and generate diverse outputs. Matrix operations are crucial for encoding and decoding data between the original space and the latent space.

Application in Healthcare with VAE: A pharmaceutical company uses VAEs to generate novel molecular structures with desired properties. By encoding existing drug molecules into a latent space, the VAE can explore this space to generate new candidate molecules that potentially have therapeutic effects.

All these examples are just the tip of the iceberg. Linear algebra plays an important role in countless applications across data science and AI. By understanding its core concepts, you'll be equipped to not only use existing algorithms but also contribute to the development of new and innovative solutions.

Practical Tips, Tools, and Resources for Learning Linear Algebra

I often get asked about the best resources for learning linear algebra and specifically what book to read to master it. My advice, as someone who's gone through the traditional academic route of textbooks and countless theoretical examples: don't feel obligated to read those massive linear algebra textbooks cover to cover.

They are valuable resources, but not the most efficient way to learn if your goal is to apply linear algebra in your data science career.

Instead, focus on a clear, guided, and time-efficient approach to learning the theory that you'll actually use. Then, prioritize practical application: learn how to implement these concepts in Python and utilize them in machine learning, deep learning, and other areas. This is a far more effective use of your time.

So, where should you start? The answer is to understand the essentials and implement these concepts with clear guidance. This will help save your time and make it easier to learn effectively.

First of all, make sure you read through the roadmap and watch the accompanying video that I included above. And then you can move on to the following:

Fundamentals of Linear Algebra: 25+ Hour Course

If you're overwhelmed by dense textbooks or endless theoretical examples, you're not alone. Linear algebra can be intimidating, but it's a crucial foundation for anyone working in data science and AI.

LunarTech's concise, career-focused course will equip you with the skills you need to excel in data science and AI. Try it now – it's included in our LunarTech Max plan at the moment. You can sign up for the Fundamentals of Linear Algebra 25+h Course here.

Image

Source: Fundamentals to Linear Algebra 25+h Course

  • Undergraduate Students: Ace your linear algebra exams and build a strong foundation for further study.

  • Working Professionals: Gain the skills you need to understand, create, and implement cutting-edge AI and machine learning algorithms.

Whether you're a student looking for a clear and concise approach to linear algebra or a professional aiming to advance your career in AI and data science, this course will equip you with the knowledge and skills you need to succeed.

Free Linear Algebra Crash Course – 7 Hours

This shorter, demo version of the main course is perfect for learners who need a quick yet comprehensive overview of the key concepts in linear algebra. It’s great as a refresher or for those who need to understand the basics before diving into more complex topics, and is a starting point to learn Linear Algebra.

You can check out this Linear Algebra Crash Course - Mathematics for Machine Learning and Generative AI [Full 7h] to get started.

freeCodeCamp Linear Algebra Course and Textbook

You can also check out this free freeCodeCamp course that covers key Linear Algebra topics like Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. There are many practical examples, and the course encourages you to work through each of them to solidify your knowledge.

There's also a link to download the professor's textbook if you're interested in that.

Connect with Me

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Want to discover everything about a career in Data Science, Machine Learning and AI, and learn how to secure a Data Science job? Download this free Data Science and AI Career Handbook.

Thank you for choosing this guide as your learning companion. As you continue to explore the vast field of machine learning, I hope you do so with confidence, precision, and an innovative spirit. Best wishes in all your future endeavors!

]]> <![CDATA[ How to Calculate Definite and Indefinite Integrals in Python ]]> https://www.freecodecamp.org/news/calculate-definite-indefinite-integrals-in-python/ 66d460c5bd438296f45cd3a4 Mon, 31 Jul 2023 18:02:27 +0000 By Roy Chng

Python is a versatile programming language that offers libraries and tools for scientific computing and mathematical calculations.

Many essential mathematical operations frequently involve definite and indefinite integrals. In this article, we will explore how to perform these calculations using Python.

How to Calculate Single Variable Definite Integrals

Install SciPy

Before we start, we need to install the SciPy module. It provides a collection of Mathematical algorithms and functions that we'll use.

You can do this by running the following command in a terminal:

pip install scipy

To calculate single variable definite integrals, we need to first import quad from scipy.integrate. It is a general purpose function used to calculate single variable definite integrals.

from scipy.integrate import quad

Elementary Functions

From there, we'll need to define the integrand as a function in Python.

For example, if we wanted to calculate the integral of x-squared, we would define the integrand as a Python function like so:

def integrand(x):
    return x**2

Once we define the integrand, we can calculate the definite integral using the quad function like this:

print(quad(integrand, 0, 1))
# (0.33333333333333337, 3.700743415417189e-15)

In the above code, 0 represents the lower limit of integration and 1 represents the upper limit of integration. They can be any other number.

Image Result of the of integral of x^2 from 0 to 1 with the associated code

In this example, we calculate that the estimated result of the integral from 0 to 1 of x-squared is approximately 0.333 with an absolute error of roughly 3.7e-15.

The quad function returns a tuple of an estimation of the definite integral followed by the absolute error of the estimation.

What the quad function does is essentially evaluate the integrand function at multiple different values between our limits of integration to be able to calculate an estimate of the integral.

Another example would be if I wanted to calculate the integral of (x+1)/x**2. We would first define it as a function in Python, and pass it into the quad function along with the limits of integration:

from scipy.integrate import quad

def integrand(x):
    return(x+1)/x**2

print(quad(integrand, 1, 2))
# (1.1931471805599452, 1.3246594716242401e-14)

Image Result of the of integral of (x+1)/x^2 from 1 to 2 with the associated code

In this example, we calculate that the estimated result of the integral from 1 to 2 of x +1 all over x-squared is approximately 1.19 with an absolute error of roughly 1.32e-14.

Other Common Functions

If we wanted to use common mathematical functions such as sin(x) or log(x), we can use another Python package for scientific computing – NumPy. You can install the package using the following command:

pip install numpy

By importing it, we have access to these common functions which we can use in our integrand:

from scipy.integrate import quad
from numpy import log, sin

def integrand(x):
    return log(sin(x))

print(quad(integrand, 0, 2))
# (-1.1022223889049558, 1.2237126744196256e-15)

Image Result of the of integral of log(sin(x)) from 0 to 2 with the associated code

In this example, we calculate that the estimated result of the integral from 0 to 2 of log(sin(x)) is approximately -1.10 with an absolute error of roughly 1.22e-15.

A full list of mathematical functions that NumPy provides is in their documentation.

How to Use Constants

NumPy also provides useful constants such as e and pi, as well as inf. It's a floating point representation of positive infinity. We can use it to calculate a definite integral that converges.

from scipy.integrate import quad
from numpy import inf, exp

def integrand(x):
  return exp(-x)

print(quad(integrand, 0, inf))
# (1.0000000000000002, 5.842606742906004e-11)

Image Result of the of integral of e^x from 0 to infinity with the associated code

In this example, we calculate that the estimated result of the integral of e raised to the negative x from 0 to infinity is approximately 1.00 with an absolute error of roughly 5.84e-11.

How to Calculate Multi-Variable Integrals

Double Integrals

To calculate double integrals, we need to import the dblquad function from scipy.integrate:

from scipy.integrate import dblquad

We define the integrand in a similar way to definite it with one variable, only this time we specified two arguments instead.

def integrand(y, x):
    return x*y**2

We can then calculate the definite integral using the dblquad function given by scipy.

Note that the integrand is a function that needs to accept y as the first parameter and x as the second parameter.

print(dblquad(integrand, 0, 1, 2, 4))
# (9.333333333333334, 2.0679162295394134e-13)

Image Result of the integral of (xy^2) dxdy from 2 to 4 for y and integral from 0 to 1 for x with the associated code

In this example, we calculate that the estimated result of the double integral x times y-squared from x = 0 to 1 and from y = 2 to y = 4 is approximately 9.33 with an absolute error of roughly 2.07e-13.

The function requires us to pass in the integrand, and the lower and upper limits of integration for x, followed by the lower and upper limits of integration for y.

Variable Limits

To calculate integrals with variable limits, we'll need to define functions for the lower and upper limits of integration for y in terms of x:

def upper_limit_y(x):
    return x**2

def lower_limit_y(x):
    return x

def integrand(y, x):
    return x+y

print(dblquad(integrand, 0, 2, lower_limit_y, upper_limit_y))

In this example, we calculate that the estimated result of the double integral of x+y from x = 0 to x = 2, and from y = x to y = x^2 is approximately 3.2 with an absolute error of roughly 1.10e-13.

Image Result of the double integral (x+y) dydx from 0 to 2 for x and integral from x to x^2 for y with the associated code

Triple Integrals

To calculate triple integrals, we can use the tplquad function:

from scipy.integrate import tplquad

def integrand(z, y, x):
    return z*(x+y+z)

print(tplquad(integrand, 0, 1, 4, 5, 0, 1))
# (2.8333333333333335, 3.6983326566167174e-14)

The function requires us to pass in similar arguments, being the upper and lower limits of integration in x, y and z.

In this example, we calculate that the estimated result of the triple integral of z multiplied by (x+y+z) from x = 0 to x = 1, y = 4 to y = 5, and z = 0 to z = 1 is approximately 2.83 with an absolute error of 3.70e-14:

Image Result of the triple integral z(x+y+z) dxdydz from 0 to 1 for x, 4 to 5 for y and 0 to 1 for z with the associated code

How to Evaluate Single Variable Indefinite Integrals

To calculate single variable indefinite integrals with Python, we need to use the SymPy library. It's used for symbolic computation and involves exact computation using variables. To install it, install the SymPy module:

pip install sympy

Once it has been installed, we can import the Symbol and integrate methods from sympy:

from sympy import Symbol, integrate

We first need to define the variables used in the integrand:

x = Symbol('x')

After that, we can integrate the function using the integrate method that SymPy provides. It expects two arguments: the first is the integrand, and the second is the variable we are integrating with respect to.

For example, if we wanted to integrate x-squared with respect to x, we can define the integrand in Python as x**2:

print(integrate(x**2, x))
# (x**3)/3

Image Result of the indefinite integral of x^2

In this example, we calculate the integral of x-squared which is x-cubed over 3.

Note that SymPy doesn't add the constant of integration, but it is implied.

SymPy also provides other common functions such as sin(x) and exp(x) that we can use.

Before using them, we first need to import it from sympy:

from sympy import Symbol, integrate, sin

Using the imported sin function, we can then evaluate the integral of sin(x).

x = Symbol('x')
print(integrate(sin(x), x))
# -cos(x)

In this example, we calculate the integral of sin(x) which is -cos(x):

Image Result of the indefinite integral of sin(x)

Sympy provides the full list of mathematical functions you can use in their documentation

Summary

In this tutorial, we went over the basics of how to calculate both definite and indefinite integrals in Python. We also looked at how to calculate integrals of elementary functions, ones that involved common mathematical functions, as well as using constants.

We made use of popular Python libraries for scientific comptutation and went over examples of calculating integrals.

If you enjoy my writing, consider subscribing to my YouTube channel.

Happy Coding!

##

]]> <![CDATA[ How to Use Graph Theory to Build a More Sustainable World ]]> https://www.freecodecamp.org/news/the-value-of-graph-theory-within-sustainability/ 66baef4fd453cb5eb7951596 Fri, 19 Aug 2022 20:50:13 +0000 Discrete mathematics is an area of math based on the study of formal structures whose nature is fundamentally separate and distinct.

This means it focuses on integers and natural sets of numbers, shapes, and other objects that you can count finitely or distinguish from one another. It models reality in a manner specifically suited to certain real-world applications.

From industry and logistics to computer science and telecommunications, having a quantized representation of everything around us has led to magnificent advances in our understanding and control of the physical world.

It's important to have at least a rough idea of the main distinctions between discreteness and continuity to address graph theory. But these aren't very well-known concepts to people outside the world of mathematics.

At first, continuous math is the one predominantly taught in the education system due to its versatility, usefulness, and practicality in most areas.

It’s based on the analysis of real numbers and functions that encapsulate mappings between these quantities, along with the notion of the infinitesimal change of a variable. This results in a series of tools like limits or derivatives that constitute calculus.

On the other hand, the discrete paradigm is more straightforward and intuitive, with the exception of a few cases. And its finiteness is given by the primordial element constituting it – sets.

Among the most notorious use areas are those whose main components imply algorithms and data structures. Although the use cases of math are not what most people think they are.

In the real world, we don’t often face problems in the same way as in the education system. Indeed, discrete ways of approaching riddles and modeling the input data we need to come up with a solution are more usual than continuous ones, especially regarding system optimization issues.

For this reason, we should reconsider the role of this way of doing mathematics since it involves the development of critical/computational thinking. This is crucial for the current era in which we are surrounded by technology. It also involves the improvement of problem-solving skills, making it possible for us to face any new challenges.

By doing so, we can see how relevant it is to apply a solid mathematical foundation to common global threats that are increasing daily, like misinformation, lack of fluency in handling technology, geopolitical instability, and even climate change.

Notwithstanding the apparent remoteness between the latter issue and graph theory itself, we should think about the way we live and the system by which our civilization is maintained as we know it.

Goals of this Article

This article aims to explain graph theory, one of the most significant components of all discrete mathematics, in an intuitive, simple, and visual way. I'll also try to guide its use towards the development of new disruptive techniques applicable in areas such as environmental care, necessary to preserve and regenerate our nature.

Effectively achieving this will not only foster curiosity or inspire readers who may intend to continue learning, but will also contribute to a further rising in society’s awareness about sustainability issues. This will increase the likelihood that in the future, the problems that scientists predict to be threatening to our existence and the existence of life on the planet will be curbed, thanks to scientific knowledge and specifically the contribution of graph theory.

Still, given its broad scope, it will be impossible to explain graph theory entirely in this article. So, I'll focus on the visual side of any explanation over the formal one since you can easily consult that in any textbook. That will also provide a different point of view from certain definitions.

Also, it's essential that we treat the idea of a graph as comprehensively as possible. We'll focus on its history, representation, and most descriptive properties instead of advanced concepts like singular cycles. This will help you grasp the kernel of graph theory and prepare you to learn these advanced concepts more easily.

Here's what we'll cover:

  1. Basic Elements of Graph Theory
  2. History of Graph Theory
  3. Definition of a Graph
  4. Representations of Graphs
  5. Properties of Graphs
  6. Algorithms and Graph Theory
  7. Why Are Graphs Important in Achieving Sustainability?
  8. Conclusion

Basic Elements of Graph Theory

Whether you are new to Graph Theory or already know something about it, reviewing the basics is always worthwhile.

First, let’s introduce the idea of a “graph” with a usual representation you may have seen:

Image Example of an arbitrary graph

Above, you have a graph where we can see, at the most fundamental level, two different building blocks: vertices (shown as circles) and edges (shown as lines connecting circles).

You can create a structure with those elements that can encapsulate the functioning of many systems present in our life that we don’t even realize.

But, most surprising of all is that graph theory as a whole is derived from such a simple concept as objects linked to each other.

History of Graph Theory

To understand the origin of this idea, we have to look back to the 18th century, when Leonhard Euler solved the famous Seven Bridges of Königsberg problem.

By that time, the city was crossed by the Pregel river, generating four pieces of land interconnected with seven bridges, as seen below:

Image _Image extracted from [here](https://en.wikipedia.org/wiki/File:Konigsbergbridges.png" rel="noopener)

The task consisted of finding a path that crosses all bridges without passing by the same bridge twice, starting and ending at the same point.

At first, with so few bridges, it may be easy to find a brute force solution by trying combinations of paths. But, since we don’t know if a feasible solution exists, it’s helpful to formalize the problem elements and correctly prove its solvability before starting any process.

Also, if the number of bridges increases, it will become much more complex to solve, as the combinations increase remarkably fast.

Image Königsberg problem displayed as a graph

As seen above, Euler represented land areas with graph vertices (also called nodes) and bridges with edges, concluding that it was impossible to have such a traversal through the graph.

Briefly, if we look at the number of edges incident to each vertex, we will see that every value is odd for every node, meaning that the graph does not have an eulerian cycle. This means that it’s not an eulerian graph, and we can’t positively prove the problem.

Nevertheless, this approach represented a breakthrough in the mathematical conception of various questions that were yet unsolvable. Euler’s contributions to the elaboration of this theory, which has been perfected and broadened over the years, made him one of the most influential mathematicians of his time.

Definition of a Graph

Now that you know what a graph looks like drawn on a diagram, let’s review the official formal definition:

A graph G is a pair of sets (V, E) where V is a non-zero set containing the graph’s vertices and E is a set made of element pairs belonging to V.

Image Formal definition of a graph with its corresponding sets

Above, we represented the two main components of a graph in two corresponding sets, one for vertices V and another for edges E. So, our graph G is ultimately an ordered pair of these sets. But before we continue, we must look inside those sets to see what they look like and understand why.

On the one hand, V is a collection of items v in which each element contains the necessary data to define a vertex. Abstractly they are called with the letter v and a numerical subindex.

But in practice, they can be complex objects holding parameters, profiles, and so on.

On the other hand, the set of edges E is a little more complicated to define since it needs to determine the connections between vertices. In this case, the elements are unordered pairs of vertices from set V such that each pair is of the form {x, y}.

Image Both formal and graphical representations of a graph

To familiarize yourself with these structures above, you have an arbitrary, fully defined graph with its respective sets. In V, you can see all the vertices numbered from 1 to 5 and placed in the upper diagram in a specific distribution, but you can arrange them according to your needs.

Meanwhile, in E, you can observe all the edges (lines) establishing an interconnection link between vertices.

The appropriate terminology to address this link is the following: for instance, if we have the edge {v1, v4}, we call it incident to v1 and v4. Also, those vertices are denoted to be adjacent since an edge links them.

As you may notice, there isn’t an edge {v4, v1} in E. But to find an explanation for this phenomenon, we have to introduce the main distinction that generates two classes of graphs.

The first one (undirected), to which the above examples belong, includes all graphs whose edges can be traversed in both directions. This makes them unordered pairs of vertices.

On the other hand, we can have a graph in which all its edges can only be traversed in one direction, that is, from one vertex to another exclusively. Thus, its vertex pairs on E set must be ordered, meaning that going from v1 to v4 is not the same as going from v4 to v1. This second class is known as a directed graph.

Image Example of a directed graph.

Before learning how to represent a graph computationally to perform operations on it, you need to understand the vertex degree concept.

In undirected graphs, the degree of a vertex refers to the number of edges incident to it, considering that self-connecting edges (loops) count as 2 in the total score.

By contrast, in directed graphs, we have in-degree and out-degree values for each vertex, representing the number of incoming and outcoming edges, respectively.

Representations of Graphs

Image The 2 most popular ways to computationally store a graph

Sometimes, the most intuitive solution for a problem is not always the most efficient in computer science. In this context, it generates different ways of representing a graph according to a problem’s nature.

What is an Adjacency Matrix?

An adjacency matrix is one of the most popular methods to store a graph on a computer. But its major drawback is unused memory consumption.

For directed graphs like the one above, there is a matrix size |V|x|V| (being |V| the cardinality of the vertices set, thus the number of vertices on the graph) where each element can be a 0 if there is no connection between vertices or a 1 if the row element links the column one by an outgoing edge. Also, if the graph is weighted, the 1 value is substituted with the weight parameter associated with each edge when necessary.

However, if the graph is undirected, the same criteria apply with the difference that no distinction is made between outgoing and incoming edges this time. So there will be a 1 value if an edge exists between the row and column elements.

Image Adjacency matrix element definition for each type of graph.

What is an Incidence Matrix?

Similar to the previous method, there is a matrix size |V|x|E| in which the same rules are fulfilled. The difference is that if an edge e is incoming to a vertex v, the corresponding element will be a -1 instead of 0.

How to Use Adjacency Lists

When using matrices, if the graph has many vertices but few edges (a sparse graph), the matrix will contain a high number of zeroes. This wastes a lot of memory and makes the representation inefficient in terms of space.

To solve this issue, adjacency lists appeared as an alternative replacing matrices with a combination of different data structures – arrays, and linked lists.

The kernel of this method is an array containing all the graph’s nodes. Each array element will have a linked list holding each leading node’s neighbor vertices (adjacent vertices). In the case of directed graphs, only the neighbor elements connected by an outgoing edge from the lead node will be inside the linked list.

Image Example of a graph depicted as an adjacency list

So if we have a dense graph with a high number of edges, we should store it in matrix form. This has the advantage of O(1) time complexity when checking vertex connection and matrix symmetry along the main diagonal in undirected graphs.

But, if our graph is sparse, the low density of edges make an adjacency list the best choice to depict it computationally.

Properties of Graphs

Like any other mathematical object, graphs have specific properties that make them unique and functional for their purposes. Some have to do with their composition, others with topology, and even accessibility.

Undoubtedly, the most relevant properties concern traversals, since they allow us to model and optimize real-world scenarios.

What is a graph traversal?

First, we need a starting node v1 and an ending node v2 to traverse a graph. Then, we can define a walk from v1 to v2 as an alternate sequence of vertices and edges. There, we can go through these elements as much as we need, and there is always an edge after a vertex (except the last one).

In the case of v1 being equal to v2, the walk would be closed.

Still, we can add repetition restrictions. So if we want a walk in which no edge is repeated, it’s renamed as “trail”. Consequently, if the trail is closed, it would be denoted as “circuit”.

The same happens if we restrict vertex repetition – the walk renames to “path,” and a closed path is known as a “cycle”.

Image Formal and graphical examples of network traversal types.

Image

This traversing ability comes along with an interesting property valid for all existing undirected/directed graphs. It’s formalized as follows:

Image Formalization of the degree sum property.

This establishes that the sum of all vertex degrees equals two times the cardinality of the edge set in an undirected graph. If directed, the sum breaks into two terms, each referring to each node in and out-degree.

This is fairly straightforward to prove, because every time you add an edge to a graph, you need two vertices to build the pair of elements stored on E. So if you add a loop (edge linking a node with itself), you anyways need to define a pair of elements from V, regardless of whether they are the same.

This characteristic supports us when solving questions like:

Given a 6-regular graph (with all its vertex degrees set to 6) of n vertices, how many edges will it have?

As its resolution is immediate, going deeper when thinking about similar questions improves your understanding of its nature and why it's that way.

What is Connectivity?

Now, let’s move on to the properties related to the graph’s linking capability. Starting with an undirected graph, we can assure that a vertex v reaches u if there's a path from v to u. Also, we can look at the whole graph and define it as connected if every pair of vertices in it is indeed connected.

Being connected is often associated with the uniqueness of its components. That is, if we end up with a disconnected graph, its number of components will always be greater than 1.

You can imagine a component as a zone of the graph isolated and disconnected from the rest of the vertices. And this, if we consider a graph, will be connected and will only have one connected component as if it were a connected graph.

Image Example of a 2 component made graph.

In contrast, when dealing with directed graphs, two vertices u and v are said to be strongly connected if they can reach each other and weakly connected if they are connected on the underlying (all edges replaced by undirected ones) graph.

As you can imagine, these properties generate many possibilities and new characteristics to consider.

To briefly mention, we can take advantage of the discrete nature of graphs to remove nodes and edges from them. Therefore, concepts such as articulation points or bridges emerge as one of the simplest ways to study a graph’s weak points.

An articulation point is a vertex that, if we remove from the graph together with all its incident edges, the graph will increase its connected components.

Likewise, a bridge is just an edge meeting the same previous condition with the difference that no vertex is removed from the graph.

Image Visual example of an articulation point and a bridge.

As an extension to the properties section, it’s worth mentioning some tools and characteristics of graphs that will help us recognize the key of the algorithms we will see later:

What are Subgraphs?

Their name is an appropriate indicator of what subgraphs are, since it is quite illustrative. A subgraph is a collection of vertices and edges that we can extract from an arbitrary graph G to form another graph, usually undersized.

Formally, a graph H is a subgraph of G if it’s formed by a subset of vertices of G and similarly a subset of edges of G, with every edge being a valid pair of nodes.

Image Example of a subgraph validity

The number of classifications and research we can perform about subgraphs makes it impossible to cover everything here. But the basis for further learning we'll start with the following ideas about its morphology, topology, and composition.

A subgraph H spans a graph G if both have the same vertices stored on V set. In this situation, subgraph H is known as a spanning subgraph.

Image Example of a spanning subgraph.

Given a graph G, if we apply the vertex removal operation n times with n<|V|, the resulting graph will be an induced graph.

Image Steps taken to reach an induced graph.

Topology doesn't just concern subgraphs. It's also mainly studied with general graphs. So reviewing some broad classifications and features will make graph theory more manageable.

A graph is said to be complete if it’s undirected, has no loops, and every pair of distinct nodes is connected with only one edge. Also, we can have an n-complete graph Kn depending on the number of vertices.

Image Example of the first 5 complete graphs.

We should also talk about the area of graph coloring. A graph is bipartite when its nodes can be divided into two disjoint sets whose union results in the whole initial vertex set, with the condition that every edge has its extremes on both sets simultaneously. This allows for the possibility of coloring each vertex set with a different color.

Also, it can be a complete-bipartite graph if both sets are densely connected (every vertex of one set is connected with all vertices of the other collection).

Image Some examples of arbitrary complete-bipartite graphs.

You might also need to represent a graph in a plane without any of its edges intersecting. Then, if possible, the graph will be planar. To better understand the state of this characteristic, we can use Kuratowski’s theorem. It involves advanced concepts like isomorphism and homomorphism concerning k5 complete and k3,3 complete bipartite graphs.

Image Visual difference between planar and non-planar graphs.

What are Particular Cycles?

Finally, some graph features deserve special attention. For example, when it’s a matter of cycle finding, there's a deep relationship with vertex degrees, integral graph topology, and traversability.

To visualize this relationship, we'll return to the Königsberg problem. In it, we need to traverse all the graph’s edges without repeating any of them, starting and finishing in the same vertex.

Since graphs were new then, Euler developed a solution by defining a unique type of cycle only found in graphs meeting precise conditions – like the degree of all their nodes being even.

These cycles were named Eulerian after their creator, and every graph that has one is also called an Eulerian graph.

There are also Eulerian Paths. These remove the condition of having to start and end on the same vertex and require the graph to have exactly two odd-degree nodes, which will be the path extremes.

Image Visualization of an eulerian path.

Also, suppose we focus on contemporary issues like the traveling salesman problem (TSP), an NP-Hard problem mainly used by delivery and logistic companies.

In that case, we will realize the relevance of the hamiltonian cycles and paths to support practical solutions to similar questions. Similar to the eulerianity, a graph is Hamiltonian if it contains a cycle in which every vertex is used instead of edges.

Image Visualization of a hamiltonian cycle.

These latter properties become challenging to deal with, given the complexity of the problems involved. Although, knowing the critical foundation supporting everything around them allows us to continue exploring with reasonable confidence.

Algorithms and Graph Theory

Once you have a solid grasp of graph theory, its elements, attributes, and tools, we should also review some basic algorithms comprising the principles of almost all other graph processes. Then we can move on to graph theory's use in climate preservation projects.

Breadth-first search algorithm

Here, we will only consider 3 algorithms since there are many types and very specialized ones for determined tasks.

To start with something simple and intuitive, we will unscramble Breadth-First Search. It's a graph traversal algorithm used to go through a graph in a breadthward motion.

In simple terms, it starts at an arbitrary vertex and iteratively visits its adjacent vertices, repeating this step until there are no more unvisited ones.

This behavior serves as the shortest path finder across all graph nodes, although you can stop the execution when a particular vertex is visited.

Image Representation of breadth-first search algorithm

Depth-first search algorithm

The second algorithm is a variant of the previous one, known as Depth First Search. Its goal is similar but is also useful when detecting cycles, connected components, topological sorting, or checking for graph bipartitions.

But the way it works differs in some aspects, like the precedence of the depth over breadth – that is, not all neighboring nodes are visited in each step. Instead, one of them is chosen for further deepening, and the process is repeated until the path reaches a dead end and recursively goes back to the starting node, visiting every vertex.

Image Representation of depth-first search algorithm

Dijkstra's Shortest-Path algorithm

Finally, the last one we will treat is Dijkstra’s algorithm, the most widespread Single Source Shortest Path problem solver ever created.

It’s designed to operate in weighted graphs with non-negative weights, and tries to find the most efficient route between 2 selected nodes.

Compared to the previous algorithms, that change increases the number of steps before completion. However, the key idea behind it is straightforward:

Image Example Graph used to explain Dijkstra's algorithm

As you can see in the above example, if we want to go from v1 to v2, we can select the edge between them and arrive at our destination after traversing six distance units.

On the other hand, if we choose to go through the v3 or v4 paths, we would be walking seven units. So we need to make a decision on whether or not to take a particular path.

On large graphs, the algorithm calculates the provisional shortest paths along every single node. It then updates these values and minimizes “distance” (given by weights) by a complete graph traversal, as you can see in this animation.

Why Are Graphs Important in Achieving Sustainability?

At this point, you may realize that Graph Theory is valuable because it can encapsulate and abstractly model problems of a nuanced nature. Especially those problems whose origin stems from society’s need to pursue a degree of globalization that brings a standard of wellness to everyone’s lives.

Yet many of us are unaware that the comfort we currently enjoy brought about by advances in communications, transport, nutrition, and entertainment requires the coordinated operation of complex systems to be in place.

So the overpopulation experienced since the twentieth century causes these systems to be so massive that they entail a severe environmental impact based on CO2 emissions and the systematic dumping of waste into natural environments.

Graphs can help with transportation of goods

In this context, everything involving the transport of goods and logistics contributes a significant amount of CO2 to the atmosphere. Here is where using graphs has a clear benefit for the environment. They can find optimal paths between cities or world locations, reducing the emissions of the vehicles engaged in such transport.

For example, you can experiment with Google Maps by tracing routes between distant places. You will notice it can automatically choose an appropriate route, minimizing the corresponding environmental cost.

Google Maps is working is based on Single Source Shortest Path algorithms like Dijkstra or advanced ones such as A-star. A-star is a heuristic variant of Dijkstra. These are used in combination with other state-of-the-art graph mechanics used to add certain constraints to algorithms.

Graphs can help with waste management

Graphs also have a place in the global industry by simulating or directly managing networks, manufacturing processes, and schedules. They can potentially reduce the amount of incorrectly handled/wasted energy and resources.

It’s also worth mentioning the numerous possibilities that graphs have to offer when we deal with the problem of excessive waste accumulation.

Nowadays, it's widely believed that plants and trees are the major contributors to oxygen in our atmosphere thanks to photosynthesis. But we have to account that between 50% and 85% of the oxygen released into the atmosphere each year is produced under the sea.

Ironically, data on waste thrown into the ocean are constantly increasing as the consumer society advances on time, causing a dramatic impact on the actual lungs of our planet, as well as on the animal species it shelters.

To avoid having to decide where to dump our garbage, we can use graph theory to generate simulations of molecular physical systems, atomic structures, and chemical reactions to develop new recyclable or biodegradable materials. These would reduce the environmental impact of the products we use.

Also, these simulations have the potential to be useful in biology, where deciphering the ultimate workings of DNA can lead to better food quality, as well as more efficient mass-production methods.

Graphs can help with machine learning and AI

Finally, the most well-known application of graphs is in the field of machine learning.

Despite all the other significant uses for graphs in computer science (like communication networks, distributed systems, or data structures), machine learning has shown us with its exponential evolution over the last decade that it is a highly promising technology when tackling climate change.

Simply put, machine learning is a subset of artificial intelligence that focuses on enabling machines to learn and detect patterns on large datasets. Sometimes, this learning is inspired by natural phenomena like synapses on human neurons, resulting in new techniques such as Artificial Neural Networks.

Regarding environmental care, the ability of these techniques to analyze large amounts of data makes it possible to measure our effect on the planet better.

As a real working example, Joinus4theplanet is an initiative focused on taking advantage of social media to raise awareness about the value of sustainability. It has developed a machine learning system able to perform waste sorting with the help of convolutional models designed to process multidimensional data in order to palliate the effects of incorrect recycling.

Image Example of a social network is represented as a graph. Image from [Wikipedia.](https://en.wikipedia.org/wiki/File:SocialNetworkAnalysis.png" rel="noopener ugc nofollow)

Conclusion

If we want to maintain a considerable amount of progress inside our civilizations and provide a thriving future for the next generations, we have to consider graphs as an essential tool when rethinking the way our technological and economic systems work.

]]> <![CDATA[ What is a Factorial? How to Calculate Factorials with Examples ]]> https://www.freecodecamp.org/news/what-is-a-factorial/ 66b0c3bff8d7f56c3f394b08 Wed, 03 Aug 2022 16:32:53 +0000 A factorial is a mathematical operation that you write like this: n!. It represents the multiplication of all numbers between 1 and n.

So if you were to have 3!, for example, you'd compute 3 x 2 x 1 (which = 6). Let's see how it works with some more examples.

Definition of a Factorial

The factorial of a number is the multiplication of all the numbers between 1 and the number itself. It is written like this: n!. So the factorial of 2 is 2! (= 1 × 2).

To calculate a factorial you need to know two things:

  1. 0! = 1
  2. n! = (n - 1)! × n

The factorial of 0 has value of 1, and the factorial of a number n is equal to the multiplication between the number n and the factorial of n-1.

For example, 5! is equal to 4! × 5.

Here the first few factorial values to give you an idea of how this works:

FactorialMultiplicationResult
0!11
1!11
2!1 × 22
3!1 × 2 × 36
4!1 × 2 × 3 × 424
5!1 × 2 × 3 × 4 × 5120
6!1 × 2 × 3 × 4 × 5 × 6720
7!1 × 2 × 3 × 4 × 5 × 6 × 75040
8!1 × 2 × 3 × 4 × 5 × 6 × 7 × 840,320
9!1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9362,880

What is a Factorial Used For?

Practically speaking, a factorial is the number of different permutations you can have with n items: 3 items can be arranged in exactly 6 different ways (expressed as 3!).

For example, let's see all the arrangements you can have with the three items, A, B and C:

ABC
ACB
BAC
BCA
CAB
CBA

And in fact, 3! = 6.

How to Calculate the Factorial of 0

Looking at the factorial from this point of view, what's the factorial of 0?

Well, how many different ways can you arrange 0 elements?

There is exactly 1 way to arrange zero elements. And that's making a sequence of zero elements.

Factorial Use Cases

You typically use a factorial when you have a problem related to the number of possible arrangements. Let's look at some example problems.

Factorial example problem 1: the letters in the word "camper"

How many different ways can you arrange the letters of the word camper?

The word camper has 6 letters, so the number of possible arrangements is given by the factorial of 6: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. That would have been a pretty big number of arrangements to find by hand, wouldn't it?

Factorial example problem 2: drawing colored balls from a bag

Let's say there are three balls in a bag – one green, one blue, and one yellow.

If you draw the three balls in sequence, what chance is there that you'll get the yellow first, the green one second, and the blue one last?

Maybe now you are wondering what chances have to do with factorials – well, in a moment you will see.

There are 6 possible sequences in which the balls can be drawn: 3! = 6.

There is a chance of 1 over the total number of possibilities to get the yellow-green-blue sequence, so that is 1/(3!) or 1/6 or 16.7% chance to get the desired outcome.

How to Calculate a Factorial Programmatically with JavaScript

There are two ways to calculate factorials programmatically in JavaScript:

How to calculate a factorial in JS with recursion

Let's get back to the two things to know when calculating a factorial – that is 0! = 1 and n! = (n - 1)! × n. We can use the first one to create the base case of the recursive function, because in that case we know the result already.

function factorial(n) {
  if (n === 0) {
      return 1;
  }
}

The second thing to know about how to calculate a factorial, n! = (n - 1)! × n, can be the recursive case.

function factorial(n) {
    if (n === 0) {
        return 1;
    } else {
        return factorial(n-1) * n;
    }
}

How to calculate a factorial with a JavaScript while loop

We said before that 0! = 1. So, to calculate the factorial of a number with a loop, we can initialize a variable to 1, and multiply the numbers from n to 1 by the variable inside the loop.

In this way, if the input is higher than 1, the output will easily be 1.

function factorial(n) {
    let result = 1;
    for (n > 1) {
        result *= n;
        n--;
    }
    return result;
}

Conclusion

The factorial is a pretty important operator to know if you are interested in statistics and probabilities.

In this article you have learned a how to calculate a factorial, a simple application, and you have seen how to calculate it using JavaScript.

Have fun with it!

]]> <![CDATA[ Learn Foundational Mathematics on freeCodeCamp – Interactive Math Courses Open Beta ]]> https://www.freecodecamp.org/news/freecodecamp-foundational-math-curriculum/ 66b0c4f8c01d78b26657b49b Mon, 06 Jun 2022 21:58:53 +0000 Do you want to use Python to solve math problems? Then freeCodeCamp's new Foundational Math course will guide you through the essential math you need to know to prepare you for the Data Science curriculum.

You will get a math refresher with a twist. Instead of using a pencil and paper or a graphing calculator, you'll learn how to write the Python code to do all of the calculations for you.

Building one step at a time, you will create projects that you can use as your personal super calculator. This curriculum uses Google Colaboratory, which is like a Jupyter Notebook, to write and execute Python code in your Google Drive.

With the Google Colab app, you can even write and run the code from your phone. If you are not familiar with Google Colaboratory, here is an article about how to set it up.

This is just the beginning. We are making steady progress on building out additional math and Python-focused courses, as part of our upcoming Data Science Curriculum Expansion.

Why is Math Useful for Developers?

This Algebra-level math will directly relate to the calculations you will do for Statistics and Data Analysis in your job as a developer. We call it "Foundational Math" because you will build upon these skills when developing other applications.

Working with coordinates will help you to plan the layout of a scene or a page. Distance formulas and angle calculations will help you with game development (and proximity detection).

Incrementing and testing numerical values in loops will help you to make predictions based on initial values and trends. Solving for unknown values will help you with financial planning and business software development.

Because any aspect of code (or life) can be numerical, math is the language to describe relationships. You will find deeper wisdom in the numbers.

What We'll Cover in This Course

Throughout this course, you will learn how to write code to display the graph of almost any function, calculate almost any formula, and “solve for x.” You will also be building an extensive resource that you can use and modify in the future.

By the end of the course, you will have a solid understanding of key concepts in math, and you will also have your own library of Colab Notebooks that apply these concepts.

Here is an example of an early step:

Image

Like many freeCodeCamp courses, this course has five parts. The five parts to this course each lead to a certification project that aggregates what you've learned in the preceding steps.

Here is an example, from close to the end of the course, to illustrate how the learning still proceeds in manageable steps:

Image

Python Math Course Outline

In this course, you will go through the major math topics from Algebra to the Y axis (the Z axis will appear in a subsequent course). This course is still in beta testing, so I invite you to try it out and leave comments in the forum.

Eventually, I want people to discuss the math and create Python scripts for each other. If you have any basic knowledge of Python, you will be able to navigate this course.

Here is a description of each part, with links to the Colab notebooks:

Part 1 – Math in Python

In the beginning, you'll build on a basic understanding of Python to show you how to handle input and apply mathematical formulas. You will even learn how to “solve for x” with code.

Part 2 – Graphs

In this part, you'll focus on graphing. You will learn how to plot points, lines, and functions. You will also learn how to create tables and interactive graphs.

Part 3 – Advanced Math

Now you'll dive deeper with polynomials, exponential functions, and more complex formulas. You will write code to calculate mortgage payments, estimate retirement account balances, and do other useful things.

Part 4 – Angles and Trigonometry

This part will teach you how to write code for lines and angles on graphs. You will use trigonometry to find distances, midpoints, and angles. Animated graphs included.

Part 5 – Sequences and Summation

This section will walk you through sequence and series calculations, then goes into probability. Along the way, you will learn about summation and counting principles.

Beyond the Jupyter Notebook

The latest version of the Texas Instruments TI-84 calculator can run Python code! The TI-84 is the “go-to” calculator for students in any class that involves graphing and formulas. You can write the programs on your computer and run them on your calculator. In this course, you will develop your math and Python skills to do so.

Now, we are still calling this the “beta version” of the course, and even that might be a little optimistic. You are getting early access to this content that we will continue to improve, and I welcome your feedback.

My goal is that this is just the beginning, and we all create more notebooks and scripts and functions in the Google Colaboratory and share them with each other.

By the way, I'm Ed, and I have been teaching high school students for 18 years and college students for 15 years. I created this course. You can email me your thoughts at ed@freecodecamp.org and I'll share them with freeCodeCamp's team of teachers. Thanks for joining me on this journey.

]]> <![CDATA[ Scientific Computing in Golang with the Gonum Package ]]> https://www.freecodecamp.org/news/scientific-computing-in-golang-using-gonum/ 66d45edd4a7504b7409c33d7 Mon, 04 Apr 2022 15:49:21 +0000 By Ukeje Chukwuemeriwo Goodness

In this article, I'll introduce you to Gonum, a package you can use to perform scientific computations in the Go programming language.

Here's what we'll cover in this intermediate tutorial:

  • What is Gonum?
  • Why use Gonum
  • How to install and set up Gonum
  • How to perform statistical operations using Gonum
  • How to perform matrix operations using Gonum
  • Other scientific computations supported by Gonum.

Prerequisites:

  • Knowledge of functional programming in Golang.
  • A Golang IDE with Go installed (I use Goland and Go 1.17.6, but you can use any other)

What is Gonum?

Screenshot-from-2022-03-14-05-30-53-1

Gonum, short for Go Numerical, is a Golang package built and designed by gonum.org to make scientific computations easier in the Go programming language.

The Gonum package is similar to Numpy in the Python programming language. Numpy currently has more to offer with more functionalities than Gonum, but Gonum's features are improving consistently.

The Gonum package supports functionalities for various scientific computations like Linear Algebra, Calculus, Statistics, graphs, and many others.

In this article, we'll go over various functions and use-cases of Gonum.

Why Use Gonum?

  • The speed and concurrency Golang offers.
  • Golang programs are easier to maintain.
  • Gonum contains more mathematical operations than the Go standard library.
  • Gonum is optimized for scientific calculations across various fields.

How to Get Started With Gonum

To get started with Gonum, you'll need to install the package from Github on your terminal using the command:

go get -u gonum.org/v1/gonum/

This command should output a successful installation message. If it doesn’t, update your Go to a more recent version and try again.

Statistical Operations Using Gonum

The Gonum package provides a library for statistical calculations. This library contains a lot of functions which you can view here.

In this tutorial, I will be going over the rudimentary functions of the library, specifically for measures of central tendency (mean, median, mode).

Import the stats library in the gonum package like this:

import “gonum.org/v1/gonum/stat”

carbon--1-

  • Mean: stat.Mean returns the mean value of a slice of the float64 type. It takes in a slice and a weight which could be nil or a corresponding slice for which the slice gets weighed.
func mean() {
   values := []float64{1, 2, 3, 4, 5, 6}
   weights := []float64{1, 1, 1, 1, 1, 1} //has the same effects as nil
   fmt.Println(stat.Mean(values, weights))
}

Output: 3.5

  • Median: stat.Quantile, there is no explicit function for the median in gonums/stat. But we can use stat.Quantile by passing in a sorted slice by importing the sort module.

stat.Quantile takes a position, the slice, a cummulant kind and weight. The position argument p is a float ranging from 0 to 1, and the cumulant kind is stat.Empirical or stat.LinInterp.

In this case, we use stat.Empirical, which returns the value at the specified position p.

    import (
        "gonum.org/v1/gonum/stat"
        "sort"
    )

    func median() {
        values := []float64{10, 20, 25, 30, 45, 70, 30}
        sort.Float64s(values) //sorts the float
        fmt.Println(stat.Quantile(0.5, stat.Empirical, values, nil))
    }

Output: 30

  • Mode: stat.Mode. Just like stat.Mean, it takes in a slice of values and a weight slice, and returns the most occurring element alongside the number of occurrences of the element.
func mode() {
   values := []float64{10, 20, 25, 30, 45, 70, 30}
   fmt.Println(stat.Mode(values, nil))
}

Output: 30 2

Matrix Operations Using Gonum

Gonum supports matrix operations in the mat package. 

import “gonum.org/v1/gonum/mat”

How to Create a Matrix:

mat.NewDense is the method for creating a matrix. It takes in the dimensions of the matrix and the data to be passed in, which could be nil (a matrix with all entities equal to zero).

mat.NewDense returns a pointer to the matrix object which can be dereferenced.

The null matrix serves as the matrix for the examples in this tutorial.

func null(){
  matrix := mat.NewDense(3, 3, nil)
  fmt.Println(*matrix)
}

output:

{{3 3 [0 0 0 0 0 0 0 0 0] 3} 3 3}

How to Format the Matrix Output

Printing a Gonum matrix without formatting returns a pointer to the matrix in this format {{3 3 [0 0 0 0 0 0 0 0 0] 3} 3 3}.

To output a two-dimensional table, we use mat.Formatted which takes in the matrix object, a prefix, and a format option which in this case, we use mat.Squeeze.

func format(matrix mat.Matrix) {
    formatted := mat.Formatted(matrix, mat.Prefix(""), mat.Squeeze())
    fmt.Println(formatted)
}

Output:

⎡0  0  0⎤
⎢0  0  0⎥
⎣0  0  0⎦

How to Set Matrix Values

To input a value into a position in the matrix, we use .Set on the matrix object. matrix.Set takes in three or more arguments as thus:

matrix.Set(rowNumber, columnNumber, element).

func input(){
    matrix.Set(1, 2, 3.0)
}

Output:

⎡0  0  0⎤
⎢0  0  3⎥
⎣0  0  0⎦

How to Get the Matrix Values

Retrieving values in the matrix is done using .At on the matrix object, which takes in the row and column numbers, respectively.

Here, we retrieve the element we set in the above example:

func retriever(){
    getElement := matrix.At(1, 2)
    fmt.Println(getElement)
}

Output: 3

How to Transpose a Matrix

Transposing a matrix involves the interchange of rows and columns in a matrix such that the rows are set to columns and vice-versa.

The .T method on the matrix object transposes the matrix.

func transposer(){
    format(null.T())
}

Here, we transpose the matrix output from the set example.

Output:

⎡0  0  0⎤
⎢0  0  0⎥
⎣0  3  0⎦

Determinant of a Matrix

You can evaluate the determinant of a matrix using the method mat.Det which takes in the matrix and returns its determinant.

func determinant(){
    determinant := mat.Det(matrix)
    fmt.Println(determinant)
}

Output: 0

How to Add Rows and Columns to Matrices

You can add a new row or column using .SetRow and .SetCol. These methods take in a row number, and a slice of values of similar dimension.

This updates the rows and columns of the matrix:

values := []float64{1, 2, 3}
matrix.SetCol(0, values)
matrix.SetRow(1, values)

Output:

⎡1  0  0⎤
⎢1  2  3⎥
⎣3  0  0⎦

Other Gonum Packages

Gonum has more packages for scientific computing:

  • blas → provides interfaces for the BLAS (Basic Linear Algebra Subprograms) linear algebra standard
  • diff → Functions for differential calculus
  • graph → interfaces for graphs
  • integrate → Functions for integral calculus
  • lapack → provides interfaces for the LAPACK (Linear Algebra Package) linear algebra standard
  • mathext → Special mathematics functions that are not included in the Go standard library
  • unit → Types and constants for easy use of SI Units

Wrapping Up

In this article, you learned about scientific computations in Golang using the Gonum package.

We discussed statistical and matrix computations in Gonum and went through other scientific computation modules in the Gonum package.

Working with other modules in the Gonum package is pretty straightforward and similar to the ones we discussed here.

]]> <![CDATA[ All the Math You Need to Know in Artificial Intelligence ]]> https://www.freecodecamp.org/news/all-the-math-you-need-in-artificial-intelligence/ 66d45f31246e57ac83a2c76d Wed, 24 Nov 2021 00:12:40 +0000 I’m an AI researcher, and I’ve received quite a few emails asking me just how much math is required in Artificial Intelligence.

I won’t lie: it’s a lot of math.

And this is one of the reasons AI puts off many beginners. After much research and talks with several veterans in the field, I’ve compiled this no-nonsense guide that covers all of the fundamentals of the math you’ll need to know.

The concepts mentioned below are usually covered over several semesters in college, but I’ve boiled them down to the core principles that you can focus on.

This guide is an absolute life-saver for beginners, so you can study the topics that matter most. But it's an even better resource for practitioners, such as myself, who require a quick breeze-through on these concepts.

Note: You don’t need to know all of the concepts (below) in order to get your first job in AI. All you need is a firm grasp of the fundamentals. Focus on those and consolidate them.

1. Algebra You Need to Know for AI

Pencil on system of equations: Photo by [Unsplash](https://unsplash.com/@antoine1003?utm_source=ghost&utm_medium=referral&utm_campaign=api-credit">Antoine Dautry / <a href="https://unsplash.com/?utm_source=ghost&utm_medium=referral&utm_campaign=api-credit)

Knowledge of algebra is perhaps fundamental to math in general. Besides mathematical operations like addition, subtraction, multiplication and division, you’ll need to know the following:

2. Linear Algebra You Need to Know for AI

Linear Algebra is the primary mathematical computation tool in Artificial Intelligence and in many other areas of Science and Engineering. With this field, you need to understand 4 primary mathematical objects and their properties:

  • Scalars **** — a single number (can be real or natural).

  • Vectors — a list of numbers, arranged in order. Consider them as points in space with each element representing the coordinate along an axis.

  • Matrices — a 2-D array of numbers where each number is identified by 2 indices.

  • Tensors **** — an N-D array (N>2) of numbers, arranged on a regular grid with N-axes. Important in Machine Learning, Deep Learning and Computer Vision.

  • Eigenvectors & Eigenvalues — special vectors and their corresponding scalar quantity. Understand the significance and how to find them.

  • Singular Value Decomposition **** — factorization of a matrix into 3 matrices. Understand the properties and applications.

  • Principal Component Analysis (PCA) — understand the significance, properties, and applications.

Properties such as the Dot product, Vector product and the Hadamard product are useful to know as well.

3. Calculus You Need to Know for AI

Photo by [Unsplash](https://unsplash.com/@jeswinthomas?utm_source=ghost&utm_medium=referral&utm_campaign=api-credit">Jeswin Thomas / <a href="https://unsplash.com/?utm_source=ghost&utm_medium=referral&utm_campaign=api-credit)

Calculus deals with changes in parameters, functions, errors and approximations. Working knowledge of multi-dimensional calculus is imperative in Artificial Intelligence.

The following are the most important concepts (albeit non-exhaustive) in Calculus:

  • Derivatives — rules (addition, product, chain rule, and so on), hyperbolic derivatives (tanh, cosh, and so on) and partial derivatives.

  • Vector/Matrix Calculus — different derivative operators (Gradient, Jacobian, Hessian and Laplacian)

  • Gradient Algorithms **** — local/global maxima and minima, saddle points, convex functions, batches and mini-batches, stochastic gradient descent, and performance comparison.

4. Statistics & Probability Concepts You Need to Know for AI

Photo by [Unsplash](https://unsplash.com/@tamiminaser?utm_source=ghost&utm_medium=referral&utm_campaign=api-credit">Naser Tamimi / <a href="https://unsplash.com/?utm_source=ghost&utm_medium=referral&utm_campaign=api-credit)

This topic will probably take up a significant chunk of your time. Good news: these concepts aren’t difficult, so there’s no reason why you shouldn’t master them.

  • Basic Statistics — Mean, median, mode, variance, covariance, and so on.

  • Basic rules in probability **** — events (dependent and independent), sample spaces, conditional probability.

  • Random variables **** — continuous and discrete, expectation, variance, distributions (joint and conditional).

  • Bayes’ Theorem — calculates validity of beliefs. Bayesian software helps machines recognize patterns and make decisions.

  • Maximum Likelihood Estimation (MLE) **** — parameter estimation. Requires knowledge of fundamental probability concepts (joint probability and independence of events).

  • Common Distributions — binomial, poisson, bernoulli, gaussian, exponential.

5. Information Theory Concepts You Need to Know for AI

Photo by [Unsplash](https://unsplash.com/@giuliamay?utm_source=ghost&utm_medium=referral&utm_campaign=api-credit">Giulia May / <a href="https://unsplash.com/?utm_source=ghost&utm_medium=referral&utm_campaign=api-credit)

This is an important field that has made significant contributions to AI and Deep Learning, and is yet unknown to many. Think of it as an amalgamation of calculus, statistics, and probability.

  • Entropy **** — also called Shannon Entropy. Used to measure the uncertainty in an experiment.

  • Cross-Entropy **** — compares two probability distributions and tells us how similar they are.

  • Kullback Leibler Divergence **** — another measure of how similar two probability distributions are.

  • Viterbi Algorithm **** — widely used in Natural Language Processing (NLP) and Speech.

  • Encoder-Decoder — used in Machine Translation RNNs and other models.

Math is Fun!

If you are terrified at the mere mention of “math”, you’re probably not going to have much fun in Artificial Intelligence.

But if you’re willing to invest time to improve your familiarity with the principles underlying calculus, linear algebra, stats, and probability, nothing — not even math — should get in the way of you getting into AI.

PS: Math really is fun. As you go deeper into math, be sure to understand the beauty of a certain math concept and how it affects something. You’ll soon share the unbridled passion that many mathematicians and AI Scientists have!

A tip: Treat mathematical concepts as a pay-as-you-go: whenever a foreign concept pops up, grab it and devour it! The guide above presents a minimal, yet comprehensive, resource to understand any kind of topic or concept in AI.

Happy learning!

]]> <![CDATA[ 460 Free Textbooks – Math, Science, and More [Online PDF for College and High School] ]]> https://www.freecodecamp.org/news/free-textbooks-math-science-and-more-online-pdf-for-college-and-high-school/ 66b8d945f805ffd579552e84 Thu, 09 Sep 2021 14:25:28 +0000 While you're in school, you will go through a number of textbooks and the cost can add up pretty quickly. Luckily, there are websites that offer free textbooks for high school and college students.

Here is a list of 460 free textbooks, arranged by topic. Once you find a topic that interests you, click that link and it will jump you down to a list of related textbooks.

High School Textbook Categories

College Textbook Categories

High school Textbooks (90)

Math

  1. Statistics
  2. Advanced High School Statistics - 2nd Edition
  3. Advanced High School Mathematics

Natural Sciences

  1. AP Physics Collection
  2. Biology for AP courses
  3. Physics
  4. Principles of Macroeconomics for AP courses 2e
  5. Principles of Microeconomics for AP courses 2e

Classical Novels

  1. Adventures of Huckleberry Finn, The
  2. Adventures of Tom Sawyer, The
  3. Aesop’s Fables
  4. Agnes Grey
  5. Alice’s Adventures in Wonderland
  6. Andersen’s Fairy Tales
  7. Anna Karenina
  8. Anne of Green Gables
  9. Around the World in 80 Days
  10. Beyond Good and Evil
  11. Bleak House
  12. Brothers Karamazov, The
  13. Christmas Carol, A
  14. Crime and Punishment
  15. David Copperfield
  16. Down and Out in Paris and London
  17. Dracula
  18. Dubliners
  19. Emma
  20. Erewhon
  21. For the Term of His Natural Life
  22. Frankenstein
  23. Great Expectations
  24. Great Gatsby, The
  25. Grimms Fairy Tales
  26. Gulliver’s Travels
  27. Heart of Darkness
  28. Hound of the Baskervilles, The
  29. Idiot, The
  30. Iliad, The
  31. Island of Doctor Moreau, The
  32. Jane Eyre
  33. Jungle Book, The
  34. Kidnapped
  35. Lady Chatterly’s Lover
  36. Last of the Mohicans, The
  37. Legend of Sleepy Hollow, The
  38. Les Miserables
  39. Little Women
  40. Madame Bovary
  41. Merry Adventures of Robin Hood, The
  42. Metamorphosis, The
  43. Middlemarch
  44. Moby Dick
  45. 1984
  46. Northanger Abbey
  47. Nostromo: A Tale of the Seaboard
  48. Notes from the Underground
  49. Odyssey, The
  50. Of Human Bondage
  51. Oliver Twist
  52. Paradise Lost
  53. Persuasion
  54. Picture of Dorian Gray, The
  55. Pollyanna
  56. Portrait of a Lady, The
  57. Portrait of the Artist as a Young Man, A
  58. Pride and Prejudice
  59. Prince, The
  60. Robinson Crusoe
  61. Scarlet Pimpernel, The
  62. Sense and Sensibility
  63. Sons and Lovers
  64. Strange Case of Dr Jekyll and Mr Hyde, The
  65. Swanns Way
  66. Tale of Two Cities, A
  67. Tales of Mother Goose, The
  68. Tarzan of the Apes
  69. Tender is the Night
  70. Tess of the d’Urbervilles
  71. Thirty Nine Steps, The
  72. Three Musketeers, The
  73. Time Machine, The
  74. Treasure Island
  75. Trial, The
  76. Ulysses
  77. Utopia
  78. Vanity Fair
  79. War and Peace
  80. Within A Budding Grove
  81. Women In Love
  82. Wuthering Heights

College TextBooks (370)

Computer Science

  1. Python for Everybody: Exploring Data Using Python 3
  2. Web Accessibility for Developers
  3. The Missing Link: An Introduction to Web Development and Programming
  4. Foundations of Computation
  5. Algorithms and Data Structures With Applications to Graphics and Geometry
  6. Think Data Structures: Algorithms and Information Retrieval in Java
  7. Open Data Structures: An Introduction
  8. Java with BlueJ
  9. Introduction to Programming Using Java - Eighth Edition
  10. A Byte of Python
  11. Eloquent JavaScript: A Modern Introduction to Programming
  12. High Performance Computing
  13. Database Design - 2nd Edition
  14. Relational Databases and Microsoft Access
  15. A First Course in Electrical and Computer Engineering
  16. How to Think Like a Computer Scientist: Learning with Python
  17. How to Think Like a Computer Scientist: C Version
  18. How to Think Like a Computer Scientist: C++ Version
  19. Programming Fundamentals - A Modular Structured Approach using C++
  20. Think Java: How To Think Like a Computer Scientist - 2e
  21. Java, Java, Java: Object-Oriented Problem Solving
  22. Think Stats: Probability and Statistics for Programmers - 2e
  23. Think Bayes: Bayesian Statistics Made Simple
  24. Think Complexity: Exploring Complexity Science with Python - 2e
  25. Digital Circuit Projects: An Overview of Digital Circuits Through Implementing Integrated Circuits
  26. Introduction to the Modeling and Analysis of Complex Systems
  27. Physical Modeling in MATLAB
  28. Spatial Thinking in Planning Practice: An Introduction to GIS
  29. Introduction to Computer Graphics
  30. Computer Networks: A Systems Approach
  31. An Introduction to Computer Networks - Second Edition
  32. Computer Networking : Principles, Protocols and Practice
  33. Signal Computing: Digital Signals in the Software Domain
  34. x86-64 Assembly Language Programming with Ubuntu
  35. Professional Web Accessibility Auditing Made Easy
  36. First Semester in Numerical Analysis with Python
  37. The Crystal Ball Instruction Manual - version 1.1 Volume One: Introduction to Data Science
  38. An Introduction to Matlab and Mathcad
  39. Yet Another Introductory Number Theory Textbook (Cryptology Emphasis Version)
  40. Delftse Foundations of Computation
  41. The Unicode cookbook for linguists: Managing writing systems using orthography profiles
  42. Implementing a One Address CPU in Logisim
  43. Embedded Controllers Using C and Arduino - 2e
  44. Introduction to MIPS Assembly Language Programming
  45. Think DSP: Digital Signal Processing in Python
  46. Information Systems for Business and Beyond
  47. Information Systems: A Manager's Guide to Harnessing Technology
  48. The Little Book of Semaphores
  49. Essentials of Geographic Information Systems
  50. Programming Languages: Application and Interpretation - Version Second Edition

Engineering

  1. A Brief Introduction to Engineering Computation with MATLAB
  2. Introduction to Industrial Engineering
  3. Fundamentals of Electrical Engineering I
  4. A First Course in Electrical and Computer Engineering
  5. Sustainability: A Comprehensive Foundation
  6. Notes on Diffy Qs: Differential Equations for Engineers
  7. Elementary Differential Equations with Boundary Value Problems
  8. Developing New Products and Services
  9. Introduction to Aerospace Structures and Materials
  10. Building Maintenance & Construction: Tools and Maintenance Tasks
  11. Direct Energy
  12. Operational Amplifiers & Linear Integrated Circuits: Theory and Application - 3e
  13. Manufacturing Processes 4-5
  14. Electromagnetics Vol 1
  15. DC Electrical Circuit Analysis: A Practical Approach
  16. AC Electrical Circuit Analysis: A Practical Approach
  17. Applied Fluid Mechanics Lab Manual
  18. Fundamentals of Microwave and RF Design
  19. Introduction to Probability
  20. Introduction to Linear, Time-Invariant, Dynamic Systems for Students of Engineering

Education

  1. Hybrid-Flexible Course Design
  2. Creating Online Learning Experiences
  3. Education for a Digital World: Advice, Guidelines and Effective Practice from Around Globe
  4. Educational Psychology
  5. Learning to Learn Online
  6. Sharing Our Knowledge: Best Practices for Supporting English Language Learners in Schools
  7. Introduction to Curriculum for Early Childhood Education
  8. Bad Ideas About Writing
  9. Write Here, Right Now: An Interactive Introduction to Academic Writing and Research
  10. Elementary Earth and Space Science Methods
  11. Mathematics for Elementary Teachers
  12. Trauma Informed Behaviour Support: A Practical Guide to Developing Resilient Learners
  13. Trauma-Informed School Practices: Building Expertise To Transform Schools
  14. Wellbeing in Educational Contexts
  15. Parenting and Family Diversity Issues
  16. Safety, Health, and Nutrition in Early Childhood Education
  17. The ELC: An Early Childhood Learning Community at Work
  18. Building Democracy for All: Interactive Explorations of Government and Civic Life
  19. Foundations of Learning and Instructional Design Technology
  20. Ethical Use of Technology in Digital Learning Environments: Graduate Student Perspectives
  21. Writing Instruction Tips For Automated Essay Graders: How To Design an Essay for a Non-human Reader
  22. Conventions 101: A Functional Approach to Teaching (And Assessing!) Grammar and Punctuation
  23. Choosing & Using Sources: A Guide to Academic Research
  24. WAC and Second-Language Writers: Research Towards Linguistically and Culturally Inclusive Programs and Practices
  25. WAC Partnerships Between Secondary and Postsecondary Institutions
  26. Working With Academic Literacies: Case Studies Towards Transformative Practice
  27. Antiracist Writing Assessment Ecologies: Teaching and Assessing Writing for a Socially Just Future
  28. Critical Expressivism: Theory and Practice in the Composition Classroom
  29. Copy(write): Intellectual Property in the Writing Classroom
  30. Teaching Autoethnography: Personal Writing in the Classroom

Math

  1. Algebra and Trigonometry
  2. Calculus Volume 1
  3. Calculus Volume 2
  4. Calculus Volume 3
  5. Multivariable Calculus
  6. College Algebra
  7. Open Resources for Community College Algebra
  8. Elementary Algebra 2e
  9. Intermediate Algebra 2e
  10. Fundamentals of Matrix Algebra
  11. Introduction to GNU Octave: A brief tutorial for linear algebra and calculus students
  12. A Cool Brisk Walk Through Discrete Mathematics
  13. Mathematical Analysis I
  14. Introductory Statistics with Randomization and Simulation - First Edition
  15. Introductory Business Statistics
  16. Introductory Statistics
  17. Statistical Thinking for the 21st Century
  18. Lies, Damned Lies, or Statistics: How to Tell the Truth with Statistics
  19. Prealgebra 2e
  20. Precalculus
  21. Geometry with an Introduction to Cosmic Topology
  22. Technical Mathematics
  23. Introduction to Game Theory: a Discovery Approach
  24. Quantitative Problem Solving in Natural Resources
  25. Euclidean plane and its relatives
  26. Proofs and Concepts: The Fundamentals of Abstract Mathematics
  27. Applied Finite Mathematics
  28. Linear Regression Using R: An Introduction to Data Modeling
  29. Applied Combinatorics
  30. Math in Society - Edition 2.5

Natural Sciences

  1. Anatomy and Physiology
  2. Human Anatomy Lab Manual
  3. A Mixed Course-Based Research Approach to Human Physiology
  4. Fundamentals of Anatomy and Physiology - Australian Edition
  5. Astronomy
  6. Biochemistry: Free For All
  7. Biology 2e
  8. Concepts of Biology
  9. Principles of Biology: Biology 211, 212, and 213
  10. Microbiology
  11. General Microbiology - 1st Edition
  12. A Primer for Computational Biology
  13. From Growing to Biology: Plants 1e
  14. A Student's Guide to Tropical Marine Biology
  15. Chemistry 2e
  16. Chemistry: Atoms first 2e
  17. College Physics
  18. Foundations of Neuroscience
  19. Foundations of Epidemiology
  20. Radiation Safety
  21. Environmental Toxicology
  22. Geological Structures: a Practical Introduction
  23. Physical Geology - First University of Saskatchewan Edition
  24. Climate Toolkit: A Resource Manual for Science and Action
  25. Introduction to Climate Science - 1st Edition
  26. Bending the Curve: Climate Change Solutions
  27. Chemical Biology & Biochemistry Laboratory Using Genetic Code Expansion Manual
  28. Biotechnology Foundations - 2nd Edition
  29. An EKG Interpretation Primer
  30. Introduction to Oceanography

Medicine

  1. Nutrition: Science and Everyday Application - beta v 0.1
  2. Human Nutrition - 2020 Edition
  3. Concepts of Fitness and Wellness
  4. Histology and Embryology for Dental Hygiene
  5. Nursing Pharmacology
  6. Nursing Care at the End of Life: What Every Clinician Should Know
  7. Transitions to Professional Nursing Practice - 2nd Edition
  8. Safety, Health, and Nutrition in Early Childhood Education
  9. The Complete Subjective Health Assessment
  10. Evidence-Based Massage Therapy
  11. A Laboratory Guide to Clinical Hematology
  12. Undergraduate Diagnostic Imaging Fundamentals
  13. Public Health Ethics: Global Cases, Practice, and Context
  14. Literature Reviews for Education and Nursing Graduate Students
  15. Health Case Studies
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  17. Field Trials of Health Interventions: A Toolbox
  18. Clinical Procedures for Safer Patient Care
  19. Creative Clinical Teaching In The Health Professions
  20. Veterinary Epidemiology: Principles and Methods

Social Sciences

  1. American Government 2e
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  3. Introduction to Sociology 2e
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  5. Principles of Economics 2e
  6. Principles of Macroeconomics 2e
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  8. Psychology 2e
  9. Psychology: The Science of Human Potential
  10. Abnormal Psychology - 2nd Edition
  11. Introduction to Community Psychology
  12. An Introduction to Psychological Statistics
  13. Human Behavior and the Social Environment I
  14. Human Behavior and the Social Environment II
  15. Language and Culture in Context - A Primer on Intercultural Communication
  16. Intercultural Communication
  17. Contemporary Families: An Equity Lens
  18. Child Growth and Development
  19. Child, Family, and Community
  20. Parenting and Family Diversity Issues
  21. Interpersonal Communication: A Mindful Approach to Relationships
  22. Attenuated Democracy: A Critical Introduction to U.S. Government and Politics
  23. Foundations of Social Work Research
  24. Introduction to Human Geography - 2nd Edition
  25. Beyond Race: Cultural Influences on Human Social Life
  26. Classical Sociological Theory and Foundations of American Sociology
  27. Speak Out, Call In: Public Speaking as Advocacy
  28. A Quick Guide to Quantitative Research in the Social Sciences
  29. The Balance of Personality
  30. The Art of Being Human: A Textbook for Cultural Anthropology

Humanities

  1. Stand up, Speak out: The Practice and Ethics of Public Speaking
  2. U.S. History
  3. World History: Cultures, States, and Societies to 1500
  4. Modern World History
  5. International Relations
  6. Becoming America: An Exploration of American Literature from Precolonial to Post-Revolution
  7. American Literature I: An Anthology of Texts From Early America Through the Civil War
  8. Spanish I: Beginning Spanish Language and Culture
  9. Linguistics for Teachers of English
  10. The Changing Story: digital stories that participate in transforming teaching & learning
  11. Essentials of Linguistics
  12. Technical Writing
  13. Let's Get Writing!
  14. You, Writing! A Guide to College Composition
  15. Writing for Success
  16. College ESL Writers: Applied Grammar and Composing Strategies for Success
  17. Introduction to Sociology - 2e
  18. Introduction to Philosophy: Philosophy of Religion
  19. Introduction to Philosophy: Philosophy of Mind
  20. English Composition: Connect, Collaborate, Communicate
  21. Studying the Bible: The Tanakh and Early Christian Writings
  22. Philosophical Ethics
  23. Ethics for A-Level
  24. Steps to Success: Crossing the Bridge Between Literacy Research and Practice
  25. Exploring Public Speaking - 4th Edition
  26. Plato's 'Republic': An Introduction
  27. Decoding the 1920s: A Reader for Advanced Learners of Russian
  28. An Introduction to Technical Theatre
  29. Theatrical Worlds
  30. Understanding Basic Music Theory
  31. Music Theory for the 21st-Century Classroom
  32. Music on the Move
  33. Introduction to Art: Design, Context, and Meaning
  34. Guide to Ancient Egyptian Art
  35. Guide to Ancient Roman art
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  37. Moving Pictures: An Introduction to Cinema
  38. Naming the Unnameable: An Approach to Poetry for New Generations
  39. Prose Fiction: An Introduction to the Semiotics of Narrative
  40. Mindful Technical Writing: An Introduction to the Fundamentals
  41. Explorations 1: Grammar for the Experienced Beginner
  42. Introduction to Logic and Critical Thinking
  43. Writing and Literature: Composition as Inquiry, Learning, Thinking, and Communication
  44. British Literature I Anthology: From the Middle Ages to Neoclassicism and the Eighteenth Century
  45. British Literature II: Romantic Era to the Twentieth Century and Beyond
  46. Perspectives: An Open Invitation to Cultural Anthropology
  47. A Guide to Good Reasoning: Cultivating Intellectual Virtues - Second edition, revised and updated
  48. Inferring and Explaining
  49. Exploring Perspectives: A Concise Guide to Analysis
  50. EmpoWord: A Student-Centered Anthology & Handbook for College Writers

Business

  1. Introduction to Business
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  10. Business Writing Style Guide
  11. Financial Management for Small Businesses: Financial Statements & Present Value Models
  12. Intermediate Microeconomics with Microsoft Excel - 2nd Edition
  13. Business Computers 365
  14. Problem Solving in Teams and Groups
  15. Digital Accessibility as a Business Practice
  16. Management Communication
  17. Introduction to Financial Accounting: U.S. GAAP Adaptation
  18. eMarketing - The Essential Guide to Online Marketing
  19. Fundamentals of Infrastructure Management
  20. An Introduction to Cooperation and Mutualism
  21. Financial Strategy for Public Managers
  22. Entrepreneurship and Innovation Toolkit
  23. Business Plan Development Guide
  24. Project Management
  25. Introducing Marketing
  26. Fundamentals of Business - 3rd Edition
  27. Business Law and the Legal Environment
  28. Good Corporation, Bad Corporation: Corporate Social Responsibility in the Global Economy
  29. Guidelines for Improving the Effectiveness of Boards of Directors of Nonprofit Organizations
  30. Small Business Management in the 21st Century
  31. The Sustainable Business Case Book
  32. Advanced Business Law and the Legal Environment
  33. Electronic Commerce: The Strategic Perspective
  34. Developing New Products and Services
  35. Focusing on Organizational Change
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  38. Managerial Accounting
  39. Business English for Success
  40. Human Resource Management
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  44. The Power of Selling
  45. Principles of Marketing
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  47. Money and Banking
  48. Risk Management for Enterprises and Individuals
  49. Exploring Business
  50. International Finance: Theory and Policy

Law

  1. Law School Materials for Success
  2. Introduction to Criminal Investigation: Processes, Practices and Thinking
  3. Ethics in Law Enforcement
  4. Criminal Law
  5. Intellectual Property: Law & the Information Society—Cases and Materials
  6. Wetlands Law: A Course Source
  7. First Amendment: Cases, Controversies, and Contexts - Second Edition
  8. The Indigo Book: A Manual of Legal Citation
  9. Fundamentals of Business Law
  10. Business Law I Essentials
  11. Law 101: Fundamentals of the Law
  12. Liberty, Equality and Due Process: Cases, Controversies, and Contexts in Constitutional Law
  13. Business Law and the Legal Environment
  14. Introduction to the American Criminal Justice System
  15. The Law of Trusts
  16. Sources of American Law: An Introduction to Legal Research - 4th Edition
  17. Basic Income Tax - 8th Edition
  18. What Color is Your C.F.R.?
  19. U.S. Federal Income Taxation of Individuals 2020
  20. Introduction to Contracts, Sales and Product Liability
  21. Law for Entrepreneurs
  22. Legal Aspects of Corporate Management and Finance
  23. Legal Aspects of Marketing and Sales
  24. Introduction to Basic Legal Citation
  25. Contract Doctrine, Theory & Practice Volume 1
  26. United States Copyright Law
  27. Land Use
  28. Introduction to the Law of Property, Estate Planning and Insurance
  29. Law of Commercial Transactions
  30. United States Securities Law

Journalism and Communication

  1. Exploring Public Speaking - 4th Edition
  2. Stand up, Speak out: The Practice and Ethics of Public Speaking
  3. Communication in the Real World: An Introduction to Communication Studies
  4. Launch! Advertising and Promotion in Real Time
  5. Business Communication for Success
  6. Understanding Media and Culture: An Introduction to Mass Communication
  7. Information Strategies for Communicators
  8. Tools for Podcasting
  9. Digital Foundations: Introduction to Media Design with the Adobe Creative Cloud - Revised Edition
  10. Humans R Social Media - Open Textbook Edition
  11. Writing for Strategic Communication Industries
  12. Open Technical Writing: An Open-Access Text for Instruction in Technical and Professional Writing
  13. Writing for Electronic Media
  14. Be Credible
  15. Language and Culture in Context - A Primer on Intercultural Communication
  16. Intercultural Communication
  17. Arguing Using Critical Thinking
  18. Writing Unleashed: Content and Structure - 3.0
  19. Management Communication
  20. OER: A Field Guide for Academic Librarians
  21. The Data Journalism Handbook
  22. Media, Society, Culture and You
  23. Message Processing: The Science of Creating Understanding
  24. Media Innovation and Entrepreneurship
  25. A Guide to Technical Communications: Strategies & Applications
  26. Graphic Design and Print Production Fundamentals
  27. Media Studies 101
  28. Small Group Communication: Forming & Sustaining Teams
  29. New Media Futures
  30. Communication, Affect, & Learning in the Classroom - 4th Edition
]]> <![CDATA[ How Machine Learning Uses Linear Algebra to Solve Data Problems ]]> https://www.freecodecamp.org/news/how-machine-learning-leverages-linear-algebra-to-optimize-model-trainingwhy-you-should-learn-the-fundamentals-of-linear-algebra/ 66d45f4347a8245f78752a66 Wed, 01 Sep 2021 15:47:54 +0000 Machines or computers only understand numbers. And these numbers need to be represented and processed in a way that lets machines solve problems by learning from the data instead of learning from predefined instructions (as in the case of programming).

All types of programming use mathematics at some level. Machine learning involves programming data to learn the function that best describes the data.

The problem (or process) of finding the best parameters of a function using data is called model training in ML.

Therefore, in a nutshell, machine learning is programming to optimize for the best possible solution – and we need math to understand how that problem is solved.

The first step towards learning Math for ML is to learn linear algebra.

Linear Algebra is the mathematical foundation that solves the problem of representing data as well as computations in machine learning models.

It is the math of arrays — technically referred to as vectors, matrices and tensors.

Common Areas of Application — Linear Algebra in Action

Image

Source: [https://www.wiplane.com/p/foundations-for-data-science-ml](https://www.wiplane.com/p/foundations-for-data-science-ml" rel="nofollow noopener noopener noopener noopener)

In the ML context, all major phases of developing a model have linear algebra running behind the scenes.

Important areas of application that are enabled by linear algebra are:

  • data and learned model representation

  • word embeddings

  • dimensionality reduction

Data Representation

**** The fuel of ML models, that is data, needs to be converted into arrays before you can feed it into your models. The computations performed on these arrays include operations like matrix multiplication (dot product). This further returns the output that is also represented as a transformed matrix/tensor of numbers.

Word embeddings

Don’t worry about the terminology here – it is just about representing large-dimensional data (think of a huge number of variables in your data) with a smaller dimensional vector.

Image

Natural Language Processing (NLP) deals with textual data. Dealing with text means comprehending the meaning of a large corpus of words. Each word represents a different meaning which might be similar to another word. Vector embeddings in linear algebra allow us to represent these words more efficiently.

Eigenvectors (SVD)

Finally, concepts like eigenvectors allow us to reduce the number of features or dimensions of the data while keeping the essence of all of them using something called principal component analysis.

From Data to Vectors

Image

Source: [https://www.wiplane.com/p/foundations-for-data-science-ml](https://www.wiplane.com/p/foundations-for-data-science-ml" rel="nofollow noopener noopener noopener noopener)

Linear algebra basically deals with vectors and matrices (different shapes of arrays) and operations on these arrays. In NumPy, vectors are basically a 1-dimensional array of numbers but geometrically, they have both magnitude and direction.

Image

Source: [https://www.wiplane.com/p/foundations-for-data-science-ml](https://www.wiplane.com/p/foundations-for-data-science-ml" rel="nofollow noopener noopener noopener noopener)

Our data can be represented using a vector. In the figure above, one row in this data is represented by a feature vector which has 3 elements or components representing 3 different dimensions. N-entries in a vector makes it n-dimensional vector space and in this case, we can see 3-dimensions.

Deep Learning — Tensors Flowing Through a Neural Network

We can see linear algebra in action across all the major applications today. Examples include sentiment analysis on a LinkedIn or a Twitter post (embeddings), detecting a type of lung infection from X-ray images (computer vision), or any speech to text bot (NLP).

All of these data types are represented by numbers in tensors. We run vectorized operations to learn patterns from them using a neural network. It then outputs a processed tensor which in turn is decoded to produce the final inference of the model.

Each phase performs mathematical operations on those data arrays.

Dimensionality Reduction — Vector Space Transformation

Image

Source: [https://www.wiplane.com/p/foundations-for-data-science-ml](https://www.wiplane.com/p/foundations-for-data-science-ml" rel="nofollow noopener noopener noopener noopener)

When it comes to embeddings, you can basically think of an n-dimensional vector being replaced with another vector that belongs to a lower-dimensional space. This is more meaningful and it's the one that overcomes computational complexities.

For example, here is a 3-dimensional vector that is replaced by a 2-dimensional space. But you can extrapolate it to a real-world scenario where you have a very large number dimensions.

Reducing dimensions doesn’t mean dropping features from the data. Instead, it's about finding new features that are linear functions of the original features and preserving the variance of the original features.

Finding these new variables (features) translates to finding the principal components (PCs). This then converges to solving eigenvectors and eigenvalues problems.

Recommendation Engines — Making use of embeddings

You can think of Embedding as a 2D plane being embedded in a 3D space and that’s where this term comes from. You can think of the ground you are standing on as a 2D plane that is embedded into this space in which you live.

Just to give you a real-world use case to relate to all of this discussion on vector embeddings, all applications that are giving you personalized recommendations are using vector embedding in some form.

Image

For example, the above is a graphic from Google’s course on recommendation systems where we are given this data on different users and their preferred movies. Some users are kids and others are adults, some movies were are all-time classics while others are more artistic. Some movies are targeted towards a younger audience while movies like memento are preferred by adults.

Now, we not only need to represent this information in numbers but also need to find new smaller dimensional vector representations that capture all these features well.

Image

A very quick way to understand how we can pull off this task is by understanding something called Matrix Factorization which allows us to break a large matrix down into smaller matrices.

Ignore the numbers and colors for now and just try to understand how we have broken down one big matrix into two smaller ones.

For example, here this matrix of 4X5, 4 rows, and 5 features, was broken down into two matrices, one that's 4X2 and the other that's 2X5. We basically have new smaller dimensional vectors for users and movies.

Image

And this allows us to plot this on a 2D vector space. Here you’ll see that user #1 and the movie Harry Potter are closer and user #3 and the movie Shrek are closer.

The concept of a dot product (matrix multiplication) of vectors tells us more about the similarity of two vectors. And it has applications in correlation/covariance calculation, linear regression, logistic regression, PCA, convolutions, PageRank and numerous other algorithms.

Industries where Linear Algebra is used heavily

By now, I hope you are convinced that Linear algebra is driving the ML initiatives in a host of areas today. If not, here is a list to name a few:

  • Statistics

  • Chemical Physics

  • Genomics

  • Word Embeddings — neural networks/deep learning

  • Robotics

  • Image Processing

  • Quantum Physics

How much Linear Algebra should you know to get started with ML / DL?

Now, the important question is how you can learn to program these concepts of linear algebra. The answer is you don’t have to reinvent the wheel, you just need to understand the basics of vector algebra computationally and you then learn to program those concepts using NumPy.

NumPy is a scientific computation package that gives us access to all the underlying concepts of linear algebra. It is fast as it runs compiled C code and it has a large number of mathematical and scientific functions that we can use.

Foundations of Data Science & ML — I have created a course that gives you enough understanding of Programming, Math (Basic Algebra, Linear Algebra & Calculus) and Statistics. A complete package for first steps to learning DS/ML.

👉 You can use the code **FREECODECAMP10** to get 10% off.

Check out the course outline here:

]]> <![CDATA[ Skewness and Kurtosis – Positively Skewed and Negatively Skewed Distributions in Statistics Explained ]]> https://www.freecodecamp.org/news/skewness-and-kurtosis-in-statistics-explained/ 66d460ce3a8352b6c5a2aaef Wed, 16 Jun 2021 20:56:49 +0000 By Rishit Dagli

In this article, I'll explain two important concepts in statistics: skewness and kurtosis. And don't worry – you won't need to know very much math to understand these concepts and learn how to apply them.

What are Density Curves?

Let's first talk a bit about density curves, as skewness and kurtosis are based on them. They're simply a way for us to represent a distribution. Let's see what I mean through an example.

Say that you need to record the heights of a lot of people. So your distribution has let's say 20 categories representing the range of the output (58-59 in, 59-60 in ... 78-79). You can plot a histogram representing these categories and the number of people whose height falls in each category.

Image Histogram of height vs population

Well, you might do this for thousands of people, so you are not interested in the exact number – rather the percentage or probability of these categories.

I also explicitly mentioned that you have a rather large distribution since percentages are often useless for smaller distributions.

If you use percentages with smaller numbers I often refer to it as lying with statistics – it's a statement that is technically correct but creates the wrong impression in our minds.

Let me give you an example: a student is extremely excited and tells everyone in his class that he made a 100% improvement in his marks! But what he doesn't say is that his marks went from a 2/30 to 4/30 😂.

I hope you now clearly see the problem of using percentages with smaller numbers.

Coming back to density curves, when you are working with a large distribution you want to have more granular categories. So you make each category which was 1 inch wide now 2 categories each (\frac{1}{2}) inch wide. Maybe you want to get even more granular and start using (\frac{1}{4}) inch wide categories. Can you guess where I am going with this?

At a point, we get an infinite number of such categories with an infinitely small length. This allows us to create a curve from this histogram which we had earlier divided into discrete categories. See our density curve below drawn from the histogram.

Image Probability density curve for our distribution

Why go through the effort?

Great question! As you may have guessed, I like to explain myself with examples, so let's look at another density curve to make it a bit easier for us to understand. Feel free to skip the curve equation at this stage if you have not worked with distributions before.

You can also follow along and create the graphs and visualizations in this article yourself through this Geogebra project (it runs in the browser).

$$ f(x) = \frac{1}{0.4 \sqrt{2 \pi} } \cdot e^{-\frac{1}{2} (\frac{x - 1.6}{0.4})^2} $$

Image

So now what if I ask you "What percent of my distribution is in the category 1 - 1.6?" Well, you just calculate the area under the curve between 1 and 1.6, like this:

$$ \int_{1}^{1.6} f(x) \,dx $$

Image

It would also be relatively easy for you to answer similar questions from the density curve like: "What percent of the distribution is under 1.2?" or "What percent of the distribution is above 1.2?"

You can now probably see why the effort of making this making a density curve is worth it and how it allows you to make inferences easily 🚀.

Skewed Distributions

Let's now talk a bit about skewed distributions – that is, those that are not as pleasant and symmetric as the curves we saw earlier. We'll talk about this more intuitively using the ideas of mean and median.

From this density curve graph's image, try figuring out where the median of this distribution would be. Perhaps it was easy for you to figure out – the curve is symmetrical and you might have concluded that the median is 1.6 since it was symmetric about (x=1.6).

Another way to go about this would be to say that the median is the value where the area under the curve to the left of it it and the area under the curve to the right of it are equal.

Image

We're talking about this idea since it allows us to also calculate the median for non-symmetric density curves.

As an example here, I show two very common skewed distributions and how the idea of equal areas we just discussed helps us find their medians. If we tried eyeballing our median, this is what we'd get since we want the areas on either side to be equal.

Image Eyeballing the median for skewed curves

You can also calculate the mean through these density curves. Maybe you've tried calculating the mean yourself already, but notice that if you use the general formula to calculate the mean:

$$ mean = \frac{\sum a_n}{n} $$

you might notice a flaw in it: we take into account the ( x ) values but we also have probabilities associated with these values too. And it just makes sense to factor that in too.

So we modify the way we calculate the mean by using weighted averages. We will now also have a term (w_n) representing the associated weights:

$$ mean = \frac{\sum{a_n \cdot w_n}}{n} $$

So, we will be using the idea we just discussed to calculate the mean from our density curve.

You can also more intuitively understand this as the point on the x-axis where you could place a fulcrum and balance the curve if it was a solid object. This idea should help you better understand finding the mean from our density curve.

But another really interesting way to look at this would be as the x-coordinate of the point on this curve where the rotational inertia would be zero.

You might have already figured out how we can locate the mean for symmetric curves: our median and mean lie at the same point, the point of symmetry.

We will be using the idea we just discussed, placing a fulcrum on the x-axis and balancing the curve, to eyeball out the mean for skewed graphs like the ones we saw earlier while calculating the median.

Image

We will soon discuss the idea of skewness in greater detail. But at this stage, generally speaking, you can identify the direction where your curve is skewed. If the median is to the right of the mean, then it is negatively skewed. And if the mean is to the right of median, then it is positively skewed.

Later in this article, for simplicity's sake we'll also refer to the narrow part of these curves as a "tail".

What are Moments?

Before we talk more about skewness and kurtosis let's explore the idea of moments a bit. Later we'll use this concept to develop an idea for measuring skewness and kurtosis in our distribution.

We'll use a small dataset, [1, 2, 3, 3, 3, 6]. These numbers mean that you have points that are 1 unit away from the origin, 2 units away from the origin, and so on.

So, we care a lot about the distances from the origin in our dataset. We can represent the average distance from the origin in our data by writing:

$$ \frac{\sum a_n -0}{n} = \frac{\sum a_n}{n} $$

This is what we call our first moment. Calculating this for our sample dataset we get 3 but if we change our dataset and make all elements equal to 3,

$$ [1, 2, 3, 3, 3, 6] \rightarrow [3, 3, 3, 3, 3, 3] $$

you'll see that our first moment remains the same. Can we devise something to differentiate our two datasets that have equal first moments? (PS: It's the second moment.)

We will calculate the average sum of squared distances rather than the average sum of distances:

$$ \frac{\sum (a_n)^2}{n} $$

Our second moment for our original dataset is 11.33 and for our new dataset is 9. Notice that the magnitude of the second moment is larger for our original dataset than the new one. Also, we have a higher value for the second moment in the original dataset because it is spread out and has a greater average squared distance.

Essentially we are saying that we have a couple of values in our original dataset larger than the mean value, which, when squared, increases our second moment by a lot.

Here's an interesting way of thinking about moments – assume our distribution is mass, and then the first moment would be the center of the mass, and the second moment would be the rotational inertia.

You can also see that our second moment is highly dependent on our first moment. But we are interested in knowing the information the second moment can give us independently.

To do so we calculate the squared distances from the mean or the first moment rather than from the origin.

$$ \frac{\sum (a_n- \mu_{1}^{'})^2 }{n} $$

Did you notice that we also intuitively derived a formula for variance? Going forward you will see how we use the ideas we just talked about to measure skewness and kurtosis.

Intro to Skewness and Kurtosis?

Let's see how we can use the idea of moments we talked about earlier to figure out how we can measure skewness (which you already have some idea about) and kurtosis.

What is Skewness?

Let's take the idea of moments we talked about just now and try to calculate the third moment. As you might have guessed, we can calculate the cubes of our distances. But as we discussed above, we are more interested in seeing the additional information the third moment provides.

So we want to subtract the second moment from our third moment. Later, we will also refer to this factor as the adjustment to the moment. So our adjusted moment will look like this:

$$ skewness = \frac{\sum (a_n - \mu)^3 }{n \cdot \sigma ^3} $$

This adjusted moment is what we call skewness. It helps us measure the asymmetry in the data.

Perfectly symmetrical data would have a skewness value of 0. A negative skewness value implies that a distribution has its tail on the left side of the distribution, while a positive skewness value has its tail on the on the right side of the distribution.

Image Positive skew and negative skew

At this stage, it might seem like calculating skewness would be pretty tough to do since in the formulas we use the population mean ( \mu ) and the population standard deviation ( \sigma ) which we wouldn't have access to while taking a sample.

Instead, you only have the sample mean and the sample standard deviation, so we will soon see how you can use these.

What is Kurtosis?

As you might have guessed, this time we will calculate our fourth moment or use the fourth power of our distances. And like we talked about earlier we are interested in seeing the additional information this provides so we will also subtract out the adjustment factor from it.

This is what we call kurtosis or a measure of whether our data has a lot of outliers or very few outliers. This will look like:

$$ kurtosis = \frac{\sum (a_n - \mu)^4 }{n \cdot \sigma ^4} $$

A better term for what's going on here is to figure out if the distribution is heavy-tailed or light-tailed. We can compare this to a normal distribution.

If you do a simple substitution you'll see that the kurtosis for normal distribution is 3. And since we are interested in comparing kurtosis to the normal distribution, often we use excess kurtosis which simply subtracts 3 from the above equation.

Image Positive and negative kurtosis (Adapted from Analytics Vidhya)

This is us essentially trying to force the kurtosis of our normal distribution to be 0 for easier comparison. So, if our distribution has positive kurtosis, it indicates a heavy-tailed distribution while negative kurtosis indicates a light-tailed distribution. Graphically, this would look something like the image above.

Sampling Adjustment

So, a problem with the equations we just built is that they have two terms in them, the distribution mean ( \mu ) and the distribution standard deviation ( \sigma ). But we are taking a sample of observations so we do not have the parameters for the whole distribution. We'd only have the sample mean and the sample standard deviation.

To keep this article focused, we will not be talking in detail about sampling adjustment terms since degrees of freedom is not in the scope of this article.

The idea is to use our sample mean ( \bar{x} ) and our sample standard deviation ( s ) to estimate these values for our distribution. We will also have to adjust our degree of freedom in these equations for it.

Don't worry if you don't understand this concept completely at this point. We can move on anyway. This leads to us modifying the equations we talked about earlier like so:

$$ skewness = \frac{\sum (a_n - \bar{x})^3 }{s^3} \cdot \frac{n}{(n-1)(n-2)} $$

$$ kurtosis = \frac{\sum (a_n - \bar{x})^4 }{s^4} \cdot \frac{n(n+1)}{(n-1)(n-2)(n-3)} - \frac{3(n-1)^2}{(n-2)(n-3)} $$

How to Implement this in Python

Finally, let's finish up by seeing how you can measure skewness and kurtosis in Python with an example. In case you want to follow along and try out the code, you can follow along with this Colab Notebook where we measure the skewness and kurtosis of a dataset.

It is pretty straightforward to implement this in Python with Scipy. It has methods to easily measure skewness and kurtosis for a distribution with pre-built methods.

The below code block shows how to measure skewness and kurtosis for the Boston housing dataset, but you could also use it for your own distributions.

from scipy.stats import skew
from scipy.stats import kurtosis

skew(data["MEDV"].dropna())
kurtosis(data["MEDV"].dropna())

Thank you for reading!

Thank you for sticking with me until the end. I hope you have learned a lot from this article.

I am excited to see if this article helped you better understand these two very important ideas. If you have any feedback or suggestions for me please feel free to reach out to me on Twitter.

]]> <![CDATA[ Bayes' Rule – Explained For Beginners ]]> https://www.freecodecamp.org/news/bayes-rule-explained/ 66d4608fc7632f8bfbf1e47f Mon, 29 Mar 2021 13:00:00 +0000 By Peter Gleeson

Bayes' Rule is the most important rule in data science. It is the mathematical rule that describes how to update a belief, given some evidence. In other words – it describes the act of learning.

The equation itself is not too complex:

Probability of event A given event B equals Prior probability of event A times Probability of event B given A, divide by marginal probability of event B The equation: Posterior = Prior x (Likelihood over Marginal probability)

There are four parts:

  • Posterior probability (updated probability after the evidence is considered)
  • Prior probability (the probability before the evidence is considered)
  • Likelihood (probability of the evidence, given the belief is true)
  • Marginal probability (probability of the evidence, under any circumstance)

Bayes' Rule can answer a variety of probability questions, which help us (and machines) understand the complex world we live in.

It is named after Thomas Bayes, an 18th century English theologian and mathematician. Bayes originally wrote about the concept, but it did not receive much attention during his lifetime.

French mathematician Pierre-Simon Laplace independently published the rule in his 1814 work Essai philosophique sur les probabilités.

Today, Bayes' Rule has numerous applications, from statistical analysis to machine learning.

This article will explain Bayes' Rule in plain language.

Conditional probability

The first concept to understand is conditional probability.

You may already be familiar with probability in general. It lets you reason about uncertain events with the precision and rigour of mathematics.

Conditional probability is the bridge that lets you talk about how multiple uncertain events are related. It lets you talk about how the probability of an event can vary under different conditions.

For example, consider the probability of winning a race, given the condition you didn't sleep the night before. You might expect this probability to be lower than the probability you'd win if you'd had a full night's sleep.

The probability of winning the race, given a full night's sleep is 30%. The probability of winning the race, given no sleep is only 5%.

Or, consider the probability that a suspect committed a crime, given that their fingerprints are found at the scene. You'd expect the probability they are guilty to be greater, compared with had their fingerprints not been found.

The notation for conditional probability is usually:

P open brackets A vertical bar B close brackets P(A|B)

Which is read as "the probability of event A occurring, given event B occurs".

An important thing to remember is that conditional probabilities are not the same as their inverses.

That is, the "probability of event A given event B" is not the same thing as the "probability of event B, given event A".

To remember this, take the following example:

The probability of clouds, given it is raining (100%) is not the same as the probability it is raining, given there are clouds.

(Insert joke about British weather).

Bayes' Rule in detail

Bayes' Rule tells you how to calculate a conditional probability with information you already have.

It is helpful to think in terms of two events – a hypothesis (which can be true or false) and evidence (which can be present or absent).

However, it can be applied to any type of events, with any number of discrete or continuous outcomes.

Probability the hypothesis is true, given the evidence is present equals the prior probability of the hypothesis being true times the likelihood of the evidence being present given the hypothesis is true, divided by the marginal probability of the evidence being present under any circumstance

Bayes' Rule lets you calculate the posterior (or "updated") probability. This is a conditional probability. It is the probability of the hypothesis being true, if the evidence is present.

Think of the prior (or "previous") probability as your belief in the hypothesis before seeing the new evidence. If you had a strong belief in the hypothesis already, the prior probability will be large.

The prior is multiplied by a fraction. Think of this as the "strength" of the evidence. The posterior probability is greater when the top part (numerator) is big, and the bottom part (denominator) is small.

The numerator is the likelihood. This is another conditional probability. It is the probability of the evidence being present, given the hypothesis is true.

This is not the same as the posterior!

Remember, the "probability of the evidence being present given the hypothesis is true" is not the same as the "probability of the hypothesis being true given the evidence is present".

Now look at the denominator. This is the marginal probability of the evidence. That is, it is the probability of the evidence being present, whether the hypothesis is true or false. The smaller the denominator, the more "convincing" the evidence.

Worked example of Bayes' Rule

Here's a simple worked example.

Your neighbour is watching their favourite football (or soccer) team. You hear them cheering, and want to estimate the probability their team has scored.

Step 1 – write down the posterior probability of a goal, given cheering

Step 2 – estimate the prior probability of a goal as 2%

Step 3 – estimate the likelihood probability of cheering, given there's a goal as 90% (perhaps your neighbour won't celebrate if their team is losing badly)

Step 4 – estimate the marginal probability of cheering – this could be because:

  • a goal has been scored (2% of the time, times 90% probability)
  • or any other reason, such as the other team missing a penalty or having a player sent off (98% of the time, times perhaps 1% probability)

Now, piece everything together:

Probability of goal, given cheering equals prior probability of goal times probability of cheering given goal, divided by probability of cheering given goal plus probability of cheering given no goal. Equals 0.02 times 0.9 over 0.02 times 0.9 plus 0.98 times 0.01 = 64.7 percent

Use cases for Bayes' Rule and next steps

Bayes' Rule has use cases in many areas:

  • Understanding probability problems (including those in medical research)
  • Statistical modelling and inference
  • Machine learning algorithms (such as Naive Bayes, Expectation Maximisation)
  • Quantitative modelling and forecasting

Next, you'll discover how Bayes' Rule can be used to quantify uncertainty and model real world problems. Then, how to reason about "probabilities of probabilities".

The final step will cover how various computational tricks let you make use of Bayes' Rule to solve non-trivial problems.

]]> <![CDATA[ How to Get Started with Algorithmic Trading in Python ]]> https://www.freecodecamp.org/news/how-to-get-started-with-algorithmic-trading-in-python/ 66d45f4a706b9fb1c166b971 Mon, 04 Jan 2021 17:56:40 +0000 When I was working as a Systems Development Engineer at an Investment Management firm, I learned that to succeed in quantitative finance you need to be good with mathematics, programming, and data analysis.

Algorithmic or Quantitative trading can be defined as the process of designing and developing statistical and mathematical trading strategies. It is an extremely sophisticated area of finance.

So, the question is how do you get started with Algorithmic Trading?

I am going to walk you through five essential topics that you should study in order to pave your way into this fascinating world of trading.

I personally prefer Python as it offers the right degree of customization, ease and speed of development, testing frameworks, and execution speed. Because of this, all these topics are focused on Python for Trading.

1. Learn Python Programming

In order to have a flourishing career in Data Science in general, you need solid fundamentals. Whichever language you choose, you should thoroughly understand certain topics in that language.

Here’s what you should look to master in the Python ecosystem for data science:

  • Environment Setup — this includes creating a virtual environment, installing required packages, and working with Jupyter notebooks or Google colabs.

  • Data Structures — some of the most important pythonic data structures are lists, dictionaries, NumPy arrays, tuples, and sets. I’ve collected a few examples in the linked article for you to learn these.

  • Object-Oriented Programming — As a quant analyst, you should make sure you are good at writing well-structured code with proper classes defined. You must learn to use objects and their methods while using external packages like Pandas, NumPy, SciPy, and so on.

The freeCodeCamp curriculum also offers a certification in Data Analysis with Python to help you get started with the basics.

Learn How to Crunch Financial Data

Data analysis is a crucial part of finance. Besides learning to handle dataframes using Pandas, there are a few specific topics that you should pay attention to while dealing with trading data.

How to exploring data using Pandas

One of the most important packages in the Python data science stack is undoubtedly Pandas. You can accomplish almost all major tasks using the functions defined in the package.

Focus on creating dataframes, filtering (loc, iloc, query), descriptive statistics (summary), join/merge, grouping, and subsetting.

How to deal with time-series data

Trading data is all about time-series analysis. You should learn to resample or reindex the data to change the frequency of the data, from minutes to hours or from the end of day OHLC data to end of week data.

For example, you can convert 1-minute time series into 3-minute time series data using the resample function:

df_3min = df_1min.resample('3Min', label='left').agg({'OPEN': 'first', 'HIGH': 'max', 'LOW': 'min', 'CLOSE': 'last'})

3. How to Write Fundamental Trading Algorithms

A career in quantitative finance requires a solid understanding of statistical hypothesis testing and mathematics. A good grip over concepts like multivariate calculus, linear algebra, probability theory will help you lay a good foundation for designing and writing algorithms.

You can start by calculating moving averages on stock pricing data, writing simple algorithmic strategies like moving average crossover or mean reversion strategy and learning about relative strength trading.

After taking this small yet significant leap of practicing and understanding how basic statistical algorithms work, you can look into the more sophisticated areas of machine learning techniques. These require a deeper understanding of statistics and mathematics.

Here are two books you can start with:

And here are a couple courses that will help you get started with Python for Trading and that cover most of the topics that I’ve captured here:

You can get 10% off the Quantra course by using my code HARSHIT10.

4. Learn About Backtesting

Once you are done coding your trading strategy, you can’t simply put it to the test in the live market with actual capital, right?

The next step is to expose this strategy to a stream of historical trading data, which would generate trading signals. The carried out trades would then accrue an associated profit or loss (P&L) and the accumulation of all the trades would give you the total P&L. This is called backtesting.

Backtesting requires you to be well-versed in many areas, like mathematics, statistics, software engineering, and market microstructure. Here are some concepts you should learn to get a decent understanding of backtesting:

  • You can start by understanding technical indicators. Explore the Python package called TA_Lib to use these indicators.

  • Employ momentum indicators like parabolic SAR, and try to calculate the transaction cost and slippage.

  • Learn to plot cumulative strategy returns and study the overall performance of the strategy.

  • A very important concept that affects the performance of the backtest is bias. You should learn about optimization bias, look-ahead bias, psychological tolerance, and survivorship bias.

5. Performance Metrics — How to Evaluate Trading Strategies

It’s important for you to be able to explain your strategy concisely. If you don’t understand your strategy, chances are on any external modification of regulation or regime shift, your strategy will start behaving abnormally.

Once you understand the strategy confidently, the following performance metrics can help you learn how good or bad the strategy actually is:

  • Sharpe Ratio — heuristically characterises the risk/reward ratio of the strategy. It quantifies the return you can accrue for the level of volatility undergone by the equity curve.

  • Volatility — quantifies the “risk” related to the strategy. The Sharpe ratio also embodies this characteristic. Higher volatility of an underlying asset often leads to higher risk in the equity curve and that results in smaller Sharpe ratios.

  • Maximum Drawdown — the largest overall peak-to-trough percentage drop on the equity curve of the strategy. Maximum drawdowns are often studied in conjunction with momentum strategies as they suffer from them. Learn to calculate it using the numpy library.

  • Capacity/Liquidity — determines the scalability of the strategy to further capital. Many funds and investment management firms suffer from these capacity issues when strategies increase in capital allocation.

  • CAGR — measures the average rate of a strategy’s growth over a period of time. It is calculated by the formula: (cumulative strategy returns)^(252/number of trading days) — 1

Further Resources

This article served as a suggested curriculum to help you get started with algorithmic trading. It is a good list of concepts to master.

Now, the question is what resources can help you get up to speed with these topics?

Here are a few classic books and useful courses with assignments and exercises that I found helpful:

Data Science with Harshit

With this channel, I am planning to roll out a couple of series covering the entire data science space. Here is why you should be subscribing to the channel:

If this tutorial was helpful, you should check out my data science and machine learning courses on Wiplane Academy. They are comprehensive yet compact and helps you build a solid foundation of work to showcase.

]]> <![CDATA[ Random Number Generator: How Do Computers Generate Random Numbers? ]]> https://www.freecodecamp.org/news/random-number-generator/ 66d45d67c17d4b8ace5b9eb4 Mon, 26 Oct 2020 21:51:45 +0000 By Alexander Arobelidze

People have been using random numbers for millennia, so the concept isn't new. From the lottery in ancient Babylon, to roulette tables in Monte Carlo, to dice games in Vegas, the goal is to leave the end result up to random chance.

But gambling aside, randomness has many uses in science, statistics, cryptography and more. Yet using dice, coins, or similar media as a random device has its limitations.

Because of the mechanical nature of these techniques, generating large quantities of random numbers requires great deal of time and work. Thanks to human ingenuity, we have more powerful tools and methods at our disposal.

Methods for generating random numbers

True Random Numbers

Image _Picture of analog-input digital-output processing device. Photo by Harrison Broadbent_

Let's consider two principal methods used to generate random numbers. The first method is based on a physical process, and harvests the source of randomness from some physical phenomenon that is expected to be random.

Such a phenomenon takes place outside of the computer. It is measured and adjusted for possible biases due to the measurement process. Examples include radioactive decay, the photoelectric effect, cosmic background radiation, atmospheric noise (which we will use in this article), and more.

Thus, random numbers generated based on such randomness are said to be "true" random numbers.

Technically, the hardware part consists of a device that converts energy from one form to another (for example, radiation to an electrical signal), an amplifier, and an analog-to-digital converter to turn the output into a digital number.

What are Pseudorandom Numbers?

Image _Picture of computer code flowing through computer screen. Photo by Markus Spiske._

As an alternative to "true" random numbers, the second method of generating random numbers involves computational algorithms that can produce apparently random results.

Why apparently random? Because the end results obtained are in fact completely determined by an initial value also known as the seed value or key. Therefore, if you knew the key value and how the algorithm works, you could reproduce these seemingly random results.

Random number generators of this type are frequently called Pseudorandom number generators and, as a result, output Pseudorandom Numbers.

Even though this type of generator typically doesn't gather any data from sources of naturally occurring randomness, such gathering of keys can be made possible when needed.

Let's compare some aspects of true random number generators or TRNGs and pseudorandom number generators or PRNGs.

PRNGs are faster than TRNGs. Because of their deterministic nature, they are useful when you need to replay a sequence of random events. This helps a great deal in code testing, for example.

On the other hand, TRNGs are not periodic and work better in security sensitive roles such as encryption.

A period is the number of iterations a PRNG goes through before it starts repeating itself. Thus, all other things being equal, a PRNG with a longer period would take more computer resources to predict and crack.

Example Algorithm for Pseudo-Random Number Generator

A computer executes code that is based on a set of rules to be followed. For PRNGs in general, those rules revolve around the following:

  1. Accept some initial input number, that is a seed or key.
  2. Apply that seed in a sequence of mathematical operations to generate the result. That result is the random number.
  3. Use that resulting random number as the seed for the next iteration.
  4. Repeat the process to emulate randomness.

Now let's look at an example.

The Linear Congruential Generator

This generator produces a series of pseudorandom numbers. Given an initial seed X0 and integer parameters a as the multiplier, b as the increment, and m as the modulus, the generator is defined by the linear relation: Xn ≡ (aXn-1 + b)mod m. Or using more programming friendly syntax: Xn = (a * Xn-1 + b) % m.

Each of these members have to satisfy the following conditions:

  • m > 0 (the modulus is positive),
  • 0 < a < m (the multiplier is positive but less than the modulus),
  • 0 ≤ b < m (the increment is non negative but less than the modulus), and
  • 0 ≤ X0 < m (the seed is non negative but less than the modulus).

Let's create a JavaScript function that takes the initial values as arguments and returns an array of random numbers of a given length: 

    
    // x0=seed; a=multiplier; b=increment; m=modulus; n=desired array length; 
    const linearRandomGenerator = (x0, a, b, m, n) => {
        const results = []
        for (let i = 0; i < n; i++) {
            x0 = (a * x0 + b) % m
            results.push(x0)
        }
        return results
    }

The Linear Congruential Generator is one of the oldest and best-known PRNG algorithms.

As for random number generator algorithms that are executable by computers, they date back as early as the 1940s and 50s (the Middle-square method and Lehmer generator, for example) and continue to be written today (Xoroshiro128+, Squares RNG, and more).

A Sample Random Number Generator

When I decided to write this article about embedding a random number generator within a web page, I had a choice to make.

I could've used JavaScript's Math.random() function as the base and generate output in pseudorandom numbers like I have in earlier articles (see Multiplication Chart - Code Your Own Times Table).

But this article itself is about generating random numbers. So I decided to learn how to gather "true" randomness based data and share my discovery with you.

So below is the "true" Random Number Generator. Set the parameters and hit Generate.

True Random Number Generator
Binary Decimal Hexadecimal

Generate

Result:

The code fetches data from one of the APIs, courtesy of Random.org. This online resource has a plethora of useful, customizable tools and comes with excellent documentation to go with it.

The randomness comes from atmospheric noise. I was able to use asynchronous functions. That is a huge benefit going forward. The core function looks like this:

    // Generates a random number within user indicated interval
       const getRandom = async (min, max, base) => {
           const response = await     fetch("https://www.random.org/integers/?num=1&min="+min+"
    &max="+max+"&col=1&base="+base+"&format=plain&rnd=new")
              return response.text() 
       }

The parameters it takes allow a user to customize random number output. For example, min and max allow you to set lower and upper limits on generated output. And base determines if the output is printed as binary, decimal or hexadecimal.

Again, I chose this configuration but there are many more available at the source.

When you click the Generate button, the handleGenerate() function is called. It in turn invokes the getRandom() asynchronous function, manages error handling, and outputs results:

    
    // Output handling
    const handleGenerate = () => {
        handleActive(generateButton)
        const base = binary.checked ? 2 : decimal.checked ? 10 : 16
        if (!minimum.value || !maximum.value) {
            prompter.style.color = 'red' 
            prompter.textContent = "Enter Min & Max values"
        } else {
            getRandom(minimum.value, maximum.value, base).then((data) => {
                resultValue.textContent = data
                prompter.textContent = ""    
            }).catch((error) => {
                resultValue.textContent = 'ERROR'
                prompter.textContent = 'Connection error. Unable to                         generate';    
            })
                handleRestart()
        }

   }

The rest of the code deals with HTML structure, appearance, and styling.

The code is ready to be embedded and used within this web page. I separated it into component parts and supplied it with detailed comments. It can easily be modified. You can also modify the functionality and styles as your needs require.

This is the link to the GitHub repo of the complete code: https://github.com/sandroarobeli/random-generator

]]> <![CDATA[ Learn Calculus 2 in This Free 7-Hour Course ]]> https://www.freecodecamp.org/news/learn-calculus-2-in-this-free-7-hour-course/ 66b20414903dc07a13516661 Mon, 14 Sep 2020 13:38:29 +0000 There are many types of software development that require high level mathematics such as graphics simulations and some types of machine learning.

We are helping people learn the mathematics they need by posting a series of college-level courses on the freeCodeCamp.org YouTube channel. Our latest free course is a complete 7-hour Calculus 2 course.

This course was created by Dr. Linda Green who teaches at the University of North Carolina at Chapel Hill. She has years of experience teaching Calculus 2 to undergraduate students.

Besides the video content, we have also included full course notes from Dr. Green as a link in the video description.

Below are all the Calculus 2 concepts you will learn about in this course.

  • Area Between Curves
  • Volumes of Solids of Revolution
  • Volumes Using Cross-Sections
  • Arclength
  • Work as an Integral
  • Average Value of a Function
  • Proof of the Mean Value Theorem for Integrals
  • Integration by Parts
  • Trig Identities
  • Proof of the Angle Sum Formulas
  • Integrals Involving Odd Powers of Sine and Cosine
  • Integrals Involving Even Powers of Sine and Cosine
  • Special Trig Integrals
  • Integration Using Trig Substitution
  • Integrals of Rational Functions
  • Improper Integrals - Type 1
  • Improper Integrals - Type 2
  • The Comparison Theorem for Integrals
  • Sequences - Definitions and Notation
  • Series Definitions
  • Sequences - More Definitions
  • Monotonic and Bounded Sequences Extra
  • L'Hospital's Rule
  • L'Hospital's Rule on Other Indeterminate Forms
  • Convergence of Sequences
  • Geometric Series
  • The Integral Test
  • Comparison Test for Series
  • The Limit Comparison Test
  • Proof of the Limit Comparison Test
  • Absolute Convergence
  • The Ratio Test
  • Proof of the Ratio Test
  • Series Convergence Test Strategy
  • Taylor Series Introduction
  • Power Series
  • Convergence of Power Series
  • Power Series Interval of Convergence Example
  • Proofs of Facts about Convergence of Power Series
  • Power Series as Functions
  • Representing Functions with Power Series
  • Using Taylor Series to find Sums of Series
  • Taylor Series Theory and Remainder
  • Parametric Equations
  • Slopes of Parametric Curves
  • Area under a Parametric Curve
  • Arclength of Parametric Curves
  • Polar Coordinates

We recommend you pull out some paper and a pencil and take physical notes – just like when you were back in a classroom.

This manual process of taking notes will help you better comprehend and retain these concepts, since Dr. Green moves fast and covers a lot of topics in a short time.

Check out our mathematics playlist if you want to brush up on the prerequisites such as Algebra, Precalculus, and Calculus 1.

You can watch the full course on the freeCodeCamp.org YouTube channel (7-hour watch).

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