## Math | All About Mathematics [November 2022]

Math or Mathematics is a body of knowledge that studies numbers. Maths topics include arithmetic, number theory, formulas & related structures.

Here we will approach Math by discussing:

- Math History
- Euclid Algorithm
- Cryptography
- Encryption Technology
- Calculus
- Topology
- Graph Theory
- Fourier Transform
- Group Theory
- Boolean Algebra
- Set Theory
- Markov Chains
- Game Theory
- Chaos Theory
- Butterfly Effect
- Geodesics
- Fermat’s Last Theorem
- Millennium Prize Problems

## Math History

Mathematics comes from Logic.

Humans have begun the use of logic in Mathematics since the 4th Century BC.

Problems of Logic involve just thinking.

It is like if A exits when B exists then it is the same as if B does not exist then A does not exist.

If the angle of 40 degrees is acute then an angle that is not acute is not 40 degrees.

In mathematics, we need a standard language to understand the theorems and logical expressions.

If you need to prove a statement that says: “If x^2 is an Even number then x is Even.”

Then at the same time, you can also prove that “If x is not an Even number then x^2 is also not an Even number.”

If you prove the first it will automatically prove the second one.

In Math, we have to understand the language and logic of every mathematical expression.

About 2000 years ago Greek mathematician Euclid published 13 books called Elements.

Which are considered the most influential textbooks ever.

There are lots of things discussed in these books.

One most important topic discussed is called Euclid Algorithm.

## Euclid Algorithm

Euclid Algorithm is the first-ever mathematical algorithm in history.

An algorithm is simply a series of steps that can solve a problem.

Algorithms often but not all the time are executed by computers.

It may take a lot of time to calculate the greatest common divisor of any two given numbers.

But this can be done by hand in minutes or seconds using Euclidean Algorithm.

This algorithm uses basic elementary school arithmetic to solve the problem.

It is used a lot in cryptography.

## Cryptography

Cryptography is a technology that is responsible for secure communication.

When you put your sensitive information online this is encrypted by a text using mathematical techniques.

This encryption turns normal text into unreadable text using a secret key.

That text can not be read or understood by any third party or by any other user.

Unless it is decrypted again to the original text by the receiving party using that encryption key.

There are many different methods to do the encryption such as RSA.

Keep the fact in mind that it is very difficult to break the larger numbers into prime factors.

Cryptography involves the study of Number Theory and Integers.

That is why Euclidean Algorithm has applications in this field.

This could encrypt passwords, credit card numbers, and all other types of sensitive data.

NSA National Security Agency in the USA employs lots of mathematicians to create those codes and also can break them.

## Encryption Technology

Encryption technology is not new.

Hundreds of years back those handwritten were also encrypted for privacy.

In 1587 these encrypted letters were sent from Mary Queen of Scots.

Those encrypted messages revealed that she had sanctioned.

And attempted assassination of Queen Elizabeth.

She was executed shortly after this message.

The discovery of PI was also done several hundred years back.

Pi was approximated as 3.16 and 3.125 earlier.

But later more precise calculations came.

Logarithms, Geometry, Algebra, Geometry, Basic Cartesian Coordinates, and Complex Numbers were discovered thousands of years back today.

In the 3rd century BC radius of the Earth was approximated using deductive reasoning, basic geometry, and shadow measuring techniques with approximately 99% accuracy.

## Calculus

Calculus was discovered in the 17th Century.

Calculus entirely changed Math and Physics.

Newton analyzed rates of change at any given instant of time.

That instant rate of change is what we call Derivative.

A derivative shows the change in a given parameter concerning another parameter at any instant of time.

Algebra can calculate the average speed if it is changing constantly.

Calculus can calculate the exact speed at any instant like radar readings.

Calculus made it easy for us to gain insight into the nature of motion.

Calculus made it easy for us to have a sense of the motion of planets.

It also made it easy for us to understand how planets change the speed of their orbits.

Other applications of Calculus include: Understanding the electromagnetic waves in terms of behavior and motion.

Calculation of Moment of Inertia.

Calculating the Resistance of an object to rotation.

Calculation of work done on a particle that is moving inside a complex vector field.

Calculus is also used to maximize profits in economics and finance.

Calculus is also used to calculate fusion rates in chemistry.

So if you study like Engineering, Physics, Maths, Chemistry, Biology, Business you are going to study Calculus.

## Topology

Like Geometry, Topology is also the study of shapes.

Geometry deals with length angles and areas of a shape whereas Topology studies connectedness holes and compactness of a shape.

Like how many holes a shape has?

This is a problem of Topology.

A donut and a coffee mug are the same things in Topology.

Topology deals with complex shapes and higher dimensional shapes and knots etc.

Topology allows you to understand:

Cutting a Mobius strip in half.

Turning sphere inside out without cutting it.

Topology applies to Physic: Quantum Field Theory Cosmology General Relativity: According to general relativity space-time is a 4-dimensional Lorentzian manifold.

Analysis of this involves Topology.

Motion Planning Robotics: Visualizing all the possible states of a robot.

Configuring space is modeled using Topology.

Topology in C**omputer Science**: Topology is used to model: The network connectivity and data flow.

Topology in Biology is used to model Enzymes and to do an analysis that how enzymes cut and reconnect the DNA.

Topology addresses the questions like whether can we morph one object into another or whether the two objects are homeomorphic.

Topology deals with the bending and stretching of a particular space.

Because bending and Stretching do not change the properties of Space in consideration.

## Graph Theory

In 1736 Leonard Euler published the first scientific paper in graph theory about 7 bridges of Konigsberg.

He provided a network of 7 bridges and asked a simple question.

He asked if it was possible to cross each of those 7 bridges at once without going to the water.

Euler proved it impossible using graph theory.

The problem of 7 bridges is not the Graph Theory but in advanced Mathematics, the field of Topology also studies Graph Theory.

In graph theory graph is made of nodes and edges that connect.

Graph theory has lots of applications in Computer Engineering.

Nodes of graphs can represent people while the connection can represent compatibility.

Dating sites use graph theory algorithms to do the best matching.

The nodes of the graph could be cities while the edges are routes that connect them.

Here we can calculate the shortest possible path between any two points.

Graph theory is used in Social Network Analysis where graphs represent how we are connected in social media websites and other social structures there.

This analysis is used in security applications to map the information against criminal gangs and terrorist organizations.

In the ’90s two computer scientists Larry Page and Sergey Brin represented nodes as websites on the internet network and the connections between any two websites as edges.

Several connections to a website show the quality of that webpage.

More connections better the quality.

This is what we call Backlinks in Search Optimization SEO.

This graph theory is now the backbone of the most famous search engine Google.

## Fourier Transform

In 1822 Joseph Fourier published a paper about heat flow.

While working on heat flow he revealed a discovery that has lots of applications in modern-day technologies.

Fourier Transform is used to write any function as a sum of sine and cosine functions.

No matter how weird a function looks like we can break it into the sum of sine and cosines.

Fourier explained that we can make any function we want by taking some of those sine and cosine functions with the right amplitudes and frequencies and adding them up all.

Fourier Transform has a lot of applications but it is applied mainly in Signal Processing and Quantum Mechanics.

Real-world signals like radar signals or a signal from a digital image and signals of sound or light are more complex signals.

These real-world signals are very complicated.

We use Fourier Analysis to break that complicated signal into a simpler trig function that makes up the signal.

This breakdown of that signal could reveal more information about the signal which is not seen originally.

Like the higher or lower frequency sources in the signal.

## Group Theory

The fundamentals of a physical field known as Group Theory appears at the start of the 9th Century.

Group Theory is the study of Groups.

A group is a set of elements that involve some operations and satisfy certain logical conditions.

A set of summations of integers is a group.

A set whose members have these four properties of Closure, identity Element, Inverse, and Associativity is called a Group.

If you add any two integers you will always end up getting an integer as a result that will be a member of that Group.

The order of addition does not matter while adding two Integers.

This group is commutative.

Because if you swap any two members of the group you will get the same answer.

If the results change by swapping the members of the group then that group is called the Abelian group.

Rubik Cube manipulations are Abelian group.

Rubik Cube with its unique properties can be manipulated by applying this Group Theory.

You do not need Group Theory to solve a Rubik Cube but it gives you more insight into the mathematics involved in the Rubik Cube.

All groups are a set but not all sets are groups.

Group Theory gave us much more insight into the Mathematics of Symmetry.

The mathematics of Symmetry has many applications in chemistry.

Groups are used to classify Crystal Structures and Symmetries inside Chemical Compounds and molecules.

Group Theory is also applied in Public Key Cryptography.

Group Theory has many applications in Physics.

Noether’s Theorem explains how the symmetry of a physical system corresponds with the law of conservation.

Einstein’s General Theory of Relativity can be understood better with Noether’s theorem and the Group Theory.

## Boolean Algebra

Boolean Algebra was discovered in 1837.

It is an algebra using only 1s and 0s.

Boolean Algebra has many applications and Computer Engineering.

It is also used in Integrated Circuits ICs.

Microprocessors are designed with this binary Algebra.

Boolean Algebra is used to reduce the number of logic gates in electronic circuits.

Those logic gates store and transmit ones and zeros in a computer.

The less number of logic gates makes computers faster.

In 1874 George Cantor published a paper about the property of a collection of all real algebraic numbers.

Here another branch of Math called Set Theory starts.

A set in mathematics is just a collection of objects.

Examples of sets are a set of people, a set of colors, a set of even numbers and a prime number, etc.

Each member of a set is called an element of that set.

## Set Theory

Set Theory deals with the union of sets, the intersection of sets, subsets, and supersets.

Set Theory has applications in all other fields we discussed like Graph Theory, Group Theory, and Topology.

If you flip coins every single event is independent.

It does not depend on your previous spin.

## Markov Chains

In early 1900, a statistical model called Markov Chains was designed which showed that the probability was depending only upon the previous event.

That model included a state space and some transition probabilities.

If you want to analyze the population of New York and Los Angles cities.

Let’s say the people are allowed to move in between two cities.

For instance, if every year10% of people who are residents of Los angles move to New York remaining 90%.

And 15% of people move from New York to Los Angles every year.

Here your current location matters.

It can be predicted that ether you are going to move or not depends on your current location.

We can determine how this system evolves with time and whether it is going to reach a steady-state or not using linear algebra.

Thermodynamics represents the details of an unknown system.

Modern speech recognition system.

Page Ranking Systems used by Google.

## Game Theory

In 1928 John Van Neumann started this field of statistics and mathematics called Game Theory.

In this theory, the strategies for two-person zero-sum games, or the games in which one player’s gain is balanced with the loss of another player, were developed.

Game Theory is about logical decision-making and handling competitive situations.

One of the most famous games which came after game theory is Prisoner’s dilemma.

Game Theory has applications in Computer Science and Economics.

Game Theory is used in Cloud Computing to model the interaction between Cloud Servers to minimize costs and maximize the quality of service.

In Economics Game Theory is used in Auctions, Mergers, and Acquisitions.

In 1880 a mathematician named Henry Poincare worked on Three-Body Problem.

This problem deals with the study of the motion of three-point masses Moon, Earth, and the Sun in the presence of their gravitational forces on each other.

Analysis of any two orbiting bodies can be done using Newton’s laws of motion.

But when the third body joins it becomes a lot more difficult to perform the analysis.

Henry found that a slight change in their positions and initial velocities can cause a big difference in their behavior concerning time.

This problem of Game Theory gave birth to Chaos Theory.

## Chaos Theory

Chaos theory deals with the dynamic systems which are sensitive to their initial conditions.

Chaos Theory did not progress much at the beginning because there was no efficient computational Power.

Edward Lorenz 1960 found his interest in Chaos Theory while working on weather prediction.

He observed that a slight change in initial conditions produced huge changes in the outcome of his computer simulations.

## Butterfly Effect

The butterfly effect is a metaphorical example.

If a butterfly flaps its wings on one side of the Atlantic Ocean, it can cause a tornado on the other side of the ocean a few weeks later.

A minor change in environmental conditions can make a big difference.

This is the concept of Chaos Theory.

Chaos Theory can be used to analyze the system of the double pendulum.

A double pendulum is used in robotics to predict the behavior of the motion of a robot concerning time.

Chaos Theory has also applications in Cryptography.

## Geodesics

Geodesics are curves.

The curves are used to represent the shortest possible path between any two points on a curved three-dimensional surface.

Geodesics explains how ants living on that surface would perceive it to be straight.

It also explains why airplane routes are curved on maps.

Geodesics is a lot important to explain and understand Einstein’s General Theory of Relativity.

## Fermat’s Last Theorem

This theorem which is related to the surface is very easy to understand but it remained unsolved for 300 years.

This theorem states that there are no such integers A B and C such that A^n + B^n = C^n is true.

Where n is an integer that must be greater than 2.

This theorem was proved true by Andrew Wiles after 300 years in 1994.

The proof of Fermat’s Last Theorem involves a lot of rigorous mathematics more than 100 pages.

## Millennium Prize Problems

Millennium Prize Problems are 7 mathematical problems that were decided in 2000 at a conference.

Each of those problems has a 1 million dollar prize for anyone who can solve it.

In the last 20 years, only 1 out of those 7 could be solved.

That problem was solved in 2003 with Poincare Conjecture Theorem.

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