Chapter 2: Number Theory

Number Theory is one of the most ancient branches of mathematics, studied since ancient times and serving as the foundation for many mathematical concepts. Personally, I've always appreciated its purity. But what does it entail?

This field focuses on natural numbers, specifically positive integers, often referred to as counting numbers. Despite nearly 4000 years of study, one might assume we've fully explored these seemingly simple numbers. If it were a craftsman's job with a 4000-year deadline to comprehend the numbers 1, 2, 3, ..., many would decline the offer!

However, the simplicity of these numbers masks their complexity. A significant aspect of Number Theory revolves around the multiplicative nature of natural numbers. These numbers possess a unique structure, akin to DNA, formed from basic elements known as prime numbers—just as DNA is composed of fundamental molecules. To grasp natural numbers, one must understand prime numbers.

Prime numbers, defined as those greater than 1 with no divisors other than 1 and themselves, begin with the sequence 2, 3, 5, 7, 11, etc. Since the Greeks began their study around 300 BC, the quest for a general formula for the nth prime has been a mathematical pursuit akin to the Holy Grail.

For instance, consider the number 12. Its prime factorization is {2, 2, 3}, expressed as 12 = 2⋅2⋅3 = 2²⋅3. This factorization is unique to 12, and all natural numbers can be broken down into their prime components.

Two fundamental questions arise:

1. Is there a discernible pattern in the distribution of primes among natural numbers?
2. How many prime numbers exist?

The consensus is that there isn't a clear pattern. Primes become increasingly scarce as one progresses along the number line, making the second question less straightforward. The gaps between consecutive primes can grow arbitrarily large, but around 300 BC, Euclid established that there are infinitely many primes—a cornerstone of Number Theory, still taught today.

We also have asymptotic formulas that approximate the number of primes below a given number. Gauss conjectured that primes grow similarly to the function x/log(x) (where log refers to the natural logarithm), and this was validated at the end of the 19th century. The quest for better approximations continues, leading to the still-unsolved Riemann Hypothesis.

Within Number Theory, various subfields exist, such as analytic number theory and algebraic number theory. Instead of detailing each, I'll explain the essence of analysis and algebra.

Before proceeding, let’s acknowledge a famous quote from a mathematical luminary:

"Mathematics is the queen of the sciences, and number theory is the queen of mathematics."

— Carl Friedrich Gauss

Chapter 3: Geometry

Geometry, alongside number theory, is among the oldest mathematical disciplines, traditionally focused on shapes, sizes, distances, and angles. It finds application across the natural sciences, which is why we still teach children about triangles, circles, and lines. For instance, in Einstein's theory of relativity, gravity is viewed as a consequence of the geometry of a four-dimensional manifold known as spacetime.

Historically, geometry was one of the first mathematical fields to be axiomatized by the ancient Greeks, serving as the foundation for modern mathematics. Classically, there are three primary types of geometry. By slightly altering the axioms, we arrive at spherical geometry, where the angles in a triangle exceed 180 degrees; hyperbolic geometry, where they are less than 180 degrees; and the familiar Euclidean geometry (flat geometry) taught in schools.

Projective geometry allows parallel lines to intersect at a "point at infinity," which is crucial in diophantine equations and elliptic curve theory. Geometry is closely tied to trigonometry, which examines angle functions defined by the unit circle, often a source of challenges for students.

When mathematicians refer to geometry, they typically mean differential or algebraic geometry. Differential geometry studies local properties of shapes in multiple dimensions using smooth functions, while algebraic geometry focuses on shapes defined by solutions to multivariate polynomial equations, known as algebraic varieties, employing principles from abstract algebra.

"Geometry is the archetype of the beauty of the world."

— Johannes Kepler

Chapter 4: Algebra

Algebra represents the abstraction of numbers. In elementary algebra, we analyze arithmetic operations using symbols called variables in place of numbers. As we advance to abstract algebra, we explore generalized concepts of symmetries termed groups and the broader classification of number systems known as rings, along with other constructions.

Moreover, we study vector spaces and linear operators, termed matrices, forming the foundation of linear algebra. Algebra is often regarded as the language of mathematics, essential for solving equations and facilitating progress.

The earliest practitioners of algebra included the Egyptians, Babylonians, and Ancient Greeks, but it was Persian mathematician al-Khwarizmi's influential 9th-century work that distinguished algebra as its own field. Over centuries, Persian, Arabic, and Indian mathematicians advanced this discipline, while Europe lagged behind. In the 13th century, traders from the Arab world introduced algebra to Europe, although the church suppressed knowledge for about 300 years.

Following algebra's liberation from ecclesiastical constraints, Europe experienced a mathematical renaissance. René Descartes significantly contributed by merging geometry and algebra through the Cartesian coordinate system, linking the two fields forever.

The interplay between algebra and geometry is a fundamental concept still taught in schools today. In universities, it evolves into algebraic geometry, where we analyze geometric shapes formed by polynomial equations through abstract algebraic techniques such as rings and ideals.

"Algebra is the intellectual instrument created to clarify the quantitative aspects of the world."

— Alfred North Whitehead

Chapter 5: Analysis and Calculus

Analysis and calculus focus on functions, particularly differentiable functions and their properties. Analysis dissects functions to understand their characteristics, while calculus pertains to differentiating and integrating functions using various methodologies.

Differential equations play a crucial role in applied mathematics, serving as vital tools in physics and engineering, as nearly all physical laws can be expressed in their terms. Complex analysis, which involves functions of complex variables, presents a vastly different and richer theory compared to real analysis. Its power allows real-world problems to be resolved through complex analysis. In combining complex analysis with number theory, we delve into analytic number theory, revealing the secrets of natural numbers via holomorphic properties.

Blending analysis with geometry gives rise to differential geometry, where we study shapes with calculus principles. Analysis also encompasses measure theory, which generalizes concepts of area and volume, forming the foundation of probability theory—closely linked with integration theory.

"Calculus is the most powerful weapon of thought yet devised by the wit of man."

— Wallace B. Smith

Chapter 6: Topology

Topology is a fundamental field in mathematics where we examine "shapes," focusing not on size, angle, or smoothness as geometry does, but rather on classifying shapes based on stretching, bending, and gluing, avoiding tearing and cutting. The maps of interest here are continuous, preserving topological features rather than geometric ones.

A classic example is the humorous assertion that a coffee cup and a doughnut are topologically equivalent since one can be transformed into the other through stretching. The essential characteristic is the hole!

Topology gains strength when combined with abstract algebra tools, leading to algebraic topology. Each shape class possesses algebraic symmetries known as groups, which translate into topological features. Structure-preserving maps between groups relate to maps between shapes that maintain their topology.

Within topology, there are specialized areas, such as knot theory, where mathematicians study mathematical knots and other exotic subjects.

"Topology is precisely the discipline that allows the transition from local to global."

— Rene Thom

Chapter 7: Discrete Mathematics

Discrete mathematics encompasses several sub-disciplines, from combinatorics and graph theory to mathematical logic and axiomatic set theory, all dealing with non-continuous mathematical objects.

Combinatorics is the art of counting, utilizing techniques in probability and related fields, focusing on choices, combinations, and permutations. In graph theory, we study relationships among objects, where only connections matter, disregarding size or metrics. A familiar example is social networks, where relationships (edges) represent friendships without considering physical distances.

Discrete mathematics can merge with continuous mathematics in concrete mathematics, employing polynomials and power series, where the intersection lies in the interplay between discrete coefficients and continuous functions.

This field includes various subjects, such as groups, rings, and fields from abstract algebra and mathematical logic. Mathematical logic focuses on the language and foundations of mathematics, rigorously formulating mathematical statements and definitions of truth. Set theory, a fundamental theory, examines collections of objects, including numbers, exploring various sizes of infinity.

"The emphasis on mathematical methods appears to shift more towards combinatorics and set theory, moving away from the algorithms of differential equations that dominate mathematical physics."

— John von Neumann

Final Thoughts

I hope this article illuminates the diverse branches of mathematics and their interconnections. I would have valued such an overview when I began my mathematical journey, as it would have provided direction for my studies.

One key realization that emerges from this overview is that many concepts across different subfields are essentially the same; we merely approach them from various perspectives. Universal properties seem to exist across disciplines.

For instance, while studying vector spaces, the real focus lies on functions that preserve their structure—linear maps (matrices). In abstract algebra, we examine structure-preserving maps called homomorphisms, while for topology, we look at continuous maps. This interconnectedness extends to set theory with functions.

The study of these abstract structures falls under category theory, an abstraction of mathematics itself. Some describe mathematics as a castle built on true statements, others as a web of interconnected ideas, while some debate its classification as a science.

I contend that mathematics embodies the art of knowledge, truth, elegance, and beauty. As Stefan Banach once said:

"Mathematics is the most beautiful and most powerful creation of the human spirit."

— Stefan Banach

Thank you for reading.

The second video titled "Every Type of Math Explained in 9 Minutes" concisely breaks down various mathematical concepts, making them accessible and engaging for viewers.