Mathematics #
Introduction #
Mathematics is a field of knowledge that encompasses the study of numbers, formulas, structures, shapes, spaces, and changes in quantities. It includes major sub-disciplines such as number theory, algebra, geometry, and analysis. The essence of mathematics lies in the discovery of properties of abstract objects and the use of pure reason to prove them. These objects can be abstractions from nature or entities defined by certain properties, known as axioms. Mathematics plays a crucial role in various fields including natural sciences, engineering, medicine, finance, computer science, and social sciences. While some areas of mathematics are closely tied to their applications, others are developed independently, often finding practical applications later. The truths of mathematics are independent of scientific experimentation, making it a unique blend of abstract reasoning and practical application.
Mathematics is not only a tool for solving practical problems but also a language that describes the universe. It provides a framework for understanding patterns, relationships, and structures, enabling us to make sense of the world around us. From the symmetry of a snowflake to the complexity of a computer algorithm, mathematics reveals the underlying order and beauty of the natural and man-made worlds. Its principles are universal, transcending cultural and linguistic boundaries, and its applications are limitless, driving innovation and progress in countless domains.
Documents #
- Continuous-time Markov Chain
- A Continuous-time Markov Chain (CTMC) is a stochastic model used to represent systems that transition between states continuously over time, where the probability of transitioning to any particular state depends solely on the current state and the amount of time spent in that state, not on the sequence of past states.
- Lumpability
- Lumpability refers to the property of a Markov chain where a coarse-grained version of the chain retains the Markov property. In simpler terms, it means that when you group states of a Markov chain into larger “lumps,” the resulting chain still behaves like a Markov chain.
- Markov Chain
- A Markov chain is a statistical model that undergoes transitions from one state to another within a finite or countable number of possible states, where the probability of transitioning to any particular state depends solely on the current state and not on the sequence of states that preceded it.
- Monte Carlo Method
- The Monte Carlo method, a statistical technique using randomness, is used to estimate mathematical constants like Pi and Euler’s number by generating random numbers and applying specific mathematical operations. For example, Pi is estimated by generating random points within a unit square and checking how many fall inside a unit circle, while Euler’s number is estimated by generating random numbers, summing them until they exceed 1, and averaging the counts of numbers needed. The accuracy of these estimates increases with the number of iterations. This method is implemented in the Rust programming language using the nanorand crate for random number generation.
- Norm
- A norm is a mathematical function that assigns a positive length or size to vectors and matrices, commonly used in machine learning to measure model error.
- Spherical Linear Interpolation
- A method used in computer graphics to interpolate between two points on a sphere. It’s particularly useful for smooth transitions.
- Frusta
- A frustum refers to the portion of a solid—usually a cone or a pyramid—that remains after cutting off the top by a plane parallel to the base.
- Mandelbrot Set
- A fascinating concept in the field of complex dynamics. It includes all complex numbers, $c$, for which this sequence does not diverge to infinity.
History #
Ancient Mathematics #
The earliest evidence of mathematical knowledge dates back to the ancient Sumerians, who developed a complex system of metrology around 4000 to 3500 BC. From these beginnings, mathematics evolved and spread to ancient Egypt, where complex mathematical systems were used in the construction of pyramids, astronomy, and accounting. The Rhind Papyrus, dating back to around 1650 BC, demonstrates the Egyptians’ knowledge of fractions, algebra, and geometry.
In ancient Mesopotamia, the Babylonians developed a sophisticated sexagesimal (base-60) numeral system, which is still reflected in our modern measurements of time and angles. They also made significant advances in algebra, developing methods for solving quadratic equations. The Babylonians were able to solve problems involving the Pythagorean theorem long before Pythagoras was born.
In ancient China, the “Nine Chapters on the Mathematical Art” compiled by the 2nd century BC became a standard mathematical text for centuries. It included methods for finding areas of figures, solutions to systems of linear equations, and the use of negative numbers.
In ancient Greece, mathematicians like Pythagoras, Euclid, and Archimedes made significant contributions. Pythagoras is often credited with the famous Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:
$$c^2 = a^2 + b^2$$
where $c$ represents the length of the hypotenuse, and $a$ and $b$ represent the lengths of the other two sides.
Euclid’s “Elements”, a collection of definitions, postulates, propositions, and mathematical proofs, is considered one of the most influential works in the history of mathematics, shaping the field of geometry for centuries to come. The work includes the famous parallel postulate, which can be stated as:
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
Archimedes, another Greek mathematician, made significant contributions to calculus and is famous for his approximation of pi (π). He showed that π is between 3 10/71 and 3 1/7 by inscribing and circumscribing a circle with 96-sided regular polygons. In modern notation, this can be expressed as:
$$3.140845 < \pi < 3.142857$$
Archimedes also derived the formula for the volume of a sphere:
$$V = \frac{4}{3}\pi r^3$$
where $r$ is the radius of the sphere.
Medieval Mathematics #
During the Middle Ages, significant mathematical development took place in the Islamic world. Persian and Arab mathematicians such as Al-Khwarizmi, Omar Khayyam and Al-Kindi made significant contributions to mathematics, algebra, and geometry.
Al-Khwarizmi’s work on algebra was groundbreaking, and he is often considered the “father of algebra”. He introduced the concept of solving quadratic equations, which can be represented in the general form:
$$ax^2 + bx + c = 0$$
where $a$, $b$, and $c$ are constants and $x$ is the variable to be solved for.
Omar Khayyam extended this work, developing methods for solving cubic equations geometrically. He also made significant contributions to the parallel postulate theory. Khayyam’s solution to the cubic equation can be represented as:
$$x^3 + a^2x = b$$
Al-Kindi made important contributions to cryptography, developing techniques for frequency analysis that were unrivaled for centuries.
In India, mathematicians like Aryabhata, Brahmagupta, and Bhaskara II made important advances. Aryabhata provided methods for solving linear and quadratic equations, as well as for finding square and cube roots. Brahmagupta, in his work “Brahmasphutasiddhanta”, provided rules for computing with zero and negative numbers. He also gave the general solution of the linear Diophantine equation:
$$ax + by = c$$
where $a$, $b$, and $c$ are integers.
Bhaskara II’s most famous work, “Lilavati”, contained many problems dealing with arithmetic, algebra, and geometry. He is credited with discovering the general solution of the Pell equation:
$$x^2 - ny^2 = 1$$
where $n$ is a given positive integer and $x$ and $y$ are the variables to be solved for.
Renaissance and Early Modern Mathematics #
The Renaissance saw a significant rebirth of mathematical research and understanding, with pioneers like Galileo Galilei and Isaac Newton leading the way. Galileo’s work on kinematics laid the foundation for Newton’s laws of motion. He derived the equation for the distance traveled by a uniformly accelerating object:
$$d = \frac{1}{2}at^2$$
where $d$ is the distance, $a$ is the acceleration, and $t$ is the time.
Newton and Leibniz’s development of calculus opened up new applications for mathematical principles, particularly in the realm of physics. The fundamental theorem of calculus, which links the concept of integrating a function with the concept of differentiating a function, can be stated as:
$$\int_a^b f’(x) dx = f(b) - f(a)$$
where $f’(x)$ is the derivative of $f(x)$, and the left side represents the definite integral of $f’(x)$ from $a$ to $b$.
Newton’s method for finding roots of a function, a powerful technique in numerical analysis, can be expressed as:
$$x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)}$$
René Descartes introduced the Cartesian coordinate system, bridging algebra and geometry. This led to the development of analytic geometry, expressed by the general equation of a straight line:
$$y = mx + b$$
where $m$ is the slope and $b$ is the y-intercept.
Blaise Pascal and Pierre de Fermat laid the foundations of probability theory. The binomial probability formula they developed is:
$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$
where $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success on each trial.
Leonhard Euler, one of the most prolific mathematicians in history, made substantial contributions to various fields of mathematics. He introduced many modern mathematical terms and notations, including the concept of a function. His famous identity, connecting exponential and trigonometric functions, is expressed as:
$$e^{i\pi} + 1 = 0$$
Euler also discovered the formula for polyhedra:
$$V - E + F = 2$$
where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces of a convex polyhedron.
Modern Mathematics #
In the 19th and 20th centuries, mathematics saw rapid expansion and abstraction. Georg Cantor developed set theory, introducing the concept of different sizes of infinity. His famous diagonal argument showed that the set of real numbers is uncountable, which can be expressed using set notation as:
$$|\mathbb{R}| > |\mathbb{N}|$$
where $\mathbb{R}$ represents the set of real numbers and $\mathbb{N}$ represents the set of natural numbers.
Cantor also introduced the concept of cardinal numbers to compare the sizes of infinite sets. For any set $A$, its power set (the set of all subsets of $A$) has a strictly greater cardinality:
$$|P(A)| > |A|$$
This leads to an infinite hierarchy of infinities.
David Hilbert posed 23 problems at the International Congress of Mathematicians in 1900, which significantly influenced 20th-century mathematics. One of these, the continuum hypothesis, can be stated using the cardinal number notation:
$$2^{\aleph_0} = \aleph_1$$
where $\aleph_0$ is the cardinality of the natural numbers and $\aleph_1$ is the next larger cardinal number.
In the field of topology, which studies properties of space that are preserved under continuous deformations, the famous Poincaré conjecture, proved in 2003 by Grigori Perelman, states that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. This can be expressed in mathematical notation as:
If $M$ is a 3-dimensional manifold such that every simple closed curve in $M$ can be continuously deformed to a point, then $M$ is homeomorphic to the 3-sphere.
Emmy Noether made fundamental contributions to abstract algebra and theoretical physics. Her famous theorem relating symmetries to conservation laws can be stated as:
For every continuous symmetry of the laws of physics, there must exist a conserved quantity.
John von Neumann’s work on game theory introduced the minimax theorem, which can be expressed as:
$$\max_x \min_y f(x,y) = \min_y \max_x f(x,y)$$
where $f(x,y)$ is a function representing the payoff for strategies $x$ and $y$.
Current Day #
Computational mathematics has emerged as a crucial field, with the advent of powerful computers enabling complex calculations and simulations. The Riemann Hypothesis, one of the most famous unsolved problems in mathematics, concerns the distribution of prime numbers and can be stated in terms of the Riemann zeta function:
$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$
The hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2.
Recent developments include the proof of Fermat’s Last Theorem by Andrew Wiles in 1995, which states that no three positive integers $a$, $b$, and $c$ can satisfy the equation:
$$a^n + b^n = c^n$$
for any integer value of $n$ greater than 2.
The classification of finite simple groups, a major achievement in group theory, was completed in 2004. This monumental work spans thousands of pages and classifies all finite simple groups into 18 infinite families and 26 sporadic groups.
In the field of number theory, the Green-Tao theorem, proved in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. This can be expressed as:
For every natural number $k$, there exist arithmetic progressions of primes with $k$ terms.
The Poincaré conjecture, one of the Millennium Prize Problems, was proved by Grigori Perelman in 2003. The conjecture can be stated as:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
In 2009, Mikhail Gromov was awarded the Abel Prize for his revolutionary contributions to geometry. His concept of hyperbolic groups has had a profound impact on geometric group theory.
The development of quantum algorithms, such as Shor’s algorithm for integer factorization, has opened up new areas of research at the intersection of mathematics and quantum computing. Shor’s algorithm can factor an integer $N$ in time $O((\log N)^3)$, exponentially faster than the best known classical algorithms.
In 2018, Sir Michael Atiyah announced a proposed proof of the Riemann hypothesis, although this proof has not been verified by the mathematical community.
Mathematics continues to play a vital role in advancing our understanding of the universe, from the smallest subatomic particles to the largest cosmic structures. Its applications in fields such as artificial intelligence, cryptography, and quantum computing are shaping the future of technology and society. The interplay between pure and applied mathematics remains a driving force in mathematical research and innovation.