MATHEMATICS

Definition:

Mathematics is the science and study of quality, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

Areas of mathematics:

At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 mathematics subject classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic and geometry. Several other first-level areas have “geometry” in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations.

Number theory

This is the ulam spiral which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and little wood’s conjecture.

Number theory began with the manipulation of numbers that is, natural numbers and later expanded to integers and rational numbers Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations. Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to pierre de fermat and Leonhard Euler .

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat’s last theorem This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew wiles, who used tools including scheme theory from algebraic geometry category theory and homological algebra Another example is Goldbach’s conjecture . which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach it remains unproven despite considerable effort.

Geometry

On the surface of a sphere, Euclidian geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.

Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as line’s angles and circle’s, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.

A fundamental innovation was the ancient Greeks’ introduction of the concept of proof’s which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or are part of the definition of the subject of study (axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book elements

Algebra

Algebra became an area in its own right only with Francois viete (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas

Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations(a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy noether (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)

Calculus and analysis:

A cauchy sequence consists of elements that become arbitrarily close to each other as the sequence progresses (from left to right).

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, “calculus” refers mainly to the elementary part of this theory, and “analysis” is commonly used for advanced parts.

Analysis is further subdivided into real analysis where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:

Multivariable calculus
Functional analysis, where variables represent varying functions;
Integration , measure theory and potential theory, all strongly related with probability theory ;
Ordinary differential equations
Partial differential equations

Discrete mathematics:

A diagram representing a two-state Markov chain. The states are represented by ‘A’ and ‘E’. The numbers are the probability of flipping the state.
Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms especially their implementation and computational complexity play a major role in discrete mathematics.^[
The four colour theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P verses NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.

Statistics and other decision sciences

Whatever the form of a random population distribution (μ), the sampling mean (x̄) tends to a Gaussian distribution and its variance (σ) is given by the central limit theorem of probability theory. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments. The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation.

Sir C.V. Ramanujan:

Srinivasa Ramanujan, (born December 22, 1887, Erode , India—died April 26, 1920, Kumbakonam), Indian mathematician whose contributions to the theory of numbers include pioneering discoveries of the properties of the partition function.

When he was 15 years old, he obtained a copy of George Shoobridge Carr’s Synopsis of Elementary Results in Pure and Applied Mathematics, 2 vol. (1880–86). This collection of thousands of theorems, many presented with only the briefest of proofs and with no material newer than 1860, aroused his genius. Having verified the results in Carr’s book, Ramanujan went beyond it, developing his own theorems and ideas. In 1903 he secured a scholarship to the University Of Madras but lost it the following year because he neglected all other studies in pursuit of mathematics.

Ramanujan continued his work, without employment and living in the poorest circumstances. After marrying in 1909 he began a search for permanent employment that culminated in an interview with a government official, Ramachandra Rao. Impressed by Ramanujan’s mathematical prowess, Rao supported his research for a time, but Ramanujan, unwilling to exist on charity, obtained a clerical post with the Madras Port Trust

In England Ramanujan made further advances, especially in the partition of numbers (the number of ways that a positive integer can be expressed as the sum of positive integers; e.g., 4 can be expressed as 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1). His papers were published in English and European journals, and in 1918 he was elected to the Royal society of London . In 1917 Ramanujan had contracted tuberculosis but his condition improved sufficiently for him to return to India in 1919. He died the following year, generally unknown to the world at large but recognized by mathematicians as a phenomenal genius, without peer since Leon Hard Euler (1707–83) and Carl Jacobi (1804–51). Ramanujan left behind three notebooks and a sheaf of pages (also called the “lost notebook”) containing many unpublished results that mathematicians continued to verify long after his death.