Supplementary mathematics/Mathematics

Mathematics is the art of calculating numbers and also studies topics such as quantity (number theory), structure (algebra), space (geometry) and variation (mathematical analysis). In fact, there is no universal definition of mathematics that everyone agrees on.

Most mathematical activities involve discovering and proving the properties of abstract objects by pure reasoning. These objects are either abstractions of nature, such as natural numbers or lines, or – in modern mathematics – entities that have certain properties called axioms. An argument consists of a set of applications of some deductive rules to already known results, including previously proven theorems, axioms, and (if abstracted from nature) some basic properties that serve as the actual starting point of the theory under consideration. is considered are taken, the result of an argument is called a theorem.

Mathematics is widely used in science to model phenomena. This allows the derivation of quantitative predictions from empirical laws. For example, the motion of the planets can be accurately predicted using Newton's law of gravitation combined with mathematical calculations. The independence of mathematical truth from any experiment shows that the accuracy of such predictions depends only on the adequacy of the model to describe reality. Incorrect predictions indicate that the mathematical models need to be improved or changed, not that the mathematics in the models themselves are wrong. For example, the precession of Mercury's perihelion cannot be explained by Newton's law of gravitation, but it is precisely explained by Newton's law of gravitation. Einstein's General Relativity - This experimental confirmation of Einstein's theory shows that Newton's law of gravitation is only an approximation, although it is accurate in everyday applications.

Mathematics is essential in many fields including natural sciences, engineering, medicine, finance, computer science and social sciences. Some areas of mathematics, such as statistics and game theory, have been developed in close connection with their applications and are often grouped under applied mathematics. Other areas of mathematics develop independently of any application (and are therefore called pure mathematics), but practical applications are often discovered later. A good example is the problem of factoring integers, which dates back to the time of Euclid, but had no practical application before being used in the RSA cryptosystem (for computer network security).

Historically, the concept of proof and the mathematical precision associated with it first appeared in Greek mathematics, particularly in Euclid's Elements. From the beginning, mathematics was basically divided into geometry and calculus (manipulation of numbers and natural fractions) until in the 16th and 17th centuries, algebra and infinitesimal calculus were introduced as new disciplines. Since then, the interaction between mathematical innovations and scientific discoveries has led to the rapid growth of mathematics. At the end of the 19th century, the fundamental crisis of mathematics led to the systematization of the axiomatic method. This in turn caused a significant increase in the number of mathematical disciplines and their applied fields. An example of this classification is mathematics courses, which lists more than sixty areas of first level mathematics.

History edit

The history of mathematics can be seen as a sequence of increasing abstractions. The first abstraction capability shared by many animals is probably the concept of number: the understanding that a set of two apples and a set of two oranges (for example) have in common, and that quantity is their number.

Prehistoric people could count both physical objects and abstract objects such as days, seasons, and years, as evidenced by woodcuts.

Evidence for more complex mathematics is not seen until 3000 BCE, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for calculations related to taxation and other economic concepts, and construction or astronomy. The oldest mathematical texts are from Mesopotamia and Egypt, dating back to 2000-1800 BC.Many early texts mention Pythagorean triads, so it seems that the Pythagorean theorem is one of the most important methods for inventing trigonometry.

This theorem has been proven many times by different geometric and algebraic methods, some of these proofs go back thousands of years. It is the oldest and most extensive mathematical development after elementary arithmetic and geometry. In historical documents, it was in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appeared. The Babylonians also used a place-value device that implemented a base 60 number device, which is still used to measure angles and time.

With the beginning of the 6th century BC, the Greek mathematicians with the Pythagoreans began a systematic study of mathematics, with the aim of knowing more about mathematics itself, which was the beginning of Greek mathematics. Around 300 BC, Euclid introduced the method of thematic principles still used in mathematics, which included definitions, principles, theorems, and proofs. His reference book known as Euclid's Principles is widely regarded as the most successful and influential reference book of all time. The greatest ancient mathematicians are often considered to be Archimedes (287-212 BC) from Syracuse. He found formulas for calculating the area and volume of rotating objects and used Afna's method to calculate the area under a parabolic curve using the sum of an infinite series. In a way that is not dissimilar to modern calculus. Other notable achievements in Greek mathematics were conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hiparchus of Nicaea (2nd century BC)), and the beginning of algebra (Diophantus, 3rd century AD).

The Indo-Arabic number system and the rules for using its operations, which are used all over the world today, were developed in India during the first millennium AD and then transferred to the Western world through Islamic mathematics. Other developments related to Indian mathematics include the modern definition of sine and cosine and the early form of infinite series.

During the Golden Age of Islam, which took place in the 9th and 10th centuries AD, mathematics saw important innovations that were based on the mathematics of the Greeks. The most important achievements of Islamic mathematics was the development of algebra. Other important achievements of mathematics in the Islamic period were the progress in spherical trigonometry and the addition of decimals to the Arabic numerical system. Many mathematicians of this period were Persian-speaking, such as Khwarazmi, Khayyam and Sharafuddin Tosi.

During the early modern era, mathematics began to develop rapidly in Western Europe. The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonard Euler was the most important mathematician of the 18th century who added several theorems and discoveries to mathematics. Perhaps the most important mathematicians of the 19th century was the German mathematician Carl Friedrich Gauss, who provided many services to various branches of mathematics such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel revolutionized mathematics by publishing his incompleteness theorems. These theorems showed that any system of principles of compatibility includes unprovable propositions.

Since then, mathematics has been widely developed, and fruitful interactions between mathematics and science have been created that benefit both. Mathematical discoveries continue to this day. According to Mikhail Suriuk, published in the January 2006 Bulletin of the American Mathematical Society, "the number of articles and books in the Mathematical Review database since 1940 (MR's first year of operation) now stands at 1.9 million, an annual increase of over 75 Thousands of items will be added to this database.The vast majority of works in this ocean contain new mathematical theorems and their proofs.

Areas of mathematics edit

Before the Renaissance, mathematics was divided into two main areas: arithmeticregarding the manipulation of numbers, and geometry, regarding the study of shapes. Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.

During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areasarithmetic, geometry, algebra, calculusendured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.

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 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 edit

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 Littlewood's Conjecture F.

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. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.

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.

Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).

Geometry edit

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 lines, angles and circles, 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 proofs, 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.

The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the three-dimensional Euclidean space.

Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This constituted a major change of paradigm: Instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.

Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.

In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space.

Today's subareas of geometry include:

  • Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
  • Affine geometry, the study of properties relative to parallelism and independent from the concept of length.
  • Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions.
  • Manifold theory, the study of shapes that are not necessarily embedded in a larger space.
  • Riemannian geometry, the study of distance properties in curved spaces.
  • Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials.
  • Topology, the study of properties that are kept under continuous deformations.
    • Algebraic topology, the use in topology of algebraic methods, mainly homological algebra.
  • Discrete geometry, the study of finite configurations in geometry.
  • Convex geometry, the study of convex sets, which takes its importance from its applications in optimization.
  • Complex geometry, the geometry obtained by replacing real numbers with complex numbers.

Algebra edit

The Rubik's Cube group is a concrete application of group theory

Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise.

Algebra became an area in its own right only with François Viète (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 ter m 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.)

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:

  • group theory;
  • field theory;
  • vector spaces, whose study is essentially the same as linear algebra;
  • ring theory;
  • commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry;
  • homological algebra;
  • Lie algebra and Lie group theory;
  • Boolean algebra, which is widely used for the study of the logical structure of computers.

The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory. The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.

Calculus and analysis edit

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 on a continuum;
  • Ordinary differential equations;
  • Partial differential equations;
  • Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.

Discrete mathematics edit

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. Algorithmsespecially their implementation and computational complexity play a major role in discrete mathematics.

The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus 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.

Discrete mathematics includes:

  • Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements

or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes

  • Graph theory and hypergraphs
  • Coding theory, including error correcting codes and a part of cryptography
  • Matroid theory
  • Discrete geometry
  • Discrete probability distributions
  • Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
  • Discrete optimization, including combinatorial optimization, integer programming, constraint programming

Mathematical logic and set theory edit

The Venn diagram is a commonly used method to illustrate the relations between sets.

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.

Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory.

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.

This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.

The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinionsometimes called "intuition"to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.

These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.

Statistics and other decision sciences edit

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 infer ence, using model selection and estimation. The models and consequential predictions should then be tested against new data.

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.

Computational mathematics edit

Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.