Mathematics

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

171 related topics

Alpha

A complex number z, as a point (black) and its position vector (blue)

Complex number

[[File:A plus bi.svg|thumb|upright=1.15|right|A complex number can be visually represented as a pair of numbers

[[File:A plus bi.svg|thumb|upright=1.15|right|A complex number can be visually represented as a pair of numbers

A complex number z, as a point (black) and its position vector (blue)
Argument φ and modulus r locate a point in the complex plane.
Geometric representation of z and its conjugate in the complex plane
Addition of two complex numbers can be done geometrically by constructing a parallelogram.
The Mandelbrot set with the real and imaginary axes labeled.
Construction of a regular pentagon using straightedge and compass.
Cayley Q8 quaternion graph showing cycles of multiplication by, and

In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted i, called the imaginary unit, and satisfying the equation

Chronology of the universe as deduced by the prevailing Big Bang theory, a result from science and obtained knowledge

Science

Systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

Systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

Chronology of the universe as deduced by the prevailing Big Bang theory, a result from science and obtained knowledge
The first diagram of an evolutionary tree made by Charles Darwin in 1837
First global view of the ozone hole in 1983, using a space telescope
Radio light image of M87* black hole, made by the earth-spanning Event Horizon Telescope array in 2019
Supply and demand curve in economics, crossing over at the optimal equilibrium
A steam turbine with the case opened, such turbines produce most of the electricity used today
A diagram variant of scientific method represented as an ongoing process
Cover of the first issue of Nature, 4 November 1869
For Kuhn, the addition of epicycles in Ptolemaic astronomy was "normal science" within a paradigm, whereas the Copernican revolution was a paradigm shift.
Marie Curie was the first person to be awarded two Nobel Prizes: Physics in 1903 and Chemistry in 1911.
Picture of scientists in 200th anniversary of the Prussian Academy of Sciences, 1900
Medal of the Nobel Prize, one of the most well-known science awards
Budget of NASA as percentage of United States federal budget, peaking at 4.4% in 1966 and slowly decline since
Dinosaur exhibit in the Houston Museum of Natural Science
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Their contributions to mathematics, astronomy, and medicine entered and shaped Greek natural philosophy of classical antiquity, whereby formal attempts were made to provide explanations of events in the physical world based on natural causes.

The Pythagorean theorem has at least 370 known proofs

Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories.

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories.

The Pythagorean theorem has at least 370 known proofs

The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

The School of Athens (1509–1511) by Raphael, depicting famous classical Greek philosophers in an idealized setting inspired by ancient Greek architecture.

Philosophy

Systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language.

Systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language.

The School of Athens (1509–1511) by Raphael, depicting famous classical Greek philosophers in an idealized setting inspired by ancient Greek architecture.
The Vinegar Tasters (Japan, Edo period, 1802-1816) by Kanō Isen'in, depicting the three main philosophical figures in East Asian thought: Buddha, Confucius and Laozi.
Statue of Aristotle (384–322 BCE), a major figure of ancient Greek philosophy, in Aristotle's Park, Stagira.
A painting of the influential modern philosopher Immanuel Kant (in the blue coat) with his friends. Other figures include Christian Jakob Kraus, Johann Georg Hamann, Theodor Gottlieb von Hippel and Karl Gottfried Hagen.
A page of The Maxims of Ptahhotep, traditionally attributed to the Vizier Ptahhotep (c. 2375–2350 BCE).
An Iranian portrait of Avicenna on a Silver Vase. He was one of the most influential philosophers of the Islamic Golden Age.
Adi Shankara is one of the most frequently studied Hindu philosophers.
The parable of the blind men and the elephant illustrates the important Jain doctrine of anēkāntavāda.
Statue of the Neo-Confucian scholar Zhu Xi at the White Deer Grotto Academy in Lushan Mountain.
Kitaro Nishida, considered the founder of the Kyoto School of philosophical thought, c. 1943.
Painting of Zera Yacob from Claude Sumner's Classical Ethiopian Philosophy.
A Tlamatini (Aztec philosopher) observing the stars, from the Codex Mendoza.
Depiction of Pachacuti worshipping Inti (god Sun) at Coricancha, in the 17th century second chronicles of Martín de Murúa. Pachacuti was a major Incan ruler, author and poet.
Mary Wollstonecraft (1759-1797) was an English writer and philosopher.
The Beijing imperial college was an intellectual center for Confucian ethics and classics during the Yuan, Ming and Qing dynasties.
Dignaga founded a school of Buddhist epistemology and logic.
The beginning of Aristotle's Metaphysics in an incunabulum decorated with hand-painted miniatures.
Thomas Hobbes, best known for his Leviathan, which expounded an influential formulation of social contract theory.
Martin Luther King Jr.

Before the modern age, the term was used in a very wide sense, which included the individual sciences, like physics or mathematics, as its sub-disciplines, but the contemporary usage is more narrow.

Detail from Raphael's The School of Athens presumed to represent Donato Bramante as Euclid

Euclid

Not to be confused with Euclid of Megara.

Not to be confused with Euclid of Megara.

Detail from Raphael's The School of Athens presumed to represent Donato Bramante as Euclid
Euclidis quae supersunt omnia (1704)
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid's construction of a regular dodecahedron.
Construction of a dodecahedron by placing faces on the edges of a cube.
19th-century statue of Euclid by Joseph Durham in the Oxford University Museum of Natural History

His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.

Portrait by Jakob Emanuel Handmann (1753)

Leonhard Euler

Portrait by Jakob Emanuel Handmann (1753)
1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, in English written as "v − e + f = 2".
Euler's grave at the Alexander Nevsky Monastery
Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.
An Euler diagram
Euler portrait on the sixth series of the 10 Franc banknote
Euler portrait on the seventh series of the 10 Franc banknote
Illustration from Solutio problematis... a. 1743 propositi published in Acta Eruditorum, 1744
The title page of Euler's Methodus inveniendi lineas curvas.
Euler's 1760 world map.
Euler's 1753 map of Africa.

Leonhard Euler (, ; 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus.

Subsets of the complex numbers

Number

Mathematical object used to count, measure, and label.

Mathematical object used to count, measure, and label.

Subsets of the complex numbers
The number 605 in Khmer numerals, from an inscription from 683 AD. Early use of zero as a decimal figure.
The natural numbers, starting with 1

In mathematics, the notion of a number has been extended over the centuries to include 0, negative numbers, rational numbers such as one half and pi, and complex numbers which extend the real numbers with a [[imaginary unit|square root of

Various examples of physical phenomena

Physics

Natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.

Natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.

Various examples of physical phenomena
Ancient Egyptian astronomy is evident in monuments like the ceiling of Senemut's tomb from the Eighteenth Dynasty of Egypt.
Ibn al-Haytham (c. 965–c. 1040), Book of Optics Book I, [6.85], [6.86]. Book II, [3.80] describes his camera obscura experiments.
The basic way a pinhole camera works
Galileo Galilei showed a modern appreciation for the proper relationship between mathematics, theoretical physics, and experimental physics.
Sir Isaac Newton (1643–1727), whose laws of motion and universal gravitation were major milestones in classical physics
Max Planck (1858–1947), the originator of the theory of quantum mechanics
Albert Einstein (1879–1955), whose work on the photoelectric effect and the theory of relativity led to a revolution in 20th century physics
The basic domains of physics
Solvay Conference of 1927, with prominent physicists such as Albert Einstein, Werner Heisenberg, Max Planck, Hendrik Lorentz, Niels Bohr, Marie Curie, Erwin Schrödinger and Paul Dirac
This parabola-shaped lava flow illustrates the application of mathematics in physics—in this case, Galileo's law of falling bodies.
Mathematics and ontology are used in physics. Physics is used in chemistry and cosmology.
The distinction between mathematics and physics is clear-cut, but not always obvious, especially in mathematical physics.
Classical physics implemented in an acoustic engineering model of sound reflecting from an acoustic diffuser
Archimedes' screw, a simple machine for lifting
Experiment using a laser
The astronaut and Earth are both in free fall.
Lightning is an electric current.
Physics involves modeling the natural world with theory, usually quantitative. Here, the path of a particle is modeled with the mathematics of calculus to explain its behavior: the purview of the branch of physics known as mechanics.
A simulated event in the CMS detector of the Large Hadron Collider, featuring a possible appearance of the Higgs boson.
Velocity-distribution data of a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate
The deepest visible-light image of the universe, the Hubble Ultra-Deep Field
Feynman diagram signed by R. P. Feynman.
A typical phenomenon described by physics: a magnet levitating above a superconductor demonstrates the Meissner effect.

Over much of the past two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the Scientific Revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right.

Figure 2. Venn diagrams for conjunction, disjunction, and complement

Boolean algebra

Figure 2. Venn diagrams for conjunction, disjunction, and complement
From left to right: AND, OR, and NOT gates.
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In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively.

An example of change ringing (with six bells), one of the earliest nontrivial results in graph theory.

Combinatorics

An example of change ringing (with six bells), one of the earliest nontrivial results in graph theory.
Five binary trees on three vertices, an example of Catalan numbers.
A plane partition.
Petersen graph.
Hasse diagram of the powerset of {x,y,z} ordered by inclusion.
Self-avoiding walk in a square grid graph.
Young diagram of a partition (5,4,1).
Construction of a Thue–Morse infinite word.
An icosahedron.
Splitting a necklace with two cuts.
Kissing spheres are connected to both coding theory and discrete geometry.

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.