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)

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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 word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.

Algebra

Quadratic formula.svg expresses the solution of the equation

Quadratic formula.svg expresses the solution of the equation

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.
A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
Linear algebra lecture at the Aalto University
Algebraic expression notation:
 1 – power (exponent)
 2 – coefficient
 3 – term
 4 – operator
 5 – constant term
 x y c – variables/constants
The graph of a polynomial function of degree 3

Algebra is one of the broad areas of mathematics.

Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.

Applied mathematics

Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.
A numerical solution to the heat equation on a pump casing model using the finite element method.
Fluid mechanics is often considered a branch of applied mathematics and mechanical engineering.
Mathematical finance is concerned with the modelling of financial markets.
The Brown University Division of Applied Mathematics is the oldest applied math program in the U.S.
Applied mathematics has substantial overlap with statistics.

Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry.

Pure mathematics studies the properties and structure of abstract objects, such as the E8 group, in group theory. This may be done without focusing on concrete applications of the concepts in the physical world.

Pure mathematics

Pure mathematics studies the properties and structure of abstract objects, such as the E8 group, in group theory. This may be done without focusing on concrete applications of the concepts in the physical world.
An illustration of the Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics.

A proof from Euclid's Elements (c. 300 BC), widely considered the most influential textbook of all time.

History of mathematics

A proof from Euclid's Elements (c. 300 BC), widely considered the most influential textbook of all time.
Geometry problem on a clay tablet belonging to a school for scribes; Susa, first half of the 2nd millennium BCE
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
The Pythagorean theorem. The Pythagoreans are generally credited with the first proof of the theorem.
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
Archimedes used the method of exhaustion to approximate the value of pi.
Apollonius of Perga made significant advances in the study of conic sections.
Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
The Hagia Sophia was designed by mathematicians Anthemius of Tralles and Isidore of Miletus.
Equipment used by an ancient Roman land surveyor (gromatici), found at the site of Aquincum, modern Budapest, Hungary
The Tsinghua Bamboo Slips, containing the world's earliest decimal multiplication table, dated 305 BC during the Warring States period
Counting rod numerals
The Nine Chapters on the Mathematical Art, one of the earliest surviving mathematical texts from China (2nd century AD).
The numerals used in the Bakhshali manuscript, dated between the 2nd century BCE and the 2nd century CE.
Explanation of the sine rule in Yuktibhāṣā
Page from The Compendious Book on Calculation by Completion and Balancing by Muhammad ibn Mūsā al-Khwārizmī (c. AD 820)
The Maya numerals for numbers 1 through 19, written in the Maya script
Nicole Oresme (1323–1382), shown in this contemporary illuminated manuscript with an armillary sphere in the foreground, was the first to offer a mathematical proof for the divergence of the harmonic series.
Portrait of Luca Pacioli, a painting traditionally attributed to Jacopo de' Barbari, 1495, (Museo di Capodimonte).
Gottfried Wilhelm Leibniz.
Leonhard Euler by Emanuel Handmann.
Carl Friedrich Gauss.
Behavior of lines with a common perpendicular in each of the three types of geometry
A map illustrating the Four Color Theorem
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star, with relativistic precession of apsides
The absolute value of the Gamma function on the complex plane.

The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past.

Onion (Allium) cells in different phases of the cell cycle. Growth in an 'organism' is carefully controlled by regulating the cell cycle.

Natural science

One of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation.

One of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation.

Onion (Allium) cells in different phases of the cell cycle. Growth in an 'organism' is carefully controlled by regulating the cell cycle.
This structural formula for molecule caffeine shows a graphical representation of how the atoms are arranged.
The orbitals of the hydrogen atom are descriptions of the probability distributions of an electron bound to a proton. Their mathematical descriptions are standard problems in quantum mechanics, an important branch of physics.
Uncrewed and crewed spacecraft missions have been used to image distant locations within the Solar System, such as this Apollo 11 view of Daedalus crater on the far side of the Moon.
The materials paradigm represented as a tetrahedron
Aristotle's view of inheritance, as a model of the transmission of patterns of movement of the body fluids from parents to child, and of Aristotelian form from the father.
Plato (left) and Aristotle in a 1509 painting by Raphael. Plato rejected inquiry into natural philosophy as against religion, while his student, Aristotle, created a body of work on the natural world that influenced generations of scholars.
Isaac Newton is widely regarded as one of the most influential scientists of all time.
The Michelson–Morley experiment was used to disprove that light propagated through a luminiferous aether. This 19th-century concept was then superseded by Albert Einstein's special theory of relativity.

As empirical sciences, natural sciences use tools from the formal sciences, such as mathematics and logic, converting information about nature into measurements which can be explained as clear statements of the "laws of nature".

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.

Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570. Billingsley erroneously attributed the original work to Euclid of Megara.

Euclid's Elements

Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570. Billingsley erroneously attributed the original work to Euclid of Megara.
A fragment of Euclid's Elements on part of the Oxyrhynchus papyri
Double-page from the Ishaq ibn Hunayn's Arabic Translation of Elementa. Iraq, 1270. Chester Beatty Library
An illumination from a manuscript based on Adelard of Bath's translation of the Elements, c. undefined 1309–1316; Adelard's is the oldest surviving translation of the Elements into Latin, done in the 12th-century work and translated from Arabic.
Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in 8:350, (2)pp. THOMAS–STANFORD, Early Editions of Euclid's Elements, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.
A page with marginalia from the first printed edition of Elements, printed by Erhard Ratdolt in 1482
The different versions of the parallel postulate result in different geometries.
An animation showing how Euclid constructed a hexagon (Book IV, Proposition 15). Every two-dimensional figure in the Elements can be constructed using only a compass and straightedge.
Codex Vaticanus 190
Propositions plotted with lines connected from Axioms on the top and other preceding propositions, labelled by book.
The Italian Jesuit Matteo Ricci (left) and the Chinese mathematician Xu Guangqi (right) published the Chinese edition of Euclid's Elements (幾何原本) in 1607.
Proof of the Pythagorean theorem in Byrne's The Elements of Euclid and published in colored version in 1847.

The Elements ( Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. undefined 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry

Geometry

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A European and an Arab practicing geometry in the 15th century
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
An illustration of Euclid's parallel postulate
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A tiling of the hyperbolic plane
Differential geometry uses tools from calculus to study problems involving curvature.
A thickening of the trefoil knot
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

Geometry is, with arithmetic, one of the oldest branches of mathematics.

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.

Integer factorization

Decomposition of a composite number into a product of smaller integers.

Decomposition of a composite number into a product of smaller integers.

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.

Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.