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 triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.

Differential geometry

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
An osculating circle

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

A point in three-dimensional Euclidean space can be located by three coordinates.

Euclidean space

Fundamental space of geometry, intended to represent physical space.

Fundamental space of geometry, intended to represent physical space.

A point in three-dimensional Euclidean space can be located by three coordinates.
Positive and negative angles on the oriented plane
3-dimensional skew coordinates

Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane (dimension two).

The regular heptagon cannot be constructed using only a straightedge and compass construction; this can be proven using the field of constructible numbers.

Field (mathematics)

The regular heptagon cannot be constructed using only a straightedge and compass construction; this can be proven using the field of constructible numbers.
The multiplication of complex numbers can be visualized geometrically by rotations and scalings.
Each bounded real set has a least upper bound.
The sum of three points P, Q, and R on an elliptic curve E (red) is zero if there is a line (blue) passing through these points.
A compact Riemann surface of genus two (two handles). The genus can be read off the field of meromorphic functions on the surface.
The fifth roots of unity form a regular pentagon.

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.

In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Linear algebra

In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Linear algebra is the branch of mathematics concerning linear equations such as:

Polynomial of degree 3: {{math|f(x) {{=}} x{{sup|3}}/4 + 3x{{sup|2}}/4 − 3x/2 − 2}} {{math|{{=}} 1/4 (x + 4)(x + 1)(x − 2)}}

Polynomial

Polynomial of degree 3: {{math|f(x) {{=}} x{{sup|3}}/4 + 3x{{sup|2}}/4 − 3x/2 − 2}} {{math|{{=}} 1/4 (x + 4)(x + 1)(x − 2)}}
Polynomial of degree 0: {{math|f(x) {{=}} 2}}
Polynomial of degree 1: {{math|f(x) {{=}} 2x + 1}}
Polynomial of degree 2: {{math|f(x) {{=}} x{{sup|2}} − x − 2}} {{math|{{=}} (x + 1)(x − 2)}}
Polynomial of degree 4: {{math|f(x) {{=}} 1/14 (x + 4)(x + 1)(x − 1)(x − 3) + 0.5}}
Polynomial of degree 5: {{math|f(x) {{=}} 1/20 (x + 4)(x + 2)(x + 1)(x − 1) (x − 3) + 2}}
Polynomial of degree 6: {{math|f(x) {{=}} 1/100 (x{{sup|6}} − 2x {{sup|5}} − 26x{{sup|4}} + 28x{{sup|3}}}} {{math|+ 145x{{sup|2}} − 26x − 80)}}
Polynomial of degree 7: {{math|f(x) {{=}} (x − 3)(x − 2)(x − 1)(x)(x + 1)(x + 2)}} {{math|(x + 3)}}

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

The Pythagorean theorem has at least 370 known proofs

Theorem

The Pythagorean theorem has at least 370 known proofs
A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
The Collatz conjecture: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers. The result is a fractal, which (in accordance with universality) resembles the Mandelbrot set.
This diagram shows the syntactic entities that can be constructed from formal languages. The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.

In mathematics, a theorem is a statement that has been proved, or can be proved.

3 + 2 = 5 with apples, a popular choice in textbooks

Addition

One of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.

One of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.

3 + 2 = 5 with apples, a popular choice in textbooks
The plus sign
Redrawn illustration from The Art of Nombryng, one of the first English arithmetic texts, in the 15th century.
A number-line visualization of the algebraic addition 2 + 4 = 6. A translation by 2 followed by a translation by 4 is the same as a translation by 6.
A number-line visualization of the unary addition 2 + 4 = 6. A translation by 4 is equivalent to four translations by 1.
4 + 2 = 2 + 4 with blocks
2 + (1 + 3) = (2 + 1) + 3 with segmented rods
5 + 0 = 5 with bags of dots
Addition with an op-amp. See Summing amplifier for details.
Part of Charles Babbage's Difference Engine including the addition and carry mechanisms
"Full adder" logic circuit that adds two binary digits, A and B, along with a carry input Cin, producing the sum bit, S, and a carry output, Cout.
Adding π2/6 and e using Dedekind cuts of rationals.
Addition of two complex numbers can be done geometrically by constructing a parallelogram.
A circular slide rule
Log-log plot of x + 1 and max (x, 1) from x = 0.001 to 1000

Addition belongs to arithmetic, a branch of mathematics.

Flowchart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" or "true" (more accurately, the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).

Algorithm

Flowchart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" or "true" (more accurately, the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).
Ada Lovelace's diagram from "note G", the first published computer algorithm
Logical NAND algorithm implemented electronically in 7400 chip
Flowchart examples of the canonical Böhm-Jacopini structures: the SEQUENCE (rectangles descending the page), the WHILE-DO and the IF-THEN-ELSE. The three structures are made of the primitive conditional GOTO (IF test THEN GOTO step xxx, shown as diamond), the unconditional GOTO (rectangle), various assignment operators (rectangle), and HALT (rectangle). Nesting of these structures inside assignment-blocks result in complex diagrams (cf. Tausworthe 1977:100, 114).
The example-diagram of Euclid's algorithm from T.L. Heath (1908), with more detail added. Euclid does not go beyond a third measuring and gives no numerical examples. Nicomachus gives the example of 49 and 21: "I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 (for this is possible); 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 (for this is possible); 7 is left, but 7 cannot be subtracted from 7." Heath comments that "The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending 'at one and the same number'."(Heath 1908:300).
A graphical expression of Euclid's algorithm to find the greatest common divisor for 1599 and 650.
"Inelegant" is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division (or a "modulus" instruction). Derived from Knuth 1973:2–4. Depending on the two numbers "Inelegant" may compute the g.c.d. in fewer steps than "Elegant".
Alan Turing's statue at Bletchley Park

In mathematics and computer science, an algorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation.

Russell in 1957

Bertrand Russell

Welsh philosopher, logician, and social critic.

Welsh philosopher, logician, and social critic.

Russell in 1957
Russell as a 4-year-old
Childhood home, Pembroke Lodge, Richmond Park, London
Russell at Trinity College in 1893
Russell with his children, John and Kate
Russell in 1954
Russell (centre) alongside his wife Edith, leading a CND anti-nuclear march in London, 18 February 1961
Plas Penrhyn in Penrhyndeudraeth
Russell on a 1972 stamp of India
Bust of Russell in Red Lion Square

As an academic, he worked in philosophy, mathematics, and logic.

Charles Babbage, sometimes referred to as the "father of computing".

Computer science

Study of computation, automation, and information.

Study of computation, automation, and information.

Charles Babbage, sometimes referred to as the "father of computing".
Ada Lovelace published the first algorithm intended for processing on a computer.

Computer science research also often intersects other disciplines, such as cognitive science, linguistics, mathematics, physics, biology, Earth science, statistics, philosophy, and logic.