Physical World/Mathematics

Geometry

Surface area of a sphere – 4 π r²

Volume of a sphere – 4/3 π r³

Surface area of a cylinder (with top and bottom) – 2 π r (r + h)

Volume of cylinder – π r² h

Surface area of a cone – π r l + π r² (where ‘l’ is length of side of cone)

Volume of a cone – 1/3 π r² h

Frustrum – cone formed by the cutting of a plane parallel to the base

A polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit

A polygon has a minimum of three sides

Regular polygon – triangle, square, pentagon, hexagon, octagon etc.

Sum of exterior angles of a polygon is 360^o

Area of a triangle = (base x height) / 2

The circumcentre of a triangle can be found as the intersection of the three perpendicular bisectors

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Each median of a triangle passes through the triangle's centroid

Internal angles of a pentagon add up to 540 degrees. Each interior angle is 108 degrees

Internal angles of a hexagon add up to 720 degrees. Each interior angle is 120 degrees

Heptagon – 7-sided polygon

Internal angles of an octagon add up to 1080 degrees. Each interior angle is 135 degrees

Nonagon – 9-sided polygon

Hendecagon – 11-sided polygon

Heptadecagon – 17-sided polygon

Enneadecagon – 19-sided polygon

Hectogon – 100-sided polygon

Chiliagon – 1000-sided polygon

Enneagram is a nine-pointed geometric figure. It is sometimes called a nonagram

In North America, the term trapezium is used to refer to a quadrilateral with no parallel sides. A four-sided figure with one pair of parallel sides is referred to as trapezoid

Digon – a polygon with two sides (edges) and two vertices. Not generally recognized as a polygon in the Euclidean plane

Parallelogram – a quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure

Rhombus – a quadrilateral whose four sides all have the same length

Kite, or deltoid, is a quadrilateral with two pairs of congruent adjacent sides

A polyhedron is a solid in three dimensions with flat faces, straight edges and sharp corners or vertices

Surface area of a pyramid – a² + 2 a h (where ‘a’ is side along the base)

Volume of a pyramid – 1/3 a h

An n-sided pyramid is a polyhedron formed by connecting an n-sided polygonal base and a point, called the apex by n triangular faces (n ≥ 3). In other words, it is a conic solid with polygonal base

Cuboctahedron – a polyhedron with eight triangular faces and six square faces

5-cell – a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentatope, or hyperpyramid

Inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces

Circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices

In Euclidean geometry, a Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex. Five solids meet those criteria, and each is named after its number of faces

Tetrahedron – any polyhedron having four plane faces

Hexahedron – a cube

Octahedron has 8 identical equilateral triangular faces

Dodecahedron has 12 regular pentagonal faces, 20 vertices and 30 edges

Icosahedron has 20 identical equilateral triangular faces, 12 vertices and 30 edges

Conic section (or just conic) – a curve that can be formed by intersecting a cone with a plane, e.g. circle, ellipse, parabola, hyperbola

Area of a circle – π r²

Chord – a line segment whose endpoints lie on the circle

Sector – a region bounded by two radii and an arc lying between the radii

Segment – a region bounded by a chord and an arc lying between the chord's endpoints

Annulus – the area between two concentric circles

Mrs. Miniver's problem is a geometry problem about circles. Given a circle A, find a circle B such that the area of the intersection of A and B is equal to the area of the symmetric difference of A and B (the sum of the area of A − B and the area of B − A). The problem derives from A Country House Visit, one of Jan Struther's newspaper articles featuring her character Mrs. Miniver

Ellipse – a curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve

Parabola – the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface

Parabolas are frequently encountered as graphs of quadratic functions such as y=x²

Hyperbola – the curve of intersection between a right circular conical surface and a plane that cuts through both halves of the cone

Hyperbola represents the function f(x) = 1/x

Eccentricity – a ratio describing the shape of a conic section; the ratio of the distance between the foci to the length of the major axis

Witch of Agnesi is a type of curve

Catenary – the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function

Cardioid – a heart-shaped plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius

Cycloid – the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line

Asymptote – a line that a curve approaches but never touches

Bezier curve – a parametric curve important in computer graphics. Used to design automobile bodies

Tesseract, also called 8-cell or octachoron, is the four-dimensional analog of the (three-dimensional) cube, where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. Tesseract has 16 corners

Non-Euclidean geometry is the study of shapes and constructions that do not map directly to any n-dimensional Euclidean system. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines

Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which had been proved by Gauss. The fundamental object is called the Riemann curvature tensor. Example of non-Euclidean geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds. It enabled Einstein's general relativity theory

Adrien-Marie Legendre is known for his work on elliptic integrals and as the author of Elements of Geometry

Congruent – figures with the same form, coinciding when superimposed

The invention of Cartesian coordinates in the 17th century by René Descartes provided the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations

Fractal – a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole’, a property called self-similarity. The term fractal was coined by Benoît Mandelbrot

Koch snowflake – a mathematical curve and one of the earliest fractal curves to have been described

Fern fronds and romanesco broccoli show fractal properties

Types of symmetry –

Reflectional symmetry – each point on one side of the line of symmetry has a corresponding point on the other side of the line

Rotational symmetry – an image can be rotated around a central point and still match its original shape

Translational symmetry – a shape is made up of repeating units

Fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space

Tensors – geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps

Manifold – abstract mathematical space in which every point has a neighbourhood which resembles Euclidean space, but in which the global structure may be more complicated, e.g. a sphere

Catastrophe Theory – a branch of mathematical topology developed by Rene Thom which is concerned with the way in which nonlinear interactions within systems can produce sudden and dramatic effects

Torus – a doughnut-shaped surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle

Colinear – points lying in the same straight line

Coplanar – lying or occurring in the same plane. Used of points, lines, or figures

Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other

Nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points defined from the triangle. Also known as Feuerbach's circle, and Euler's circle

Mandelbrot set is defined by z = z² + c

Cathetus – either of the sides that are adjacent to the right angle in a triangle, i.e. opposite or adjacent

Klein bottle – a non-orientable surface, i.e. a surface (a two-dimensional topological space), for which there is no distinction between the ‘inside’ and the ‘outside’ of the surface

Mobius strip – a non-orientable surface with only one side and only one boundary component. The international symbol for recycling is a mobius strip

A self-similar object is exactly or approximately similar to a part of itself. Self-similarity is a typical property of fractals

Topology – is concerned with spatial properties that are preserved under continuous deformations of objects, for example, deformations that involve stretching, but no tearing or gluing

Squaring the circle is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. In 1882, the task was proven to be impossible, proving that π is a transcendental number

Locus – a collection of points which share a common property

Archimedean spiral (also known as the arithmetic spiral) is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity r = Aθ

Trigonometry

Trigonometry – from the Greek trigonon ‘three angles’ and metro ‘measure’

Hipparchus was known as ‘the father of trigonometry’

The exterior angle is the supplementary angle to the interior angle

Acute (0-90), Right (90), Obtuse (90-180) and Reflex (180-360) – types of angle

Complementary angles add up to 90^o

Supplementary angles add up to 180^o

Corresponding angles (between parallel lines) are always equal

A pair of angles opposite each other, formed by two intersecting straight lines, are called vertical angles or opposite angles

Adjacent angles are angles that share a common vertex and edge but do not share any interior points

Complementary angles are angle pairs whose measures sum to one right angle

Two angles that sum to a straight angle are called supplementary angles

A linear pair of angles are two angles that are adjacent and supplementary

A straight angle is an angle whose measure is exactly 180^o

A full angle is an angle whose measure is exactly 360^o

sine = opposite / hypoteneuse

cosine = adjacent  / hypotenuse

tangent = opposite / adjacent

SOHCAHTOA – mnemonic for remembering definitions of sine, cosine and tangent

tan A= sin A / cos A

cosecant csc(A) is the multiplicative inverse of sin(A)

secant sec(A) is the multiplicative inverse of cos(A)

cotangent cot(A) is the multiplicative inverse of tan(A)

For any angle x, sin²x + cos²x = 1

sin 0⁰ = 0, sin 90⁰ = 1

cos 0⁰ = 1, cos 90⁰ = 0

tan 0⁰ = 0, tan 90⁰ = ∞

Arcsin – the inverse function of the sine; the angle that has a sine equal to a given number, e.g. arcsin 0.629 = 0.68 radians (39^o)

Orthocentre, centroid, circumcentre, Euler line – in a triangle

Gradian, or grad – a unit of plane angle, dividing a right angle in 100

Radian – the ratio between the length of an arc and its radius

One radian is equal to 180/π degrees

π radians in a semicircle

Conjugate or explementary angles – two angles whose sum is 360°

Calculus

Differential calculus – concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behaviour of functions. This can be illustrated by the slope of a function's graph

Differentiation – finding the derivative of a function in order to determine its rate of change

Integral calculus – studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced

Integration – finding the function whose derivative is known

Derivative – rate of change of a function, equivalent to the gradient of a curve

Leibniz and Newton are usually credited with the probably independent and nearly simultaneous ‘invention’ of calculus

Fluctions – Newton’s term for calculus

Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace

Euler–Lagrange equation is a differential equation whose solutions are the functions for which a given functional is stationary

Trapezium rule – an approximate technique for calculating the definite integral

Algebra

Andrew Wiles solved Fermat’s Last Theorem in 1993. It states that it is impossible to find an integer solution to the equation xⁿ + yⁿ = zⁿ if n is greater than 2 and x, y and z are non-zero. The proof was revised by Wiles in 1994 with the help of Richard Taylor

Golden ratio – (or golden mean) usually denoted as phi, expresses the relationship that the sum of two quantities is to the larger quantity as the larger is to the smaller, c. 1.618

Greek letter phi represents the golden ratio

Fibonacci series (0, 1, 1, 2, 3, 5, 8, 13…) equates to the golden ratio

The arrangement of sunflower seeds and pine cones follow a Fibonacci series

The ratio of the golden number to one is equal of the ratio of one to ‘the golden number minus one’

Golden rectangle – a rectangle with dimensions which are of the golden ratio (i.e., 1 : 1.618..., represented by phi ). A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle

Golden spiral – a logarithmic spiral whose growth factor b is related to φ, the golden ratio. Looks like a nautilus shell

Golden spiral is a logarithmic spiral whose growth factor is the golden ratio

Fibonacci spiral is seen in nature, e.g. in nautilus shells

Fibonacci spiral approximates the golden spiral

Boolean logic is a complete system for logical operations. It was named after George Boole, an English mathematician at University College Cork who first defined an algebraic system of logic in the mid 19th century

Fuzzy logic is a form of multi-valued logic derived from fuzzy set theory to deal with reasoning that is approximate rather than precise

Associativity – within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value

Commutativity – for example a + b = b + a, thus the '+' "function" is commutative

Arithmetic Series – a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For example. the sequence 3, 5, 7, 9, 11

Geometric Series – a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54 is a geometric progression with common ratio 3

Quadratic equation – an equation in which the highest power of an unknown quantity is a square

Fourier series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines. Fourier series were introduced by Joseph Fourier for the purpose of solving the heat equation in a metal plate

Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point

Maclaurin series 1 + x + x² + x³ +… is a Taylor series

Even function – f(x) = f (-x). Symmetrical about y axis

Odd function – f(x) = -f (-x). Symmetrical about y axis, then x axis

Continuous function – a function for which, intuitively, ‘small’ changes in the input result in ‘small’ changes in the output

Mixed number – a whole number plus a fraction

Exponential function – the function e^x, where e is the number (approximately 2.718) such that the function e^x is its own derivative

e^x = 1 + x + x²/2! + x³/3! + …

In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation

Robert Recorde was a Welsh physician and mathematician. He introduced the equals sign (=) in 1557 and also introduced algebra to the UK

Obelus – division sign (/)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring

The first use of zero can be traced back to Muhammad ibn Musa al-Khwarizmi, who is credited as the father of algebra

Vector – has magnitude and direction

Hilbert space extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions

Unit vector – a vector whose length is 1. A unit vector is often denoted by a lowercase letter with a ‘hat’ (^ symbol). (x,y,z) axes are represented by (i,j,k) unit vectors

Cross product or vector product is a binary operation on two vectors in three-dimensional space

The cross product is defined by the formula a x b = ||a|| ||b|| sin θ n

where θ is the angle between a and b in the plane containing them, ‖a‖ and ‖b‖ are the magnitudes of vectors a and b, and n is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule

Dot product – an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the magnitudes of the two vectors and the cosine of the angle between them

Radical symbol (√) – a symbol used to indicate the square root or nth root

Quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Hamilton in 1843 and applied to mechanics in three-dimensional space

Every complex number can be represented as a point in the complex plane, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form)

Polar coordinate system – a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point is called the pole, and the ray from the pole with the fixed direction is the polar axis

Determinant – a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression

Eigenvalue – any number such that a given matrix minus that number times the identity matrix has a zero determinant

Eigenvector – a vector that when operated on by a given operator gives a scalar multiple of itself

Identity matrix – a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros

Polynomial – a mathematical function that is the sum of a number of terms

Logarithm – the number that is the exponent of another number to a specified base, e.g. the logarithm of 100 to the base 10 is 2

Common logarithm is the logarithm with base 10. It is also known as the Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use

Natural logarithm – the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718. The natural logarithm is generally written as ln x, log_e x or sometimes, if the base of e is implicit, as simply log x

The natural logarithm of a number x is the power to which e would have to be raised to equal x

The term Napierian logarithm is often used to mean the natural logarithm

Infinity symbol (∞) invented by John Wallis in 1655

A binomial is a polynomial with two terms – the sum of two monomials

Factor – one of two or more quantities that divides a given quantity without a remainder. For example, 2 and 3 are factors of 6

Primorial – similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied

Saddle point – a point of a function of two variables which is a stationary point but not a local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves down in a different direction

Pascal’s triangle – geometric arrangement of the binomial coefficients in a triangle

A function on the real numbers is called a step function (or staircase function)

Two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant. The constant is called the coefficient of proportionality or proportionality constant, e.g. k in the equation y = kx

Subtraction formula. minuend (c) − subtrahend (b) = difference (a)

Numbers

e and π are transcendental numbers, i.e. non-algebraic, cannot be expressed as solutions to a polynomial equation made up of rational numbers

e – known as Euler’s number or Napier’s constant

π – Archimedes constant. Archimedes made the first calculation of π

Permutable, Sexy, Fibonacci – types of prime numbers

The Sieve of Eratosthenes is a simple way, and the Sieve of Atkin a fast way, to compute the list of all prime numbers up to a given limit

A sexy prime is a prime number that differs from another prime number by six

Goldbach Conjecture – all even numbers > 2 are the sum of two prime numbers

A Mersenne number is a number that is one less than a power of two

Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers

3, 7, 31 – first three Mersenne primes

Chen prime – remains a prime, or the product of two primes, when two is added

Fermat number is a positive integer of the form F_n = 2(²ⁿ) + 1

The first few Fermat numbers are: 3, 5, 17, 257, 65537

Highly composite numbers have many factors, e.g. 12

With division, you divide a divisor into a dividend, and the answer is a quotient and a remainder

Proper fraction (vulgar fraction) – numerator is smaller than denominator, e.g. 2/3

Improper fraction – numerator is bigger than denominator, e.g. 22/7

Rational number – can be expressed as a fraction. A real number that can be expressed as a ratio of two integers

Irrational number – cannot be expressed as a fraction

Real numbers – all rational and irrational numbers

Natural number – a positive whole number of the type that occurs in nature, e.g. you can have two stones, but not half a stone or -2 stones. The ‘counting’ numbers 1,2,3 …

Whole number – zero and the counting numbers 0,1,2,3 …

Triangular number – one of a set of integers which correspond to the number of points required to produce a triangle, where each row of points has one more than the previous row, i.e. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55

Surd – an irrational number that can be expressed as such a root of a rational number

Perfect number – an integer which is the sum of its proper positive divisors. 6, 28, 496, 8128

2520 is the smallest number which is divisible by all the numbers from 1 to 10

A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers. The lowest happy numbers are 1, 7, 10, 13 and 19

Amicable numbers are two different numbers so related that the sum of the proper divisors of one of the numbers is equal to the other. The smallest pair of amicable numbers is (220, 284)

Friendly number is a natural number that shares a certain characteristic called abundancy, the ratio between the sum of divisors of the number and the number itself, with one or more other numbers. Two numbers with the same abundancy form a friendly pair

Pandigital number – an integer that in a given base has among its significant digits each digit used in the base at least once

Bernoulli numbers are a sequence of rational numbers with deep connections to number theory, and were discovered by Jakob Bernoulli

An imaginary number is a square root of a negative number. Imaginary numbers have the form bi where b is a non-zero real number and i is the imaginary unit, defined as the square root of -1. An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where a and b are called respectively, the ‘real part’ and the ‘imaginary part’ of the complex number

1729 is the smallest number expressible as the sum of two cubes in two different ways 1729 = 1³ + 12³ = 9³ + 10³

Harshad number – an integer that is divisible by the sum of its digits when written in that base, e.g. 42 is divisible by 4+2

Abundant number – a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number

Modulus – absolute value of a real number

The number 123.45 can be represented as a decimal floating-point number with an integer mantissa of 12345 and an exponent of −2

Vinculum – a horizontal line used in mathematical notation, e.g. 0.3333 can be represented as 0.3¯

Number systems

Long scale is the English translation of the French term echelle longue. It refers to a system of names in which every new term greater than million is 1,000,000 times the previous term: billion means a million to the power of two or a million millions (10¹²), trillion means a million to the power of three or a million billions (10¹⁸), and so on. Used in UK

Short scale is the English translation of the French term echelle courte. It refers to a system of large number names in which every new term greater than million is 1,000 times the previous term: billion means a thousand millions (10⁹), trillion means a thousand billions (10¹²), and so on. Used in USA

A prefix may be added to a unit to produce a multiple of the original unit. All multiples are integer powers of ten:

Zetta           10²¹

Yotta           10²⁴

Zepto         10^-21

Yocto         10^-24

Googol       10¹⁰⁰, 1 followed by 100 zeroes. Ten duotrigintillion

Googolplex 10 to the power of a googol, 1 followed by a googol of zeroes

Decillion     10³³ (USA) or 10⁶⁰ (UK)

Quintillion   10¹⁸ (US), 10³⁰ (British)

Octillion      10²⁷ (US), 10⁴⁸ (British)

Decillion     10³³ (US), 10⁶⁰ (British)

Denary – base 10 numeral system

Vigesimal – base 20 numeral system

Statistics

Normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions. Also known as a bell curve

Central limit theorem – if the sum of the variables has a finite variance, then it will be approximately normally distributed

Poisson distribution – a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event

Bernoulli distribution – a discrete probability distribution, e.g. a single toss of a coin

Binomial distribution – the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p

Mean – average value

Median – middle value

Mode – most common value

Mid-range – arithmetic mean of the maximum and minimum values in a data set

Standard deviation – a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be very close to the mean

Variance – one of several descriptors of a probability distribution, describing how far the numbers lie from the mean

Skewness – a measure of the asymmetry of the probability distribution of a real-valued random variable

Kurtosis – a measure of the ‘peakedness’ of the probability distribution of a real-valued random variable

Geometric mean – numbers are multiplied and then the nth root (where n is the count of numbers in the set) of the resulting product is taken

Markov chain – a discrete-time stochastic process with the Markov property. Having the Markov property means for a given process knowledge of the previous states is irrelevant for predicting the probability of subsequent states. In this way a Markov chain is ‘memoryless’: no given state has any causal connection with a previous state

A stochastic process, or sometimes random process, is the counterpart of a deterministic process (or deterministic system) in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time, in a random process there is some indeterminacy in its future evolution described by probability distributions

Karl Pearson invented the product-moment correlation coefficient in 1897, and also made the first use of the terms standard deviation and chi-square

Francis Galton created the statistical concept of correlation and widely promoted regression toward the mean

In probability theory, Bayes' theorem shows how to determine inverse probabilities

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations. Invented by Stanislaw Ulam

In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value

A box plot or boxplot is a convenient way of graphically depicting groups of numerical data through their quartiles. Box plots may also have lines extending vertically from the boxes (whiskers) indicating variability outside the upper and lower quartiles, hence the terms box-and-whisker plot and box-and-whisker diagram

A stem-and-leaf display is a device for presenting quantitative data in a graphical format, similar to a histogram, to assist in visualizing the shape of a distribution

Kriging – a method of interpolation used in spatial analysis. Named after a South African mining engineer

Extreme value theory or extreme value analysis deals with the extreme deviations from the median of probability distributions. It seeks to assess the probability of events that are more extreme than any previously observed, e.g. a 100-year flood

Set theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects

Union of the sets A and B is denoted A ∪ B

Intersection of the sets A and B is denoted A ∩ B

Relation – a set of ordered pairs

Function – a map or transformation between sets

Singleton – set with exactly one element

Tuple – an ordered list of elements

The Hull-born philosopher and mathematician, John Venn, introduced the Venn diagram in 1881. A stained glass window in Caius College, Cambridge, where Venn studied and spent most of his life, commemorates him and represents a Venn diagram

Euler diagram is a way of representing how sets overlap – similar to a Venn diagram. It is named after its inventor Leonhard Euler. Unlike a Venn diagram, it does not have to contain all possible zones (where a zone is defined as the area of intersection of the sets it shows)

Chaos theory

Chaos theory is a study of behaviour of dynamical systems that are highly sensitive to initial conditions. A small variation in these initial conditions will produce wildly differing results. Chaos is a form of dynamical instability. This sensitivity is popularly referred to as the butterfly effect

Edward Lorenz wrote Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?, which led to the term ‘butterfly effect’

Graph theory

Graph theory – the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A graph in this context is a collection of vertices or nodes and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another

The paper written by Leonhard Euler on the Seven Bridges of Konigsberg and published in 1736 is regarded as the first paper in the history of graph theory

Four colour conjecture (four colour theorem) was proved by Wolfgang Haken and Kenneth Appel in 1976. Given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the regions of the map so that no two adjacent regions have the same colour. It was the first major theorem to be proved using a computer

Game theory

Game theory is the study of strategic decision making. Specifically, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”

In combinatorial game theory, a game tree is a directed graph whose nodes are positions in a game and whose edges are moves

Nash equilibrium – in a game involving two or more players, each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally

Zero-sum games – one person's gains exactly equal net losses of the other participant or participants

Prizes

Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. Founded at the behest of Canadian mathematician John Charles Fields, the medal was first awarded in 1936, to Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas, and has been awarded every four years since 1950. On the obverse is Archimedes and a quote attributed to him written in Latin which means ‘Rise above oneself and grasp the world’

Rene Thom won the Fields Medal in 1958

Andrew Wiles was not awarded the Fields Medal in 1998 as he was over the age limit

In 2006, Grigori Perelman, credited with proving the Poincaré conjecture, refused his Fields Medal

In 2014 Maryam Mirzakhani became both the first woman and the first Iranian honored with the Fields Medal

Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. It has often been described as the ‘mathematician's Nobel prize’

Millennium Prize Problems – 7 problems, each with a $1 million reward provided by the Clay foundation:

1.    P versus NP

2.    The Hodge conjecture

3.    The Poincaré conjecture

4.    The Riemann hypothesis

5.    Yang-Mills existence and mass gap

6.    Navier-Stokes existence and smoothness

7.    The Birth and Swinnerton-Dyer conjecture

P versus NP problem asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer

The Poincaré conjecture was solved by Grigori Perelman, but he declined the award in 2010

The Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of zeroes