A number is an abstract entity that represents a count or measurement. A symbol for a number is called a numeral. The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.
Numbers can be classified into sets called number systems. The most familiar numbers are the natural numbers, which to some mean the non-negative integers and to others mean the positive integers. In everyday parlance the non-negative integers are commonly referred to as whole numbers, the positive integers as counting numbers, symbolised by . Mathematics is used in many classes throughout the course of one's education.
The integers consist of the natural numbers (positive whole numbers and zero) combined with the negative whole numbers, which are symbolised by (from the German Zahl, meaning "number").
A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face (for quotient).
Pi, represented by the Greek letter π, is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space (i.e., on a flat plane); it is also the ratio of a circle's area to the square of its radius. (These facts are reflected in the familiar formulas from geometry, C = π d and A = π r2.) In this animation, the circle has a diameter of 1 unit, giving it a circumference of π. The rolling shows that the distance a point on the circle moves linearly in one complete revolution is equal to π. Pi is an irrational number and so cannot be expressed as the ratio of two integers; as a result, the decimal expansion of π is nonterminating and nonrepeating. To 50 decimal places, π ≈ 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510, a value of sufficient precision to allow the calculation of the volume of a sphere the size of the orbit of Neptune around the Sun (assuming an exact value for this radius) to within 1 cubic angstrom. According to the Lindemann–Weierstrass theorem, first proved in 1882, π is also a transcendental (or non-algebraic) number, meaning it is not the root of any non-zero polynomial with rational coefficients. (This implies that it cannot be expressed using any closed-form algebraic expression—and also that solving the ancient problem of squaring the circle using a compass and straightedge construction is impossible). Perhaps the simplest non-algebraic closed-form expression for π is 4 arctan 1, based on the inverse tangent function (a transcendental function). There are also many infinite series and some infinite products that converge to π or to a simple function of it, like 2/π; one of these is the infinite series representation of the inverse-tangent expression just mentioned. Such iterative approaches to approximating π first appeared in 15th-century India and were later rediscovered (perhaps not independently) in 17th- and 18th-century Europe (along with several continued fractions representations). Although these methods often suffer from an impractically slow convergence rate, one modern infinite series that converges to 1/π very quickly is given by the Chudnovsky algorithm, first published in 1989; each term of this series gives an astonishing 14 additional decimal places of accuracy. In addition to geometry and trigonometry, π appears in many other areas of mathematics, including number theory, calculus, and probability.