Number

Number is a mathematical concept used to count, measure, label, and compare quantities. It is more than a symbol or numeral; it denotes an abstract idea that can be manipulated according to defined rules. Numbers underpin arithmetic, measurement, and many theoretical areas of mathematics.

Natural numbers describe counting and, in some conventions, include zero. Integers extend natural numbers to include negative values and zero. Rational numbers are ratios of integers and include terminating or repeating decimals. Real numbers comprise rational numbers together with irrational numbers, which cannot be written as a ratio; examples include sqrt(2) and pi. Complex numbers extend the real numbers by including an imaginary unit i, with i^2 = -1.

Historically, number concepts arose in ancient counting systems and tallying. The Hindu-Arabic numeral system, developed in India and transmitted to the West, introduced positional notation and the concept of zero, greatly expanding computational capability. Modern mathematics formalizes numbers through axioms and structures, such as the Peano axioms for natural numbers and the hierarchy of number systems up to the real and complex numbers.

Numbers support operations—addition, subtraction, multiplication, and division—subject to familiar rules. They carry order relations on the real and rational numbers, enabling comparisons. In higher math, numbers form algebraic structures such as fields, rings, and ordered fields; the real numbers form a complete ordered field, and the complex numbers form an algebraically closed field.

Numbers are used in science, engineering, finance, and computing to model quantities, perform measurements, and describe data. In number theory, questions about primes, divisibility, and Diophantine equations explore intrinsic properties of integers.