numbers

Numbers are abstract objects used for counting, measuring, labeling, and ordering quantities. They can be represented by symbols and written in various numeral systems, making it possible to perform operations such as addition, subtraction, multiplication, and division. Numbers provide the foundation for mathematics and for describing real-world quantities.

Common families are natural numbers N (often including 0), integers Z (positive, negative, and zero), rational numbers Q (fractions of integers), real numbers R (all rational and irrational numbers), and complex numbers C (numbers of the form a + bi). In addition, special classes include irrational numbers, which cannot be expressed as a ratio of integers (for example sqrt(2)); and transcendental numbers, which are not roots of any nonzero polynomial with rational coefficients (for example π and e).

Zero plays a central role as a number and as a placeholder in place-value notation. The decimal system is the most widely used base-10 positional notation, but many other bases exist for representing numbers.

Historically, ideas about numbers arose in various ancient cultures for counting and trade. The Hindu-Arabic numeral system, with zero as a number and the decimal place-value concept, became dominant in the medieval world and is the ancestor of most modern numeration. Numbers are central to science, engineering, computing, and almost all areas of inquiry, and they are studied in fields ranging from elementary arithmetic to advanced number theory and analysis. In higher mathematics, numbers form the elements of algebraic structures such as fields and rings, with real numbers forming a complete ordered field and complex numbers forming an algebraically closed field.