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Rational numbers (): Numbers that can be expressed as a ratio of an integer to a non-zero integer. [3] All integers are rational, but there are rational numbers that are not integers, such as −2/9. Real numbers (): Numbers that correspond to points along a line. They can be positive, negative, or zero.
The numbers d i are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by Rényi (1957) and first studied in detail by Parry (1960). Every real number has at least one (possibly infinite) β-expansion. The set of all β-expansions that have a finite representation is a subset of the ring Z[β, β −1].
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined.
Some authors use for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the group of units of ). Additionally, Z p {\displaystyle \mathbb {Z} _{p}} is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p -adic integers .
The name algebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O K.
Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field , and a field has characteristic zero if and only if it contains the rational numbers as a subfield.
For example, the integers are made by adding 0 and negative numbers. The rational numbers add fractions, and the real numbers add infinite decimals. Complex numbers add the square root of −1. This chain of extensions canonically embeds the natural numbers in the other number systems. [6] [7] Natural numbers are studied in different areas of math.
This allows identifying any natural number n with the sum of n real numbers equal to 1. This identification can be pursued by identifying a negative integer − n {\displaystyle -n} (where n {\displaystyle n} is a natural number) with the additive inverse − n {\displaystyle -n} of the real number identified with n . {\displaystyle n.}