## Rational numbers

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.

Integers are also rational numbers

Since q may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode ℚ ); it was thus named in 1895 by Peano after quoziente, Italian for "quotient".

Examples: 1/2, 1/3, 1/4 are rational numbers

Decimal form of rational numbers

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.

Irrational numbers

A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.

More detailed definitions (not clearly explained)

The rational numbers can be formally defined as the equivalence classes of the quotient set (Z × (Z \ {0})) / ~, where the cartesian product Z × (Z \ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n is not 0 (n ≠ 0), and "~" is the equivalence relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0.

In abstract algebra, the rational numbers together with certain operations of addition and multiplication form a field. This is the archetypical field of characteristic zero, and is the field of fractions for the ring of integers. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.

In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals.

Zero divided by any other integer equals zero, therefore zero is a rational number (but division by zero is undefined).