Rationals

Rationals, or rational numbers, are numbers that can be expressed as a ratio a/b where a and b are integers and b is not zero. The set of all rational numbers is denoted by Q. Every integer is rational, since n = n/1.

A rational number can be written in reduced form a/b with gcd(|a|,|b|) = 1 and b > 0. This reduced representation is unique; different pairs can denote the same rational, such as 2/4 and 1/2.

Decimal expansions of rationals either terminate or repeat. A fraction in lowest terms has a terminating decimal if and only if its denominator has no prime factors other than 2 and 5. Every terminating or repeating decimal represents a rational number.

Q is closed under addition, subtraction, multiplication, and division by a nonzero rational. It forms a field under the usual operations and can be ordered consistently with these operations, making Q an ordered field. Integers are embedded in Q as rationals with denominator 1.

One standard construction views rationals as equivalence classes of ordered pairs (a, b) with b ≠ 0, identifying (a, b) with (ka, kb) for nonzero integers k. Another is to reduce fractions to lowest terms. The set Q is countable and dense in the real numbers: between any two real numbers there exists a rational.

Rationals play a central role in arithmetic, algebra, and number theory. They appear in fractions, ratios, and approximations, and contrast with irrational numbers, which cannot be expressed as a ratio of integers.