denominators

Denominators are the bottom components of fractions. In the usual notation a/b, b is the denominator and indicates into how many equal parts the whole is divided. The numerator, a, is the part being considered. The value of the fraction depends on both, with the denominator constraining how finely the whole is partitioned.

The denominator must be nonzero. A zero denominator is undefined. If a fraction is written with a negative denominator, the minus sign is typically moved to the numerator to produce the standard form, e.g., -3/4 rather than 3/(-4). Multiplying numerator and denominator by the same nonzero number yields an equivalent fraction with a different denominator.

To perform operations on fractions, fractions are often rewritten with a common denominator. The least common denominator (LCD) is the smallest positive number that is a common multiple of the denominators involved. Adding or subtracting fractions uses the common denominator, while multiplication multiplies the numerators and denominators directly. Reducing a fraction to lowest terms involves dividing numerator and denominator by their greatest common divisor (gcd), which simplifies the expression.

In decimal representation, a reduced fraction has a finite decimal expansion if and only if its denominator, after removing common factors with the numerator, has no prime factors other than 2 and 5. The denominator also appears in probability as the size of the sample space, and more generally in ratios as the divisor in a relationship between two quantities.