The denominator is the part of a fraction that appears beneath the fraction bar (or to the right of the slash). It shows into how many equal parts the whole is divided. The number or expression above the bar is the numerator; together they represent the ratio numerator/denominator. For a quick reference to fraction notation see fraction and for the complementary term see numerator.

Basic rules and examples

A denominator must never be zero because division by zero is undefined. When the bar is horizontal the denominator sits below; when written with a slash it follows the slash. Examples:

  • In 3/4 the denominator is 4.
  • In 23 + 76/100 the denominator of the fractional part is 100.
  • In 15/(35 + 7) the denominator is the parenthesized expression (35 + 7).
  • In a/(6b + c) the denominator is the algebraic expression (6b + c), which may restrict allowed values of variables.

Properties, operations and algebraic considerations

When simplifying a fraction, common factors of numerator and denominator can be canceled. Addition and subtraction of fractions usually require a common denominator; the least common denominator (LCD) is the smallest shared multiple that makes addition straightforward. Multiplying fractions multiplies numerators together and denominators together. Dividing by a fraction is equivalent to multiplying by its reciprocal, which swaps numerator and denominator.

In algebra, denominators can be polynomials or other expressions. Any values that make the denominator zero are excluded from the domain of the rational expression; these are often noted as restrictions before simplification. For example, a/(6b + c) is undefined when 6b + c = 0.

Denominators also appear in mixed numbers and improper fractions: the denominator stays the same when rewriting between forms, while the numerator changes to reflect whole parts. Understanding denominators is essential for fraction arithmetic, rational functions, proportions, probability, and many applied problems in science and engineering.

Note: The English word "denominator" derives from Latin roots meaning to name or designate the parts; in modern usage it refers specifically to the divisor part of a fraction.