Numerator (mathematics)

Definition and basic properties

The numerator may be any number or expression: an integer, rational number, real number, complex number, polynomial, or other algebraic expression. A numerator equal to zero makes the entire fraction equal to zero provided the denominator is not zero. Division by zero is undefined, so a nonzero numerator over a zero denominator has no value in standard arithmetic. The numerator can be positive or negative; a negative sign is commonly placed in front of the numerator, in front of the denominator, or before the whole fraction, but the meaning is equivalent after simplification.

Role in arithmetic and simplification

When performing operations with fractions the numerator behaves differently depending on the operation. For multiplication, numerators multiply with numerators and denominators with denominators. For division, the second fraction is inverted so its numerator becomes a denominator and vice versa. For addition and subtraction the fractions must share a common denominator; numerators are then added or subtracted while the common denominator is retained. Reduction or cancellation removes common factors that appear both in numerator and denominator, typically by dividing both by their greatest common divisor; this produces an equivalent fraction with a smaller numerator.

Examples and types

• Proper fraction: numerator smaller than the denominator (e.g., 3/5).
• Improper fraction: numerator greater than or equal to the denominator (e.g., 7/4), often convertible to a mixed number.
• Zero numerator: 0/7 = 0 (valid), but 0/0 is indeterminate.
• Algebraic numerator: expressions such as (x^2-1)/(x+1) show the numerator can be factored and partially canceled with the denominator when allowed by domain considerations.

Algebraic and advanced contexts

In symbolic mathematics the numerator may be a polynomial, rational function, matrix element or other object. Cancelling factors between numerator and denominator is valid algebraically only when those factors are nonzero for the values of the variable under consideration; care is required to preserve domain restrictions. In calculus and analysis, comparing numerator and denominator growth rates helps determine limits of rational functions. In computer representations and typesetting the numerator may be rendered differently (superscripted or left of a slash) depending on notation conventions.

History, terminology and notable facts

The English word "numerator" comes from Latin roots related to number and counting. The role of placing one quantity over another to indicate division has been used in many mathematical traditions; the current horizontal bar and oblique-slash notations developed gradually with printing and algebraic practice. In education the numerator is often emphasized as "how many" of the equal parts named by the denominator, a useful interpretation for fractions of physical quantities. For further basic definitions, see resources on real numbers and related topics such as real numbers.