A fraction is a mathematical expression that denotes a part of a whole or a ratio between two quantities. It is conventionally written with two integers separated by a horizontal line: the numerator (top) and the denominator (bottom). Together they indicate how many parts of a specified size are considered, for example 3/4 means three parts out of four equal parts.

Basic structure and types

The numerator counts selected parts; the denominator gives the number of equal parts that make up the whole. Common types include:

  • Proper fractions — numerator smaller than denominator (e.g., 2/5).
  • Improper fractions — numerator greater than or equal to denominator (e.g., 7/4).
  • Mixed numbers — combination of an integer and a proper fraction (e.g., 1 3/4).
  • Unit fractions — numerator equals one (e.g., 1/3); historically important in early arithmetic.

Operations and conversions

Basic arithmetic follows standard rules: to add or subtract, express fractions with a common denominator; to multiply, multiply numerators and denominators; to divide, multiply by the reciprocal of the divisor. Fractions can be simplified by cancelling common factors to reach lowest terms. They convert to decimals and percentages: for example, 1/4 = 0.25 = 25%.

History and notation

Representations of parts have existed across many cultures. Ancient societies used unit fractions, sexagesimal fractions, or other systems before the widespread adoption of the modern numerator-over-denominator form. The present notation and manipulation rules developed gradually alongside positional number systems and algebraic methods.

Applications and noteworthy facts

Fractions are central in arithmetic, measurement, probability, ratios, and algebra. Every rational number can be expressed as a fraction of integers. Some fractions correspond to repeating decimals while others terminate. Related concepts include continued fractions, which provide useful approximations, and rational approximations used in computations and engineering. Understanding fractions is a key step in mathematical literacy and problem solving.