Overview

A fraction is a way to represent a number as a ratio of two integers: the numerator (top) and the denominator (bottom). It indicates how many equal parts are being considered out of a whole divided into a specific number of parts. Fractions are one of several representations of rational numbers and appear throughout arithmetic, algebra, measurement and everyday situations such as recipes, finance and construction.

Parts and notation

A typical fractional notation is written as a/b where a is the numerator and b is the denominator (b ≠ 0). The numerator counts the selected parts; the denominator indicates how many equal parts make the whole. Fractions can be written with a horizontal vinculum (a line between numbers) or with a slash (1/4). Variants include mixed numbers (e.g., 1 1/2) and decimal or percent equivalents.

Types and examples

Common categories of fractions:

  • Proper fractions: numerator smaller than denominator (3/5).
  • Improper fractions: numerator equal to or larger than denominator (7/4).
  • Mixed numbers: an integer plus a proper fraction (1 3/4), often converted to an improper fraction for computation.
  • Equivalent fractions: different numerators and denominators that represent the same value (1/2 = 2/4 = 50/100).

Simplifying and comparing

Fractions are simplified (reduced) by dividing numerator and denominator by their greatest common divisor so the denominator is as small as possible. To compare fractions, techniques include bringing them to a common denominator or converting to decimal form. Equivalent fractions are created by multiplying or dividing numerator and denominator by the same nonzero integer.

Arithmetic with fractions

Basic operations follow standard rules:

  1. Addition/Subtraction: convert to a common denominator, then add or subtract numerators.
  2. Multiplication: multiply numerators and multiply denominators, then simplify.
  3. Division: multiply by the reciprocal of the divisor (a/b ÷ c/d = a/b × d/c).

Visual models and applications

Fractions are often taught with pie charts, number lines and area models that show parts of a whole. They are essential in measurements (length, time, volume), probabilities, ratios, rates and in representing slopes or gradients. Engineers, chefs and financiers routinely use fractional values when exact ratios are required.

History and notable facts

Practices for writing and using fractions date back to ancient civilizations. Egyptian mathematics favored unit fractions (fractions with numerator 1), while other cultures used common fractions or sexagesimal partitions. In modern mathematics, any rational number can be expressed as a fraction of two integers; continued fractions and decimal expansions are alternative representations that reveal different properties of rational and irrational numbers.