Reciprocal

Reciprocal Definition

A reciprocal is the flipped version of a number—specifically, the number you multiply by to get a product of 1. For any nonzero number, its reciprocal is 1 divided by that number. In mathematical terms, the reciprocal of x is written as 1 ⁄ x. Multiplying a number by its reciprocal always equals 1:

x × (1 ⁄ x) = 1

For whole numbers, finding the reciprocal involves writing the number as a fraction and then inverting it. For example, the reciprocal of 4 is 1⁄4, and the reciprocal of 5⁄7 is 7⁄5. Zero does not have a reciprocal because no number exists that you can multiply by 0 to get 1.

Reciprocals play a key role in division and algebra. Dividing by a number is the same as multiplying by its reciprocal. This makes reciprocals especially useful in solving equations, simplifying fractions, and working with ratios and rates.

Example

If you’re scaling a recipe and want to double the number of servings, you multiply each ingredient by 2. To reverse the change and get back to the original serving size, you multiply everything by the reciprocal—1⁄2—effectively undoing the doubling.