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Reciprocals (Mathematics Glossary)

What Is a Reciprocal?

The reciprocal of a quantity is the result of dividing 1 by that quantity.

Reciprocal of 5 is 1 over 5

Dictionary Definition

The Oxford English Dictionary defines a reciprocal as "based on an inverse relationship."

Real Examples of Reciprocals

The reciprocal can't be found for numbers, letters and more complicated quantities, like terms and expressions.

Type of Quantity Quantity Reciprocal
Numbers 2 1 over 2
3 1 over 3
8 1 over 8
Letters a 1 over a
x 1 over x
Terms 2x 1 over 2 x
t squared 1 over t squared
Expressions x + 1 1 over x plus 1
a plus b 1 over a

More Real Examples of Reciprocals

The slider below shows more real examples of reciprocals.

Reciprocals of Fractions

To find the reciprocal of a fraction, 1 is divided by the fraction.

Dividing 1 by a fraction turns the fraction upside down.

Read more about how to find the reciprocal of a fraction

Interactive Test

Here's a second test on reciprocals.
Here's a third test on reciprocals.


Exponent Notation For a Reciprocals

To denote the reciprocal of a quantity, place -1 to the top-right of the quantity. This is an exponent of -1.

5 to the minus 1 equals 1 over 5

Read more about exponents of -1

Multiplying a Quantity With Its Reciprocal Equals 1

Multiplying a quantity with its reciprocal equals 1.

2 times 1 over 2 equals 1. x times 1 over x equals 1

Be Careful With Notation

Sometimes brackets are needed to indicate which part of a term is a reciprocal.

For example,

is different from