# Reciprocal

The reciprocal of a number is 1 divided by the number. All numbers have a reciprocal, except for 0, since the reciprocal of 0 is undefined. Reciprocals can be useful for working with fractions, particularly fractional expressions in algebra where reciprocals can be used to simplify expressions to solve equations. The reciprocal of a non-zero integer such as 5 is:

Notice that if you take the reciprocal of a reciprocal, you get the original number (1/5 is the reciprocal of 5 and vice versa). Thus, the reciprocal is also referred to as the multiplicative inverse; it is what you need to multiply a given number (or expression) by for the product of the two numbers to be 1. 5 can also be written in fraction form as 5/1. Since

we can confirm that is the reciprocal of 5. Multiplying any expression by its reciprocal will result in 1.

If a number is written in fraction form, you can also think of the reciprocal as "flipping" the number. Flipping refers to switching the position of the numerator and denominator of a fraction as if you were flipping the fraction upside down. When taking the reciprocal of whole numbers (like 5), we need to add the fraction bar and the number 1, as we did above. For decimals, we can either convert the decimal to fraction form, or divide 1 by the decimal using long division or another method. The result is the reciprocal.

Example

Find the reciprocals of the following numbers.

1. :

Since it is already in fraction form, we can simply switch the position of the numbers in the numerator and denominator of the fraction to find that the reciprocal of the fraction is:

2. 1.25:

The reciprocal can be determined by dividing 1 by 1.25:

Test this by multiplying 1.25 by 0.8:

We also could have converted 1.25 to an improper fraction, then flipped the improper fraction:

The reciprocal is therefore 4/5, which is equal to 0.8.

Using exponents, given a number, n, the reciprocal can also be written as n^{-1}. Raising any expression to the power of -1 is the equivalent of taking its reciprocal.