Reciprocals and Inverse Numbers

When working with algebra reciprocal numbers are common, and when working in algebra inverse numbers are too. Get help with both here.

Reciprocals and the multiplicative inverse

The second type of opposite number has to do with multiplication and division. It's called the multiplicative inverse, but it's more commonly called a reciprocal.

To understand the reciprocal, you must first understand that every whole number can be written as a fraction equal to that number divided by 1. For example, 6 can also be written as 6/1.

6=6
1

Variables can be written this way too. For instance, x = x/1.

x=x
1

The reciprocal of a number is this fraction flipped upside down. In other words, the reciprocal has the original fraction's bottom number—or denominator—on top and the top number—or numerator—on the bottom. So the reciprocal of 6 is 1/6 because 6 = 6/1 and 1/6 is the inverse of 6/1.

Below, you can see more reciprocals. Notice that the reciprocal of a number that's already a fraction is just a flipped fraction.

5y1
5y
181
18
34
43

And because reciprocal means opposite, the reciprocal of a reciprocal fraction is a whole number.

17
7
12
2
125
25

From looking at these tables, you might have already noticed a simpler way to determine the reciprocal of a whole number: Just write a fraction with 1 on top and the original number on the bottom.

Decimal numbers have reciprocals too! To find the reciprocal of a decimal number, change it to a fraction, then flip the fraction. Not sure how to convert a decimal number to a fraction? Check out our lesson on converting percentages, decimals, and fractions.

Using reciprocals

If you've ever multiplied and divided fractions, the reciprocal might seem familiar to you. (If not, you can always check out our lesson on multiplying and dividing fractions.) When you multiply two fractions, you multiply straight across. The numerators get multiplied, and the denominators get multiplied.

42 =8
5315

However, when you divide by a fraction you flip the fraction over so the numerator is on the bottom and the denominator is on top. In other words, you use the reciprocal. You use the opposite number because multiplication and division are also opposites.

4÷2=43=12
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