What Is a Reciprocal in Math?

Reciprocals are two numbers that have one as their product when multiplied with each other. They are most often expressed as proper or improper fractions.

To find the reciprocal of a given fraction, simply invert the fraction’s numerator and denominator. This means a fraction’s numerator becomes its reciprocal’s denominator, and its denominator becomes its reciprocal’s numerator. Obtain the reciprocal by flipping over the original number.

To get the reciprocal of a mixed number, first convert it into an improper fraction so the numerator and denominator can be interchanged. For example, to find the reciprocal of 3 ¼, rewrite it as 13/4. Then reverse the numerator and denominator to get the reciprocal, 4/13. Using reciprocals becomes necessary when one must rely on division to solve problems involving fractions.

For a word-problem example of real-world reciprocal use, suppose someone had 2 ¼ pizza left over after a party, was no longer hungry and wanted to invite some friends to finish it off, knowing each friend was capable of eating three-fourths of a pizza. Determining how many friends to invite is a matter of dividing 2 ¼ by three-fourths. When 2 ¼ is changed into an improper fraction, it becomes 9/4. To divided this by three-fourths, it must be multiplied times the reciprocal of three-fourths, which is four-thirds.

The product of 9/4 times four-thirds is 36/12. By dividing this fraction's numerator by its denominator, it becomes evident that the host can invite three friends over to finish the leftover pizza.