What Are Opposite Reciprocals?

By Pete Collins

The opposite reciprocal is a concept in mathematics that can be used to explain the slopes of perpendicular straight lines. Opposite reciprocals are closely related to reciprocals, or multiplicative inverses, a concept in number theory.

Multiplicative Inverses

The multiplicative inverse of a number is equivalent to its reciprocal. If you have a positive number "a" and you multiply it by another number "b" to get 1, then "b" is said to be the multiplicative inverse of "a" (and vice versa). That is:

a x b = 1

If a = 2, the necessarily b = 0.5, as

2 x 0.5 = 1

Thus 2 and 0.5 are said to be multiplicative inverses of each other.


Reciprocals are another term for multiplicative inverses. Another way of thinking about reciprocals is that the reciprocal of "a" is 1 / a. This works with our previous example because:

1/ 2 = 0.5

It can help to think about both numbers as fractions. The value 2 is the same as 2/1; the reciprocal is simply that fraction turned upside down. So 2/1 becomes 1/2, which equals 0.5.

Opposite Reciprocals

The opposite reciprocal of a number is simply its negative reciprocal. Since the reciprocal of 2 is 0.5, the negative reciprocal of 2 is -0/5.

Gradients of Perpendicular Lines

If you have two straight lines on a graph that are perpendicular (they meet at right angles), then the gradient (or slope) of one is always the opposite reciprocal of the other. So if the gradient of one line is 4/3, then the gradient of the other is -3/4. Here we have inverted the fraction (4/3 becomes 3/4) to find the reciprocal, and changed the sign (3/4 becomes -3/4) to make it the opposite reciprocal.

About the Author

Pete Collins has been writing since 2007, primarily on data analysis and environmental science topics. Collins has a Bachelor of Science in ecology from the University of East Anglia and a Master of Science in environmental biogeochemistry from Newcastle University.