Use adaptive quiz-based learning to study this topic faster and more effectively.

# Reciprocal

The reciprocal of a number is one divided by that number. It is the same as raising it to the power $-1$

The reciprocal of $3$ is a third. It can be written as $1/3$ or $3^{-1}$.

The reciprocal of a fraction is found by swapping the numerator and the denominator.

$\left(\frac{\Tblue{2}}{\Tgreen{7}}\right)^{-1} = \frac{\Tgreen{7}}{\Tblue{2}} = 3.5$

These are the main properties of the reciprocal:

• Every real number except for zero has a reciprocal.
• The reciprocal of a number has the same sign as the number.
• The reciprocal of the reciprocal is the original number.
• The reciprocal multiplied by the original number is $1$.
• The reciprocal of a positive number between $0$ and $1$ is larger than $1$, and vice versa.
• The reciprocal of a negative number between $0$ and $-1$ is less than $-1$, and vice versa.
$(-3)^{-1} = -\frac{1}{3},\; (2^{-1})^{-1} = 2,\; (0.25)^{-1} = 4,\;(-0.25)^{-1}=-4$
A graph showing $y$ and the reciprocal of $x$