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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 $$x$$
A graph showing $$y$$ and the reciprocal of $$x$$