# Inverse Property of Multiplication

**Inverse property of multiplication**:

$$$\color{purple}{a\times\frac{1}{a}=\frac{1}{a}\times a=1}$$$

$$$\frac{1}{a}$$$ is called the **multiplicative inverse** of $$$a$$$.

Inverse property is true for any real number $$$a$$$.

Notice, that we wrote, that $$${a}\times\frac{{1}}{{a}}=\frac{{1}}{{a}}\times{a}$$$. This is true, according to the commutative property of multiplication.

Actually, we already discussed the multiplicative inverse.

Yes, yes. Multiplicative inverse is just another name for reciprocal!

**Example 1.** Multiplicative inverse of $$$\frac{{5}}{{3}}$$$ is $$$\frac{{3}}{{5}}$$$, because $$$\frac{{5}}{{3}}\times\frac{{3}}{{5}}={1}$$$.

**Example 2.** Multiplicative inverse of $$$-\sqrt{{{2}}}$$$ is $$$-\frac{{1}}{\sqrt{{{2}}}}$$$, because $$${\left(-\sqrt{{{2}}}\right)}\cdot{\left(-\frac{{1}}{\sqrt{{{2}}}}\right)}={1}$$$.

**Example 3.** $$${2.57}\cdot\frac{{1}}{{2.57}}=\frac{{1}}{{2.57}}\cdot{2.57}={1}$$$ (recall, that both $$$\times$$$ and $$$\cdot$$$ denote multiplication).

**Conclusion.** Multiplicative inverse (reciprocal) of the number $$$a$$$ is a number, that is turned upside down, i.e. $$$\frac{{1}}{{a}}$$$.