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. $\frac{{1}}{{a}}$.