Properties of Multiplication

There is a unique real number $1$ such that for every real number $a$ ,

$a\cdot 1=a$ and $1\cdot a=a$

One is called the identity element of multiplication.

For all non-zero real numbers $a$ , $a\cdot \frac{1}{a}=1\text{and}\frac{1}{a}\cdot a=1$ .

$\frac{1}{a}$ is the reciprocal of $a$ .

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

For every real number $a$ ,

$a\cdot 0=0$ and $0\cdot a=0$

For all real numbers $a$ and $b$ ,

$a\cdot b=b\cdot a$

The order in which you multiply two real numbers does not change the result.

For all real numbers $a$ , $b$ , and $c$ ,

$\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$

When you multiply any three real numbers, the grouping (or association) of the numbers does not change the result.

For all real numbers $a$ and $b$ ,

$a\left(-1\right)=-a$ and $\left(-1\right)a=-a$

For every real number $a$ ,

$\left(-a\right)b=-ab,a\left(-b\right)=-ab,$ and

$\left(-a\right)\left(-b\right)=ab$