Properties of Multiplication
The following are the properties of multiplication for real numbers. Some textbooks list just a few of them, others list them all. They may have slightly different names in your textbook.
PROPERTIES
OF MULTIPLICATION


Identity Property 
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. 

Inverse Property 
For all nonzero real numbers $a$ , $a\cdot \frac{1}{a}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{a}\cdot a=1$ . $\frac{1}{a}$ is the reciprocal of $a$ . $\frac{1}{a}$ is also called the multiplicative inverse of $a$ . 

Multiplicative Property of Zero 
For every real number $a$ , $a\cdot 0=0$ and $0\cdot a=0$ 

Commutative Property 
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. 

Associative Property 
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. 

Multiplicative Property of $1$ 
For all real numbers $a$ and $b$ , $a\left(1\right)=a$ and $\left(1\right)a=a$ 

Property of Opposites in Products 
For every real number $a$ , $\left(a\right)b=ab,a\left(b\right)=ab,$ and $\left(a\right)\left(b\right)=ab$ 