# 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 non-zero 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 $\left(-a\right)\left(-b\right)=ab$