PROPERTIES OF ADDITION Identity Property There is a unique real number $0$ such that for every real number $a$ , $a+0=a$ and $0+a=a$ Zero is called the identity element of addition. Commutative Property For all real numbers $a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b$ , $a+b=b+a$ The order in which you add two numbers does not change the result. Associative Property For all real numbers $a,b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c$ , $\left(a+b\right)+c=a+\left(b+c\right)$ When you add three real numbers, the grouping (or association) of the numbers does not change the result. Property of Opposites For all real number $a$ , there is a unique real number $-a$ such that $a+\left(-a\right)=0$ and $\left(-a\right)+a=0$ A number and its opposite are called additive inverses of each other because their sum is zero. Property of Opposite of a Sum For all real numbers $a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b$ , $-\left(a+b\right)=\left(-a\right)+\left(-b\right)$ The opposite of a sum of real numbers is equal to the sum of the opposites.