Properties of Addition
The following are the properties of addition 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 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. 