Why Do Operations Have Properties?

Before continuing to learn about the properties of operations you must understand two basic questions:

What is the point of operations having properties?
→ Without operations having properties, we would not know their usage. Understanding their usage helps lay the foundation for solving word problems later on.
Why do operations have properties?
→ Operations have properties which define their usage. The usage of operations is the very essence of them, without usage properties are useless (vice versa).

With understanding properties we are able to enter the realm of higher-level thinking. 
This is because properties illustrate general cases which allow us to lead to more mathematical generalizations.

Definitions of Mathematical Properties:
Addition:

1. Commutative Property: ${\displaystyle a+b=b+a}$
2. Associative Property: ${\displaystyle (a+b)+c=a+(b+c)}$
3. Identity Property of zero: ${\displaystyle 0+a=a(=a+0)}$
4. Inverse Property: For every member a, there is - a such that ${\displaystyle a+(-a)=0}$

Multiplication:

1. Commutative Property: ${\displaystyle ab=ba}$
2. Associative: ${\displaystyle (ab)c=a(bc)}$
3. Identity: ${\displaystyle a\cdot 1=1\cdot a=a}$
4. Inverse Property, for every ${\displaystyle a\neq 0}$, there is ${\displaystyle ({\dfrac {1}{a}})}$ (or ${\displaystyle a^{-1}}$), such that a ${\displaystyle ({\dfrac {1}{a}})=1.}$

The general rule for the inverse property of multiplication is if when you multiply two numbers and the product is 1, then what you multiplied must be multiplicative inverses or reciprocals of each other.

It is important you remember that connecting addition and multiplication is the:
→ Distributive Rule: ${\displaystyle a(b+c)=ab+ac.}$

Often rules are final factor of what you get as your answer:
An example of this is shown below:
${\displaystyle a\cdot 0=0}$, because ${\displaystyle a=a\cdot 1=a\cdot (1+0)=a\cdot 1+a\cdot 0.}$
Now subtract a from both sides to get ${\displaystyle 0=0+a\cdot 0=a\cdot 0.}$

Properties of Addition and Subtraction

Sure, you've added 4 + 3 = 7, but have you tried 3 + 4? You get the same answer right? Yes, because of the commutative law of addition.

For any numbers, a and b, a plus b is equal to b plus a. From this we know that addition is commutative, meaning that the operation of addition can be performed in any order with the same result.

What about 5 + 6 = 11? That's right, 6 + 5 = 11 as well.

There is another property of addition, the associative property. Here is an algebraic example:

For any numbers, a , b, and c, a + (b + c) = (a + b) + c

Can the same be said about subtraction? Well, let's try it... 7 - 5 = 2, does 5 - 7 = 2? Well, no actually. Because, if you look at a number line you will notice that when you subtract 5 - 7 you go below 0 into the realm of the negative numbers. Specifically, your solution is -2. Even though -2 is same in absolute value as 2, it isn't the same number. Therefore, subtraction is not commutative. Is it associative? No, because the associative law depends on the commutative law in order to work(since it really is just an extension of the commutative law.)

Associative

The Associative property of real numbers is: ${\displaystyle (a+b)+c=a+(b+c)}$ for all real numbers a,b,c
This implies that the order in which addition is done does not matter. Note however this only applies to addition and not to subtracting numbers.

Multiplication shares the same property. ${\displaystyle (ab)c=a(bc)}$ As in addition the groups or the association of the numbers in the parenthesis change. The actual order of the numbers remains the same.