Arithmetic/Properties of Operations/Why Do Operations Have Properties?

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

Multiplication:

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

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