Properties of Real Numbers

Closure Property
If you start with two real numbers and add them together, you'll get a real number.
Similarly, if you start with two real numbers and multiply them together, you will get
a real number.

Example:
2 and 3 are real numbers. When you add them, the result is 5, which is a real number.
When you multiply them, you get 6, which is a real number.
The set of real numbers is "closed" under the operations of addition and multiplication.
This means that if you add or multiply with real numbers, you'll never end up with a
result that is "outside" the set of real numbers.

Commutative Property of Addition
               x + y  =  y + x
Order doesn't matter when you add numbers.

Example:
7 + 8 + 9  = 8 + 9 + 7
    15 + 9  = 17 + 7
         24   =  24

Commutative Property of Multiplication
               x · y  =  y · x
Order doesn't matter when you multiply numbers.

Example:   
  5 · 68  =  68 · 5
    340   =  340

Associative Property of Addition
(a + b) + c = a + (b + c)

It doesn't matter how you group things when you're adding.

Associative Property of Multiplication
It doesn't matter how you group factors when you are multiplying.

Example:
   

Identity Property
For each operation (addition and multiplication), there is a special number,
called the identity, because if you perform the operation with it, you get the
identical thing back again. Nothing has changed.

For addition, the identity is 0.

For multiplication, the identity is 1.

Inverse Property
For each operation (addition and multiplication), there is a number called the inverse,
such that if you perform the operation with it, you get the identity.

For addition, the inverse is also called the opposite.

For multiplication, the inverse is also called the reciprocal.

The operation of subtraction is defined as adding the additive inverse.
In words,  x minus y is    defined    to be x + the opposite of y.
In symbols,
                    x - y = x + (-y)

The operation of division is defined as multiplication by the multiplicative inverse.
In words, x divided by y is   defined   to be x times the reciprocal of y.
In symbols,
                  

Distributive Property
                   a(b + c) = ab + ac

Multiply everything that is inside the parentheses by what's outside the parentheses.


Example:
                  3(4 + 7) = (3 · 4) + (3 · 7) = 12 + 21 = 33


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