The closure property states that if we take a real number and we add/multiply it to another real number, the result will be a real number.

Example: If I take a real number and add it to another real number, according to the closure property, the result will be a real number.
Example: If I take a real number and multiply it by another real number, according to the closure property, the result will be a real number


The identity property states that if you add/multiply a real number by the identity, the result will be the real number you started with.

There is one identity in each operation. In addition the identity is 0. Any value added by 0 will be the original value (5 + 0 = 5).
In multiplication, the identity is 1, for any value multiplied by 1 is itself (5 x 1 = 5).
The job of the inverse is to do nothing.


The inverse property states that if you add/multiply a real number by its inverse, the result will be the identity.
Here, each number has its own inverse, the number that gets it back to the identity.

In addition, opposites act as inverses, for when you add two opposites, you get 0 (the identity of addition).
Example: -2 + 2 = 0;   3.4 + -3.4 = 0;   So, in addition, -2 is 2's inverse. Likewise, 2 is -2's inverse.

In multiplication, reciprocals act as inverses, form when you multiply two reciprocals, the result is 1 (the identity of multiplication).
To make a reciprocal of a value, simply switch the numerator and denominator.
Example:
          
The job of the inverse is to get you to the identity.


The commutative property states that we can add/multiply two real numbers in any order.
Example:   4 + 5 = 5 + 4;    2 x 6 = 6 x 2;   8 + -8 = -8 + 8


The associative property states that we can add/multiply a group of three real numbers by starting with either the first two values (as normal) or we can start with the last two numbers (the second and third numbers).
Example:   (2 + 3) + 4 = 2 + (3 + 4);     (5 x 3) x 8 = 5 x (3 x 8)
By continuously using a combination of the commutative and associative properties, we can add/multiply a group of values in any order.


The distributive property allows us to multiply values using a combination of multiplication and addition.
For example, if multiplying 5 and 19, you can break up 19 into an addition problem, say (10 + 9). The resulting problem is 5 x (10+9).
The distributive property allows us to "distribute" the multiplication throughout the addition problem. Simply, we "distribute" the 5 to every number in the addition problem by way of multiplication. So, 5 x (10 + 9) = (5 x 10) + (5 x 9).  Now the result is found by multiplying 5 and 10 then adding it to the result of 5 multiplied by 9. 50 + 45 = 95. (This same problem could have been made 5 x (20 - 1) --- See it?)
This property allows us to make easier problems so that the multiplication can be done mentally.

Try it!
What is 9 x 24? Instead of writing it down in an elementary fashion (though it works), try doing it in your head.
9 times 20 is 180, 9 times 4 is 36. 180 plus 36 is 216 - DONE!

What about 13 x 199? Try changing it to 13 x (200 - 1). 13 times 200 is 2600. 13 times 1 is 13. Now this time subtract (notice the subtraction sign).
2600 minus 13 is 2587 -- DONE! (Admittedly, this does take practice).