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).
