Sometimes it is necessary to add long strings of numbers without a calculator.
For example, one might be asked to find 48 + 33 + 52 + 11 + 17. This sum is difficult
to compute without a calculator, but the task can be made a lot easier by
knowing some simple properties of addition. In this section, we will focus on
these properties, which will help make "mental math" easier and will be useful
in later sections of Pre-Algebra.

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Commutative Property

The Commutative Property states that for any numbers *a* and *b*,
the following is always true:

*a* + *b* = *b* + *a*

For example,

3 + 5 = 5 + 3. We can see that this is true because

3 + 5 = 8 and

5 + 3 = 8,
so

3 + 5 and

5 + 3 are equal to each other. Another way to think of the commutative
property is the following: if you have a quarter and a dime in your pocket, and you
add them together, you will come up with the same amount of
money whether you add the quarter to the dime or the dime to the quarter.

By the commutative property, if we add two or more numbers, we can always add
them in any order. This is useful because it might be easier to add numbers in
a different order than the order given. In our example above, it takes a long
time to add the numbers from left to right (try it). However, because addition
has the commutative property, we can switch the order of the numbers in the
expression:

48 + 33 + 52 + 11 + 17 = 48 + 52 + 33 + 17 + 11

This new expression is easier to evaluate, because

48 + 52 = 100 and

100 + 33 + 17 = 150.
It is easier to add numbers to numbers which end in "0". This expression can be made even
easier to evaluate with the associative property:

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Associative Property

The Associative Property states that for any numbers *a*, *b*, and
*c*, the following is always true:

(*a* + *b*) + *c* = *a* + (*b* + *c*)

For example,

(2 + 4) + 7 = 2 + (4 + 7). We can see that this is true because

(2 + 4) + 7 = 6 + 7 = 13 and

2 + (4 + 7) = 2 + 11 = 13, so

(2 + 4) + 7 and

2 + (4 + 7)
are equal to each other. Or we can once again think about it using the
example of coins: if I have a nickel and a dime in my left pocket and a
quarter in my right pocket, I will have the same amount of money if I
take the dime out of my left pocket and put it in my right pocket with the quarter.

Not only can we add numbers in any order, we can also add pairs of numbers
within the expression before adding them all together. In other words, we can
put parenthesis around any two (or more) numbers and add those numbers
separately. Using our example above, we can rearrange the numbers using the
commutative property and then use the associative property to add them in
pairs:

48 + 52 + 33 + 17 + 11 = (48 + 52) + (33 + 17) + 11 = 100 + 50 + 11

It's a lot easier to add these three numbers in one's head than to add the
original five numbers one by one, and both methods yield the same answer--161.

The Commutative Property of Addition can be remembered by remembering that when
only addition is involved, numbers can move ("commute") to anywhere in the
expression. The Associative Property of Addition can be remembered by
remembering that any numbers that are being added together can "associate" with
each other. Another good rule of thumb is, when trying to decide which
properties to use, look for numbers that add up to multiples of 10; these should
be added first because they are easy to add to other numbers.