Properties of Integers:
D. Commutative Law of Addition
The order makes no difference when adding two
integers.
Examples: 3 +
2 and 2 + 3 both
equal 5.
-2
+ 4 and 4 + -2 both equal
2.
6 - 4 (careful here...)
6 - 4 IS 6 + -4 and -4 +
6 both equal 2.
-5
- 10: -5 + -10 and -10 + -5 both equal -15.
E. Commutative Law of Multiplication
The order makes no difference when multiplying two
integers.
Examples: 3 *
2 and 2 * 3 both equal 6.
-2
* 4 and 4 * -2 both equal -8.
F. Associative Law of Addition
When adding 3 integers together, it makes no
difference which two in sequence you add first.
Example: 5 + 2 + 1
add 5 + 2 first, then add 1, that is, 7 + 1 = 8
or
add 2 + 1 first, then add 5, that is, 5 + 3 = 8
Grouping symbols (), {}, and [], are used to
indicate which operation should be done first. So, we can say for the
preceeding example,
(5 + 2) + 1 is
the same as 5 + (2 + 1)
G. Associative Law
of Multiplication
When multiplying 3 integers together, it makes no
difference which two in sequence you multiply first.
Example: 2 * 3 * 4
multiply 2 by 3 first, then multiply by 4, that is, 6 * 4 = 24
or multiply 3 by 4 first, then myltiply by 2, that is 2 * 12
= 24
Using grouping symbols, (2 * 3) * 4 is
the same as 2 * (3 * 4).
H. Distributive Law
The product of one integer with the sum of two or more integers is the sum
of the products of the first integer with the other integers, taken one at a
time.
Example: 2 * (3 + 4) is
the same as 2*3 + 2*4
Let's take a closer look,
2 * (3 + 4) the parenthesis indicate 3+4 should be done first
so we have 2 * (7) and we can drop the parenthesis
since there is no other operation to do inside the parenthesis (THIS IS MOST
IMPORTANT! there is no + or - sign inside the parenthesis, if there were, the
parenthesis CANNOT be removed, yet.) So we have 2 * 7 = 14
This distributive law states that we can distribute
2 across the + sign within the parenthesis and thereby remove the
parenthesis and get
2*3 + 2*4. So which
operation comes first? the * or the +? Since multiplication is a higher
order operation than addition (recall that multiplication is defined in terms
of addition), we first multiply then we add. So, we have 6 + 8 = 14.
This law is usually called the distributive law of
multiplication over addition to emphasize that addition must be within the
grouping symbols.
In other words, 2 * (3 + 4)
= 2*3 + 2*4 = 14
We cannot
distribute across multiplication,
that is 2 * (3 * 4) is NOT 2*3 *
2*4
2 * 12 is NOT 6 * 2 * 4
24 is NOT 12
* 4 = 48!
Example: 5 * (6 - 2)
5 * (6 + -2)
5*6 + 5 * -2
30 + -10
20
5*(6-2) = 5*(4) = 5*4 = 20.
(This law is heavily used, in
reverse, when factoring terms into expressions.)
Practical use:
Suppose you were told to
multiply 34 by 7, knowing this distributive law and that 34 is 30 + 4 we
have 30*7 + 4*7 = 210 + 28 = 238. One can easily do such multiplications
in one's head.
Example: 215 * 6
= 1200 + 60 + 30 = 1290
that is 200 * 6 + 10 * 6 + 5*6
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