29 terms

# Elementary Math chapter 8.2 Multiplication

Multiplication of Integers
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3 × (-4) 3 bad checks
3 × (-4) (-4) (-4) = - 12
Number line
-4 -4 -4 left from zero
The first column remains 3 throughout
3 × 4 = 12
3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
The first column remains 3 throughout
3 × (-1)= -3
3 × (-2)= - 6
3 × (-3) = - 9
3 × (-4) = - 12
The first column remains (-3)
(-3) × 3 = - 9
(-3) × 2 = - 6
(-3) × 1 = - 3
(-3) × 0 = 0
(-3) × (-1) = 3
(-3) × (-2) = 6
(-3) × (-1) = 3
The sign on the second number determine the number of chips
4 × (-3) Since the first number in this combination (4) is positive we combine 4 groups of (-3) red chips
- 4 × 3, in this case
The first number is (-4) is negative, which indicates that we should "take away 4 groups of black chips).
How to do it
Add an equal number of red and black chips to the set.
After taking away 4 groups of 3 black chips, the resulting set has 12 red chips or a value of -12.
how cont.
12 blacks + 12 reds inserted.
Take away 4 groups of 3 blacks
-12 remain ( 12 red chips remain)
Multiplication of Integers
Let a and b be any integers
Multiplying by 0 (zero)
a × 0 = 0 = 0 × a
Multiplying two positives
If a and b are positive, they are multiplied as whole numbers
Multiplying a positive and a negative (+) × (-)
If a is positive and b is positive (thus (-b) is negative) , then
a (a-b) = - (ab),
where ab is the whole-number product of a and b. That is, the product of a positive and a negative is NEGATIVE.
Multiplying two negatives
If a and b are positive, then
(-a) (-b) = ab,
where ab is the whole-number product of a and b. That is, the product of two negatives is POSITIVE.
Multiplying by zero
5 × 0 = 0
Multiplying two positives + × + = +
5 × 8 = 40
Multiplying a positive and a negative + × - = -
5 × (-8) = - ( 5 × 8 ) = - 40
Multiplying two negatives - × - = +
(-5 ) × (- 8) = 5 × 8 = 40
Properties of Integer Multiplication
Let a, b and c, be any integers
Closure Property for Integer Multiplication
ab is an integer
Commutative Property for Integer Multiplication
ab = ba
Associative Property for Integer Multiplication
(ab)c = a(bc)
Identity Property for Integer Multiplication
1 is the unique integer such that a × 1 = a = 1 × a for all a
Distributivity of Multiplication over Addition of Integers
Let a, b and c, be any integers then:
a (b + c) = ab + ac
Theorem
Let a be any integer. Then
a (-1) = - a
Theorem
Let a and b be any integers. Then
( -a) b = - (ab)
Theorem
Let a and b be any integers. Then
(-a)(-b) = ab for all integers a, b
is read: "the opposite of a times the opposite of b is ab"
Multiplication Cancellation Property
Let a, b and c, be any integers with c ≠ 0.
If ac = bc, then a = b
c ≠ 0 because:
3 × 0 = 2 × 0, but 3 ≠ 2!
Zero Divisors Property
Let a and b be integers.
Then ab = 0 if and only if
a = 0 or b = 0
or a and b both equal 0