Elementary Math chapter 8.2 Multiplication

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Multiplication of Integers

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

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