Multiplication of Integers

### 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)

### 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.

### 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 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!