Unlike adding and subtracting integers, multiplication and division of integers is actually an easy process. If you know how to multiply and divide positive values, you're all set. The only different part here is figuring out wheter the answer is positive or negative - and that is EASY!

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You know I can't just give you the easy trick...we must explore what is going on so that we can make connections to future topics.
We will start with multiplication and use what we have learned there to understand division. So, what is multiplication?

Multiplication:
Multiplication is simply, repetitive addition - addition of one number over and over again.
For example: 5 x 3 is really 0 + 5 + 5 + 5 (adding 5 to zero three times). So what happens with negatives? Let's take a look.
What would -5 x 3 be? That would be 0 + (-5) + (-5) + (-5) = -15 (adding -5 to zero three times).
Here you are adding a negative amount a positive numeber of times, therefore, it will produce a negative value.

What about -5 x -3? Here you're required to add a negative amount a negative number of times.
How do you add something a negative number of times?
To add something a negative number of times is to subtract it a positive number of times.
Therefore, instead of adding -5, -3 times, we will subtract -5, +3 times. That gives us: 0 - (-5) - (-5) - (-5) = 15.

What about 5 x -3? Same as the above! Instead of adding 5 a negative 3 times, we will subtract 5 a positive 3 times.
That gives us 0 - 5 - 5 - 5 = -15.

Let's make some observations:
1. When we multiplied a positive and a positive, the product is positive.
2. When we multiplied a positive and a negative (in either order), the product was negative.
3. When we multiplied a negative and a negative, the product was positive.
4. An odd number of positives produced a negative answer, whereas an even number of negatives (0 is even) produced a positive answer.

Division:
Division is really the reversing of the multiplication process. If we multiply two values together we get a product. When we divide that product
by one of those two values, the result (quotient) must be the other value. Example: 4 x 5 = 20. When I divide 20 (the product, now the dividend) by
one of the values, say 4 (20 / 4), the result is the other value (factor) 5.   20 / 4 = 5.

Now, let's take a look with the negatives. Since -5 x 3 = -15, if I divide -15 by 3, the result must be -5.  [-15 / 3 = -5]
Since -5 x -3 = +15, if I divide 15 by -5, the result must be -3 [15 / -5 = -3]

Since the product of two negatives is positive, dividing the positive answer by one of the two negatives must give me the other negative.   + / - = -
Since the product of a positive and negative is negative, dividing the negative answer by the positive must give me the negative. Dividing the negative
answer by the negative, must give me the positive. - / - = +;  - / + = -

Observations:
1. When we divide two positives, the quotient is positive.
2. When we divide a positive and a negative (either order), the quotient is negative.
3. When we divide two negatives, the quotient is positive.
4. An odd number of negatives produces a negative quotient. An even number of negatives produces a positive quotient.

1. When multiplying / dividing an EVEN number of negatives, and zero is not present, the product / quotient will be positive.

2. When multiplying / dividing an ODD number of negatives, and zero is not present, the product / quotient will be negative.

 

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