# Multiplying and dividing Negatives

In mathematics, there are rules for multiplying and dividing positive and negative numbers.

The rules are:

1 - When the signs are different the answer is negative
2 - When the signs are the same the answer is positive

Or another way of remembering this is:

My friend's friend is my friend (positive x positive = positive)
My friend's enemy is my enemy (positive x negative = negative)
My enemy's friend is my enemy (negative x positive = negative)
My enemy's enemy is my friend (negative x negative = positive)

The main issue to remember is:

A negative x negative is a positive – 2 wrongs make a right!

My enemy’s enemy is my friend

Examples

 3times5 = 15 (signs are the same so positive) -3times5 = -15 (signs are different so negative) 3times(-5) = -15 (signs are different so negative) -3times(-5) = 15 (signs are the same so positive)

 6-:2 = 3 (signs are the same so positive) -6-:2 = -3 (signs are different so negative) 6-:(-2) = -3 (signs are different so negative) -6-:(-2) = 3 (signs are the same so positive)

NOTE:

If there were three negatives in a row:

-3times(-5)times(-5)

Then you would complete one calculation first

-3times(-5)=15

And now relook at the equation

15times(-5)

You now complete this calculation

15times(-5)=-75

So -3times(-5)times(-5)=-75

## How can a negative multiplied by a negative be a positive?

The following may help explain why:

 Running total of cash available You got paid £100 £100 Next you got paid £100 £200 The following day you were paid £100 £300 Then an electric bill came in for -£50 £250 Then another electric bill came in for -£50 £200 And the next day another electric bill came for -£50 £150

Then the electricity company wrote to you to say
have sent the bills. So they need to take back the
bills as follows:

 Subtract -3 lots of -£50 £150 £300

Summary

-3 lots of -£50 is +£150

Because -3times-£50=£150

Two negatives make a positive.

## There is another way of looking at this as follows:

Look at the following sequence:

 2times5 = 10 1times5 = 5 0times5 = 0 -1times5 = -5 -2times5 = -10 -3times5 = -15

Notice that the number on the left drops by one each time and the total on the right drops by 5 each time.

Now do the same but with a -5

 2times(-5) = -10 1times(-5) = -5 0times(-5) = 0 -1times(-5) = 5 -2times(-5) = 10 -3times(-5) = 15

Notice that the number on the left drops by one each time and the total on the right climbs by 5 each time.

Two negatives have to make a positive.