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:

Your mum says:

		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
they had made a terrible mistake and they shouldn’t
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.