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.