# 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.