# Multiplying Negatives

## When We Multiply:

Example | |||

× | two positives make a positive: | 3 × 2 = 6 | |

× | two negatives make a positive: | (−3) × (−2) = 6 | |

× | a negative and a positive make a negative: |
(−3) × 2 = −6 | |

× | a positive and a negative make a negative: |
3 × (−2) = −6 |

Yes indeed, two negatives make a positive, and we will explain **why**, with examples!

## Signs

Let's talk about **signs**.

"+" is the positive sign, "−" is the negative sign.

When a number has **no sign** it usually means that it is **positive**.

Example: **5** is really **+5**

And we can put () around the numbers to avoid confusion.

Example: **3 × −2** can be written as **3 × (−2)**

## Two Signs: The Rules

"Two like signs make a positive sign, two unlike signs make a negative sign" |

### Example: (−2) × (+5)

The signs are − and + (a negative sign and a positive sign), so they are **unlike signs** (they are different to each other)

So the result must be **negative**:

(−2) × (+5) = -10

### Example: (−4) × (−3)

The signs are − and − (they are both negative signs), so they are **like signs** (like each other)

So the result must be **positive**:

(−4) × (−3) = +12

## Why does multiplying two negative numbers make a positive?

Well, first there is the "common sense" explanation:

When I say "Eat!" I am encouraging you to eat (positive)

But when I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do **NOT** not eat!", I am saying I don't
want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you don't need to read any more.

## Direction

It is all about direction. Remember the Number Line?

Well here we have Baby Steven taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 2 steps x 3 = 6 steps forward:

Now, Baby Steven can also step backwards (he is a clever little guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this three times:

Once again Steven's Dad puts him back at the start, but facing the other way. Steven takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times:

Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times:

So, by walking backwards, while facing in the negative direction, he moves in the positive direction.

Try it yourself! Try walking forwards and backwards, then again but facing the other direction.

## More Examples

### Example: Money Owed vs Money Owned

You owe 3 people $5 each. So you are "Negative 15" (3 × −$5 = −$15).

They then say "we like you so much we forgive the debt" ... you have just had 3 subtractions of −5, so it is like you have **added $15** (−3 × −$5 = +$15).

### Example: Tank Levels Rising/Falling

The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank **3 days ago**?

We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go **back 3 days**), so the change will be:

−3 × −1,000 = +3,000

The full calculation is:

30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000

So 3 days ago there were 33,000 liters of water in the tank.

## Multiplication Table

Here is **another way** of looking at it.

Start with the multiplication table (just up to 4×4 will do):

× | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 1 | 2 | 3 | 4 |

2 | 2 | 4 | 6 | 8 |

3 | 3 | 6 | 9 | 12 |

4 | 4 | 8 | 12 | 16 |

Now see what happens when we head into **negative territory**!

Let's go **backwards** through zero:

× | 1 | 2 | 3 | 4 |
---|---|---|---|---|

-4 | -4 | -8 | -12 | -16 |

-3 | -3 | -6 | -9 | -12 |

-2 | -2 | -4 | -6 | -8 |

-1 | -1 | -2 | -3 | -4 |

0 | 0 | 0 | 0 | 0 |

1 | 1 | 2 | 3 | 4 |

2 | 2 | 4 | 6 | 8 |

3 | 3 | 6 | 9 | 12 |

4 | 4 | 8 | 12 | 16 |

Look at the "4" column: it goes **-16, -12, -8, -4, 0, 4, 8, 12, 16**. Getting 4 larger each time.

Look over that table again, make sure you are comfortable with how it works, because ...

... now we go **further to the left**, through zero:

× | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|---|---|---|---|

-4 | 16 | 12 | 8 | 4 | 0 | -4 | -8 | -12 | -16 |

-3 | 12 | 9 | 6 | 3 | 0 | -3 | -6 | -9 | -12 |

-2 | 8 | 6 | 4 | 2 | 0 | -2 | -4 | -6 | -8 |

-1 | 4 | 3 | 2 | 1 | 0 | -1 | -2 | -3 | -4 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |

2 | -8 | -6 | -4 | -2 | 0 | 2 | 4 | 6 | 8 |

3 | -12 | -9 | -6 | -3 | 0 | 3 | 6 | 9 | 12 |

4 | -16 | -12 | -8 | -4 | 0 | 4 | 8 | 12 | 16 |

Same pattern: we can follow along a row (or column) and the values change consistently:

- Follow the "4" row along: it goes
**-16, -12, -8, -4, 0, 4, 8, 12, 16**. Getting 4 larger each time. - Follow the "-4" row along: it goes
**16, 12, 8, 4, 0, -4, -8, -12, -16**. Getting 4 smaller each time. - etc...

So it all follows a neat pattern!

## What About Multiplying 3 or More Numbers Together?

Multiply two at a time and follow the rules.

### Example: What is (−2) × (−3) × (−4) ?

First multiply (−2) × (−3). Two like signs make a positive sign, so:

(−2) × (−3) = +6

Next multiply +6 × (−4). Two unlike signs make a negative sign, so:

+6 × (−4) = -24

**Result: (−2) × (−3) × (−4) = −24**