Multiplying Negatives

## When We Multiply:

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: A Video of People Running

People Run Forward, Video Normal:

All normal, people running forwards: +1 × +1 = +1

People Run Forward, but Video in **Reverse**:

Looks like people running backwards: +1 × −1 = −1

People Run **Backward**, Video Normal:

You see people running backwards: −1 × +1 = −1

People Run **Backward**, but Video in **Reverse**:

Looks like people running forwards: −1 × −1 = +1

### 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 is:

−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):

 2
 3
 4

Now see what happens when we head into **negatives**!

Let's go **backwards** through zero:

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:

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**