Maths Notes

Number

8. Negative Numbers

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8. Negative Numbers

WARNINGIf you are not concentrating, negative numbers can trip up the best of mathematicians. So… have a glass of water, shake all other thoughts out of your head, sit down, take a deep breath, and let’s begin…

The Number LineThe key to negative numbers is the number line.

Now, I like to think of the number line going up and down, so when you add you go up, and when you subtract, you go down. Kind of like a thermometer.

If you ever find yourself stuck or unsure about a negative number question, just draw yourself a very quick number line, count the spaces with your finger, and you will be fine.

I still do this, and I’m… well, quite a bit older then you.

4

3

2

1

0

-1

-2

-3

0

1

2

3

4

5

6

7

- 1

- 2

- 3

- 4

- 5

- 6

+–

Adding and Subtracting when the Signs are NOT TouchingWhere people seem to go wrong with negative numbers is that they learn the rule that two minuses make a plus.Now, this rule is a good one, but must only be used when two signs (+ or -) are touching.If no signs are touching, I would just use this rule

Example 1 2 - 7

Number Line: put your finger at 2 and move down 7 places

Rule: If no signs are touching, use a number line (on paper or in your head), or think about money!

0

1

2

3

4

5

6

7

- 1

- 2

- 3

- 4

- 5

- 6

+– Money: if I have £2 in my bank and someone takes away £7, then how much do I have?

2 – 7 = -5

Example 2 -4 +

6Number Line: put your finger at -4 and move up 6 placesMoney: if I have £-4 in my bank (I am in debt) and someone gives me £6, then how much do I have?

-4 + 6 = 2

Example 3 -1 - 4

Number Line: put your finger at -1 and move down 4 placesMoney: if I have £-1 in my bank (I am in debt) and someone takes away £4, then how much do I have?

-1 - 4 = -5

Now both these methods still work when the numbers become harder and the number line becomes too big to draw:

Example 4

56 - 89

Number Line: imagine your finger is as 56.

How far must you go down to get to zero?... 56 spaces, right?

And so, how much further down do you still have to go?… another 33 spaces!

0

56

- 33

Money: if I have £56 in my bank and someone takes away £89, then how much do I have?

56 – 89 = -33

–56

–33

Example 5

-102 + 217

Number Line: imagine your finger is as -102.

How far must you go up to get to zero?... 102 spaces, right?

And so, how much further up do you still have to go?… another 115 spaces!

0

115

- 102

Money: if I have £-102 in my bank and someone gives me £217, then how much do I have?

-102 + 217 = 115

+102

+115

Adding and Subtracting when the Signs ARE TouchingOkay, now it’s time for our rule…

Example 1 -4 + -8

Rule: If two signs are touching (+’s or –’s next to each other), then replace the two signs with one sign using these rules:

+ and - = - - and + = -+ and + = + - and - =

+

Have you spotted the touching signs?...

Using our rule, we can change + and - to -

So, our sum becomes:

-4 + -8 = -12

-4 - 8

Which is pretty easy using either number lines or money.

Example 2 5 - -6

Have you spotted the touching signs?...

Using our rule, we can change - and - to +

So, our sum becomes:

5 - -6 = 11

5 + 6

Which is pretty easy however you do it!

Example 3 -22 - - 9

Have you spotted the touching signs?...

Using our rule, we can change - and - to +

So, our sum becomes:

-22 - -9 = -13

-22 + 9

Which is pretty easy using either number lines or money

Example 4 -6 - 10

Have you spotted the touching signs?...

I hope not, because there aren’t any!

The two minuses are NOT touching

So our sum stays the same and we do it using either of our methods

-6 - 10 = -16

Multiplying and DividingAs was the case with fractions, multiplying and dividing with negative numbers is a little bit easier than adding and subtracting, but you still have to concentrate!

Example 1 -20 ÷ 4

Rule: Do the sum as normal, ignoring the plus and minus signs and write down the answer

Then, carefully count the number of minus signs in the question.

If there is one the whole answer is negative, if there are two the answer is positive, if there are three the answer is negative, four means positive, and so on…

Do the sum as normal, ignoring the minus signs

One minus makes the whole answer negative

So:

20 ÷ 4 = 5

Count the number of minus signs in the question… 1!

-20 ÷ 4 = -5

Example 2 -6 x -9

Do the sum as normal, ignoring the minus signs

Two minuses makes the whole answer positive

So:

6 x 9 = 54

Count the number of minus signs in the question… 2!

-6 x -9 = 54

Example 3-3 x -2 x -5

Do the sum as normal, ignoring the minus signs

Three minuses makes the whole answer negative

So:

3 x 2 x 5 = 30

Count the number of minus signs in the question… 3!

-3 x -2 x -5 = -30

Example 4

Do the sum as normal, ignoring the minus signs

Two minuses makes the whole answer positive

So:

Count the number of minus signs in the question… 2!

884

88 224

88 224

Tricky Questions involving Negative NumbersThe people who write maths exams are nasty. Just when you think you have got a topic sorted, they chuck in a right stinker.But do not panic. So long as you remember the rules we have discussed here, and you don’t forget old BODMAS/BIDMAS, you will be fine!

Example 1

Now, BIDMAS says we must sort out the brackets first:

2 ( 3 5)

3 5 2 So now we have:

2 2 And using our negative number rules, we should get the answer of:

= -4

Example 2

Now, BIDMAS says we must sort out the division first:

3 8 4 2

8 4 2 Putting that back in the question, we have:

3 2 2

Let’s sort those two touching signs out:

= -7

3 2 2 Using number lines, or money, we should get

Example 3

Now remember, even though we can’t see any brackets, they are hidden on the top and bottom of the fraction:

4 3

3 9

= 2

( 4 3)

( 3 9)

So, the top gives us:

4 3 12 And from the bottom:

3 9

3 9 6

Leaving us with:

12

6

Good luck with your revision!