Maths Notes
Number
8. Negative Numbers
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8. Negative Numbers
WARNING
If 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 Line
7
The key to negative numbers is the number
line.
4
3
5
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.
2
4
1
3
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.
0
-1
6
–
2
1
0
-2
-1
-3
-2
-3
-4
-5
-6
+
Adding and Subtracting when the Signs are NOT Touching
Where 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
Rule: If no signs are touching, use a number line (on paper or in your head), or
think about money!
7
Example 1
Example 2
Example 3
6
2-7
-4 + 6
-1 - 4
5
4
3
–
2
1
0
-1
+
Number Line: put
your finger at 2 and
move down 7 places
Number Line: put
your finger at -4 and
move up 6 places
Number Line: put
your finger at -1 and
move down 4 places
Money: if I have £2
in my bank and
someone takes away
£7, then how much do
I have?
Money: if I have £-4
in my bank (I am in
debt) and someone
gives me £6, then
how much do I have?
Money: if I have £-1
in my bank (I am in
debt) and someone
takes away £4, then
how much do I have?
-2
-3
-4
-5
-6
2 – 7 = -5
-4 + 6 = 2
-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
Example 5
56 - 89
–56
Number Line: imagine your
finger is as 56.
-102 + 217
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
+115
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
–33
- 33
Number Line: imagine your
finger is as -102.
115
+102
Money: if I have £56
in my bank and
someone takes away
£89, then how much
do I have?
56 – 89 = -33
- 102
Money: if I have £-102
in my bank and someone
gives me £217, then how
much do I have?
-102 + 217 = 115
Adding and Subtracting when the Signs ARE Touching
Okay, now it’s time for our rule…
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
- = + = +
Example 1
-4 + -8
Example 2
5 - -6
and
and
+ = - = +
Example 3
-22 - - 9
Example 4
-6 - 10
Have you spotted the
touching signs?...
Have you spotted the
touching signs?...
Have you spotted the
touching signs?...
Have you spotted the
touching signs?...
Using our rule, we can
change + and - to -
Using our rule, we can
change - and - to +
Using our rule, we can
change - and - to +
I hope not, because
there aren’t any!
So, our sum becomes:
So, our sum becomes:
So, our sum becomes:
The two minuses are
NOT touching
-4 - 8
Which is pretty easy
using either number
lines or money.
-4 + -8 = -12
5+6
Which is pretty easy
however you do it!
5 - -6 = 11
-22 + 9
Which is pretty easy
using either number
lines or money
-22 - -9 = -13
So our sum stays the
same and we do it
using either of our
methods
-6 - 10 = -16
Multiplying and Dividing
As 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!
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…
Example 1
Example 2
-20 ÷ 4
Do the sum as normal,
ignoring the minus signs
Example 3
-6 x -9
Do the sum as normal,
ignoring the minus signs
-3 x -2 x -5
Do the sum as normal,
ignoring the minus signs
Example 4
88
4
Do the sum as normal,
ignoring the minus signs
88  22
4
20 ÷ 4 = 5
6 x 9 = 54
Count the number of
minus signs in the
question… 1!
Count the number of
minus signs in the
question… 2!
Count the number of
minus signs in the
question… 3!
Count the number of
minus signs in the
question… 2!
One minus makes the
whole answer
negative
Two minuses makes
the whole answer
positive
Three minuses makes
the whole answer
negative
Two minuses makes
the whole answer
positive
So:
So:
So:
So:
-20 ÷ 4 = -5
-6 x -9 = 54
3 x 2 x 5 = 30
-3 x -2 x -5 = -30
88
4
 22
Tricky Questions involving Negative Numbers
The 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
Example 2
2  (3  5)
3  8  4  2
Now, BIDMAS says we
must sort out the brackets
first:
3  5  2
So now we have:
Now, BIDMAS says we
must sort out the division
first:
8  4  2
Putting that back in the
question, we have:
2  2
And using our negative
number rules, we should get
the answer of:
= -4
3  2  2
Let’s sort those two
touching signs out:
3  2  2
Using number lines, or
money, we should get
= -7
Example 3
4  3
3  9
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)
So, the top gives us:
4 3  12
And from the bottom:
3  9
 3  9  6
Leaving us with:
12
6
= 2
Good luck with
your revision!