BIRKBECK MATHS SUPPORT
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Numbers 1
In this section we will look at
- the meaning of integers
- positive and negative numbers
- adding and subtracting using the numberline
- multiplying with positive and negative numbers
- dividing with positive and negative numbers
- finally we introduce fractions and decimals
Helping you practice
At the end of the sheet there are some questions for you to practice.
Don’t worry if you can’t do these but do try to think about them. This
practice should help you improve. I find I often make mistakes the
first few times I practice, but after a while I understand better.
Videos
All the examples in this worksheet and all the answers to questions
are available as answer sheets or videos.
Good luck and enjoy!
Videos and more worksheets are available in other formats from
www.mathsupport.wordpress.com
www,mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2010
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1. Meanings of Words
Integers are the whole numbers and Zero means none.
Positive numbers (+) are numbers that are more than zero
(+1, +2, +3 and so on, this can also be written 1, 2, 3, 4, 5, and so on)
Negative numbers (-) are whole numbers that are less than zero
(-1, -2, -3, -4, -5, and so on)
Examples:
If I have 5 pound coins in a bank I can say I have +5 pounds
If I spend 3 pound I am left with +5 – 3 = +2 pound coins
If I now give away 3 pounds then I have +2 – 3 = –1 pound coins so I have –1
pound coins in total. It is difficult to draw –1 but we can invent ways to do this.
We could draw them as an empty circle, as long as we remember that this
means we owe one pound, so this is –1 pound
So if we now add a pound they cancel each other out and we get –1+1 = 0
&
=0
www,mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2010
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2. Adding and Subtracting using the Numberline
To add (+) and subtract (–) numbers it can be useful to use a numberline.
To calculate +5–3 we start at positive 5 and then move three points in the
negative direction. We finish at the number positive 2.
+5 – 3 = +2
-7 -6 -5 -4 -3 -2 -1
0
A
1
A
2
A
3 4
A
5
A
6
7
8
Examples:
1) Here we calculate +7–4 (we usually miss out the + in front of the 7 and so this
is the same as 7–4). We start at the number +7 and move 4 places in the
negative direction to arrive at +3
+7– 4 = +3
-7 -6 -5 -4 -3 -2 -1
0
A
1
A
2
A
3 4
A
5
A
6
7
8
2) Here we calculate +2–8. So we start at +2 and move 8 spaces in the negative
direction and land on the –6
–2 + 8 = 6
-7 -6 -5 -4 -3 -2 -1
0
A
1
A
2
A
3 4
A
5
A
6
7
8
3) Here we calculate –7+3. So we start at the –7 position and since we are
adding 3 we move 3 places in the positive direction and land on –4
–7+3=-4
-7 -6 -5 -4 -3 -2 -1
A
0
1
A
2
A
3 4
A
5
A
6
7
www,mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2010
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8
3. Multiplying and Dividing with Positives and Negatives
You will probably notice that generally when we write positive numbers we
usually leave out the + that shows it is positive, so ‘positive 3 multiply positive 2’
would be written+3 x +2 = +6 or more simply as 3x2=6
There are three rules for multiplying and dividing numbers.
RULE 1: A positive number multiplied or divided by another positive number
gives a positive number (+ times + gives +)
Examples
1. Two lots of three apples are six: 2 x 3 = 6
2. Ten apples divided among five kids means two apples each: 10 ÷ 5 = 2
RULE 2: A positive multiplied or divided by a negative number gives a negative
number (+ times – gives –)
Examples
1. If I borrow £10 then I have –10 pounds. So if I borrow ten pounds four times I
owe £40 so I have –40 pounds: –10 x 4 = –40
2. If five people between them borrow ten pounds then everyone owes 2
pounds. We write this as –10 divided among 5: –10 ÷ 5 = –2
RULE 3: A negative number multiplied by a negative number gives a positive
number (– times – gives +).
Examples using all three rules
1) 6 x 3 = 18
4) –2 x 1 = –2
2) –3 x 6 = –18
6) 7 x -3 = –21
3) 4 x –2 = –8
7) 5 x 3 = 15
5) –4 x –2 = 8
8) –5 x –3 = 15
www,mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2010
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4. More Types of Numbers
So far we have only considered whole numbers (integers). But now we will
introduce other types of numbers.
Real Numbers are the numbers that fit between whole numbers on the
numberline. So if I’ve got between two and three of something the number
goes between the two and the three.
So when I zoom into my number line and divide up the spaces between the
whole numbers
-2
-1
0
1
2
I can add the following numbers:
HALF = 0.5 is between zero and one
ONE and a HALF = 1.5 is between two and three
MINUS HALF = –0.5 is between zero and minus one
MINUS ONE and a HALF = –1.5 is between minus one and minus two
–1.5 + 0.5 = –1.0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
And this leads us on to
Decimals: which is a system of dividing everything into tens and we’ll cover this
in the next section on decimals and fractions.
But notice that moving one marked place along the numberline now means
adding or subtracting by half = 0.5. So above we have shown –1.5+0.5 = –1.0
www,mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2010
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5. Now your turn
Generally the more maths you practice the easier it gets. If you make mistakes
don’t worry. I generally find that if I make lots of mistakes I understand the
subject better when I have finished. If you want to see videos explaining these
ideas and showing the answers visit www.mathsupport.wordpress.com
A) Adding and subtracting numbers
1) 2+3 =
4) –2–4 =
2) 2–3 =
5) 8–12 =
3) –6+7 =
6) –2–7 =
B) Multiplying and dividing whole numbers
1) 2 x 3 =
4) –10 ÷ –5 =
2) 4 ÷ –2 =
5) –7 x 2 =
3) –4 x –2 =
6) 3 ÷ 6 =
C) Adding and subtracting with decimals
1) 1.0 + 0.5 =
4) –7.5 – 1.5 =
2) –0.5 + 2.5 =
5) 2.5 – 2.0 =
3) –3.0 + 1.5 =
6) –4.0 – 2.5 =
D) Questions combining all types in word form
1) I buy a jacket for £15 but have a credit voucher for £7. How much do I pay?
2) I have a credit note for £20 and buy a skirt for £10. Do I have to pay extra or
do I get another credit note?
3) I have an overdraft of £12.50 and get charged £1.50. How much do I owe?
4) I am £4.50 overdrawn and then spend £3.50 on my card. If I then pay £10 into
my account how much money do I have in my account?
5) Three people agree to split the cost of lunch which is £12. How much do they
pay each?
www,mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2010
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