INTEGERS
Number line
When working with integers it is a good idea to think of a number line, with zero as the
reference point. A positive number would lie above, or to the right of the reference point
and a negative number would lie below or to the left of the reference point.
negative
-3-
positive
-2
-1
0
1
2
3
Practical examples of positive, negative and zero are:

Temperature, Bank Accounts, Altitude.
Positive integers
The numbers to the right of zero are called positive integers. (They are also known as
Natural Numbers). Examples of positive integers are +1 and +2. Usually we write 1 instead
of +1 and 2 instead of +2. Therefore the integer 3 means +3 which is read as "positive 3".
Negative integers
The numbers to the left of zero are called negative integers. Examples of negative integers
are –2 and –3. Notice the "-" sign in front of each number. –1 is read as "negative 1".
Zero





0 is a trouble maker! It requires special handling that no other number requires.
It is the only number that is neither positive nor negative.
0 is the only number that can never be used as a divisor in a division.
0 is the only number that must be one of the factors if it is the product of two
numbers
0 is the only number that doesn't affect the sum (when it is one of the numbers
being added) or affect the difference (when it is one of the numbers being
subtracted)
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Updated Feb 2013
Examples
Addition
8 + 0 = 8 etc.
In general if a is any number, a + 0 = a
Subtraction
8 - 0 = 8 etc.
In general if a is any number, a - 0 = a
8  0 = 0 etc.
Multiplication
In general if a is any number, a  0 = 0
Division
= 0 etc.
In general if a is any number (except zero),
= 0
What if zero is the divisor?
For example,
i.e.
= ?
= some unknown quotient (?)

?  0 = 8,
This says that some unknown number multiplied by itself equals 8.
We know that this cannot be true because any number multiplied by zero equals zero. So it
is not possible to divide a non-zero number by zero.
Try to divide any number by zero on your calculator. The display will give the error message
- E -. In general it is impossible to divide by zero.
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Order of integers

The statement "3 is greater than 2" can be written as 3  2

Similarly the statement "1 is less than 2" can be written as 1  2

When you look at the number line you see that the integers increase towards the
right, i.e. an integer on the left is less than an integer on the right.
 -2 is to the left of 1, therefore -2 is less than 1 i.e. -2  1
 -3 is to the left of –1, therefore –3 is less than –1, i.e. -3  -1
Addition and subtraction using the number line
 When you add a positive number using the number line, you count the number of places
that you are adding to the right.
 –2 + 4 = +2
+4 means move 4 places to the right
-3
-2
-1
0
1
2
3
 When you subtract a positive number using the number line, you count the number of
places that you are subtracting to the left.
 –1 - 2 = -3
-2 means move 2 places to the left
-3
-2
-1
0
1
2
3
 When you add a negative number, it is equivalent to subtracting a positive number. i.e.
+3 + (-1) = +3 - (+1) = +2
 When you subtract a negative number, the two minuses combine and become a plus.
i.e. 2 – (-1) = 2 + 1 = 3
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Multiplying integers
The rules for multiplying integers with different signs are:

+2  +3 = +6
i.e.
+  +
= +

+2  -3 = -6
i.e.
+  -
=
-  + = -

-2  -3 = +6
i.e.
-  -
=
+
Dividing integers
The rules for dividing integers are similar to those of multiplication

6
= +2
3
i.e.

= +


6
= -2
3
i.e.
 =

6
= +2
3
i.e.





= -
= +
Place values
The number 4672 can be written as follows:
4672 = 4000 + 600 + 70 + 2
The digit 4 represents 4 thousands
The digit 6 represents 6 hundreds
The digit 7 represents 7 tens
The digit 2 represents 2 ones (or units)
4
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6
7
Units (ones)
Tens
Hundreds
Thousands
Ten
Thousands
Hundred
Thousands
Millions
Ten Millions
Hundred
Millions
Billions
We can draw up a place value table as follows:
2
Sum, difference, product, quotient and remainder
We can draw up a place value table as follows:
 To find the sum of two numbers is to add two numbers
 The difference between two numbers can be found by subtracting the smaller number
from the bigger.
 In the multiplication 6  7 = 42, we call 42 the product of 6 and 7.
 When multiplying 15 by 3, 3 is called the multiplier.
 In the division 40  5 = 8, we call 8 the quotient and 5 the divisor.
 In 13  2, the quotient is 6. However there is a remainder of 1.
Tests of Divisibility
 Divisibility by 2
Numbers that are divisible by 2 are called even numbers, therefore an even number is
divisible by 2.
 Divisibility by 3
A number is divisible by 3 if the sum of the digits is divisible by 3.
e.g. 624  6 + 2 + 4 = 12 which is divisible by 3
624  3 = 208
 Divisibility by 4
A number is divisible by 4 if the last two digits are divisible by 4.
e.g. 1924, 24 is divisible by 4, so 1924 is divisible by 4
1924  4 = 481
 Divisibility by 5
A number is divisible by 5 if the last number is either 5 or 0.
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Updated Feb 2013
 Divisibility by 6
A number is divisible by six, if it is an even number and the sum of its digits is divisible by
3.
e.g. 2688, 2688 is an even number
the sum of its digits is 2 + 6 + 8 + 8 = 24 which is divisible by 3
2688  6 = 448
 Divisibility by 9
A number is divisible by 9 if the sum of the digits is divisible by 9
e.g. 5103  5 + 1 + 0 + 3 = 9 which is divisible by 9
5103  9 = 567
 Divisibility by 10
A number is divisible by 10 if the last digit is 0.
 Divisibility by 11
A number is divisible by 11 if the difference between the sum of alternate digits is either
0 or divisible by 11.
 1925
9 + 5 = 14
1
9
2
5  14 – 3 = 11
4
3  12 – 12 = 0
1+2=3
1925  1 1 = 175
 8943
8
9
8 + 4 = 12
8943  1 1 = 813
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Updated Feb 2013
 There is a quick way to multiply by 25.
25  4 = 100
Multiply the number by 100, namely add two 0s.
Then divide by 4.
 e.g. 52  25
52  100 = 5200
5200  4 = 1300
Integer exercises
1.
2.
Find the value of:
(a)
2–4 =
(b)
-3 – 5 =
(c)
9 + (-2) =
(d)
5 – 8 – 15 =
(e)
-3 + 7 – 13 =
(f)
0–5 =
(g)
(-6) – (-1) =
(h)
(-1) + (-2) – (-3) =
(i)
7  (-6) =
(j)
2  (-3)  4 =
(k)
4  (-5)  0 =
(l)
(-3)  (-2) =
(m)
+3  (-2)  (-1) =
(n)
(-2)  (-2)  (-2) =
(o)
9  (-3) =
(p)
(-28)  4 =
(q)
(-32)  (-4) =
(r)
(-3)  (-4)  (-2) =
Fill in the spaces with the correct number
(a)
__ + 3 = -5
(b)
__ + (-5) = 2
(c)
__ - (-8) = 6
(d)
8 + __ = 2
(e)
(-1) + __ = 20
(f)
1 + (-2) - __ = 3
(g)
(-2) + __ -4 = 5
(h)
4 – (-5) + __ = 7
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Updated Feb 2013
3.
4.
5.
6.
(i)
__  3 = -15
(j)
(-6)  __ = -30
(k)
5  (-4)  __ = -60
(l)
5  6  __ = -120
(m)
__  7 = 9
(n)
__  (-8) = 3
(o)
__  (-11) = -7
(p)
__  (-12) = 6
In each number, state the place vale of the digit which is in bold.
(a)
17
(b)
412
(c)
5 327
(d)
85 436
(e)
692 745
(f)
90 678 541
State the place value of the digit 7 in each of the following numbers.
(a)
5 462 027
(b)
70 634
(c)
2 788
(d)
337 228
(e)
12 234 670
Which of the following numbers is divisible by 2, 3 or 5?
(a)
30
(b)
45
(c)
132
(d)
225
(e)
9 997
(f)
9 018
(g)
53 523
(h)
840 412
Which of the following numbers are divisible by 4 or 9?
(a)
36
(b)
63
(c)
819
(d)
396
(e)
1 936
(f)
9 009
(g)
51 450
(h)
156 816
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Updated Feb 2013
7.
Which of the following numbers are divisible by 6 or 11?
(a)
66
(b)
174
(c)
187
(d)
92
(e)
858
(f)
165
(g)
2002
(h)
106 260
(a)
Your owe the bank $5000 on a car and your bank balance statement shows $5000. You win a prize with the bank where they take away the debt and give
you a bonus of $2000. What will your new bank balance be?
(b)
Temperatures often vary due to a wind chill factor. Presume you are on top of a
mountain and the air temperature is –10oC. The wind picks up and you hear that
the wind chill factor will drop the temperature by another 18oC. What will the
new apparent temperature be?
(c)
(i) Electrons have a charge of –1. If there is a neutral zinc atom and it loses
two electrons, what will its charge be?
(ii) A neutral chlorine atom gains one electron. What will its charge be?
8.
(iii)
(d)
An iron ion with a charge of +3 gains one electron. What will its charge be?
The thermostat in an oven is malfunctioning and the following sequence occurs:
The oven is set to 220oC
It heats up to 240oC, cuts off and automatically cools by 20 oC
It then heats up by 50 oC, cuts off and cools by 30 oC
It heats up by another 40 oC, cuts off and cools by 20 oC
What is the temperature at the end of the sequence?
(e)
An oil rig diver is living under water at 500 m below sea level. From his living
chamber he dives down to work at 30 m below the living chamber. How far
below sea level is he? Write this depth as a distance relative to sea level.
(f)
Make positive or negative numbers from the following:
(i)
two kilometres east
(ii)
five kilometres west
(iii)
ten kilometres west
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Updated Feb 2013
(iv)
going up five floors
(v)
starting at sea level and descending two hundred and fifty metres
(vi)
walking down forty two steps
(vii)
a temperature of three degrees centigrade above zero
(viii)
Lake Eyre is sixteen metres below sea level
(ix)
the Mariana Trench in the Pacific Ocean is eleven thousand and
twenty two metres deep
(x)
smoking reduces life expectancy by ten years
(g)
The temperature at 6am at Oakey on 10th July was – 4.2oC. The temperature
rose 10oC during the day. What was the maximum temperature at Oakey on the
day in question?
(h)
A bank statement showed a balance of $1000.90 on 1 June. During June cheques
were written for $79, $135, $80.81, $35.95 and $435. A deposit for $255 was
also made. What balance did the account show at the end of the month?
The bank made a mistake and paid $13.50 in error instead of $135. What was
the faulty bank balance?
Integer exercises solutions
1.
(a)
(e)
(i)
(m)
(q)
-2
-9
-42
6
8
(b)
(f)
(j)
(n)
(r)
-8
-5
-24
-8
-6
(c)
(g)
(k)
(o)
7
-5
0
-3
(d)
(h)
(l)
(p)
-18
0
6
-7
2.
(a)
(e)
(i)
(m)
-8
21
-5
63
(b)
(f)
(j)
(n)
7
-4
5
-24
(c)
(g)
(k)
(o)
-2
11
3
77
(d)
(h)
(l)
(p)
-6
-2
-4
-72
3.
(a)
ten
(b)
four hundred
(c)
twenty
(d)
five thousand
(e)
ninety thousand
(f)
ninety million
(a)
seven
(b)
seventy thousand
(c)
seven hundred
(d)
seven thousand
4.
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Updated Feb 2013
(e)
seventy
5.
(a)
(e)
2,3,5
none
(b)
(f)
3,5
2,3
(c)
(g)
2,3
3
(d)
(h)
3,5
2
6.
(a)
(e)
4,9
4
(b)
(f)
9
9
(c)
(g)
9
none
(d)
(h)
4,9
4,9
7.
(a)
(e)
6,11
6,11
(b)
(f)
6
11
(c)
(g)
11
11
(d)
(h)
none
6,11
8.
(a)
$2000
(b)
-28 oC
(c)
(i) +2
(d)
260 oC
(e)
530 m below sea level.
(ii) -1
(iii) +2
(f)
(i)
+2
(ii)
-5
(iii)
-10
(iv)
+5
(v)
-250
(vi)
-42
(vii)
+3
(viii)
-16
(ix)
–11 022
(x)
-10
(g)
5.8oC
(h)
$490.14,
$611.64
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Updated Feb 2013