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EXERCISE 1: (The need for integers.)
Note:
1.
N = {natural numbers} = {1, 2, 3, 4, . . .}
W = {whole numbers} = {0, 1, 2, 3, . . . }
2.
The elements of W can be represented on a line which starts at 0 (zero) and extends to the right.
3.
0 1 2 3 4 5 6 7 8 9
When adding a whole number to, say 3, we start at 3 and count to the right.
(e.g. 3 + 1 = 4, 3 + 2 = 5, 3 + 3 = 6, 3 + 4 = 7 etc)
When subtracting a whole number from 3 we start at 3 and count to the left.
(e.g. 3 – 1 = 2, 3 – 2 = 1, 3 – 3 = 0, 3 – 4 = ?)
4.
To find the value of 3 – 4, 3 – 5, 3 – 6 etc we must extend the whole number line to the left of
zero. This gives us what we call the integer number line.
-7
-6
5.
I = {integers} = {. . . . –3, -2, -1, 0 , 1, 2, 3, . . . }
6.
Other examples of number lines which extend in opposite directions from zero are –
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
 historical time lines showing A.D. and B.C. dates;
 temperature scales showing temperatures above and below zero;
 measuring sticks (attached to wharves) showing how far the tide is above or below sea
level;
 meters (in some cars) showing whether the battery is charging or discharging.
WORKED EXAMPLE:
A golfer’s scores on the first 9 holes are recorded as 1 over, 1 under, even, 2 over, 1 under, even, 2 over, 1
under and 1 under. Her scores on the second 9 holes are recorded as 1 over, even, 1 under, even, 2 under, 1
under, even, 1 over and 1 under. Par for the whole round (18 holes) is 72.
a.
Her score for the first 9 holes is 1 over (5 over and 4 under).
b.
Her score for the second 9 holes is 3 under (2 over and 5 under).
c.
Her score for the whole round (18 holes) is 2 under i.e. 72 – 2 = 70.
ORAL EXAMPLES:
1.
The water level of Lake Manapouri is 180 metres above sea level.
a. In some parts of the lake the bottom is 70 metres above sea
level. How deep is the lake in these parts?
b. The deepest part of the lake is 70 metres below sea level.
What is the lake’s greatest depth?
2.
Augustus (the first Roman Emperor) was born in 63 BC and became emperor in 27 BC
until he died in 14 AD.
BC
200
100
0
63 BC
100
200
AD
14 AD
27 BC
DS/GRB/JMH: June 17, 2017
Page 1 of 11
a.
b.
c.
How old was he when he became emperor?
For how many years was he the emperor?
How old was he when he died?
WRITTEN EXAMPLES:
1.
The temperature of a liquid at the
start of an experiment is 15 oC.
Copy down and complete this table
by writing down the reading on the
thermometer after each temperature
change (the first one has been done
for you - use BZ for temperatures
below zero.)
rises
rises
falls
falls
falls
rises
rises
o
10 C
5 oC
20 oC
20 oC
5 oC
10 oC
20 oC
Deposit
2.
A bank statement opens with a
credit balance of $250. Copy down
and complete this table by writing
down the balance after each
transaction (the first one has been
done for you - use OD to represent
overdrawn).
3.
Reading
15 oC
25 oC
Withdrawal
$100
$50
40 oC
30 oC
20 oC
10 oC
0 oC
10 oC
20 oC
above
zero
below
zero
Balance
$250
$350
$200
$400
$50
$200
$300
Use this diagram to help you answer the question.
(Note: R4 means 4 to the right of zero. L4 means 4 to the left of zero).
Shift
Copy down and complete the table by writing
down your position after each shift starting at 0
(i.e. zero)
- the first two have been done for you.
(Note: 7R means a shift of 7 steps to the right.
3L means a shift of 3 steps to the left.)
7R
3L
6L
4L
5R
1R
Copy down and complete this "integer number line."
4.
Note:
Position
0
R7
R4
3 can be written as +3 (called positive 3) to distinguish it from - 3(called
negative 3). 3 (or +3) and - 3 are said to be opposites of each other.
DS/GRB/JMH: June 17, 2017
Page 2 of 11
5.
Copy down and complete these statements.
a.
c.
e.
g.
i.
6.
7.
the opposite of 3 is _____
the opposite of 6 is _____
the opposite of _____ is - 2
the opposite of _____ is 7
the opposite of y is _____
b.
d.
f.
h.
j.
the opposite of - 4 is _____
the opposite of - 8 is _____
the opposite of _____ is - 5
the opposite of _____ is 9
the opposite of - z is _____
Copy down and complete these statements by putting <, > or = in the
circles.
+
a. - 4 4
b.
3 -3
c.
2 +2
d.
2 2
e.
f.
3 4
g.
2 4
h.
5 3
2 5
i.
6 4
j.
8 5
k.
3 7
l.
m. 2 - 3
n.
5 4
o.
6 -7
p.
9 8
q. 6 9
r.
6 9
s.
6 -9
t.
6 -9
u. 8 7
v.
8 7
w. 8 7
x.
8 -7
+
y. +5 5
z.
5 -5
-
2
-
9
Copy down and complete these statements (using the number line drawn above or
number patterns to help you).
a.
3+ 3=
3+ 2=
3+ 1=
3+ 0=
3 + -1 =
3 + -2 =
3 + -3 =
3 + -4 =
3 + -5 =
3 + -6 =
b.
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
+ 3=
+ 2=
+ 1=
+ 0=
+ -1 =
+ -2 =
+ -3 =
+ -4 =
+ -5 =
+ -6 =
8.
Copy down and complete this addition table.
9.
Copy down and complete these statements.
a.
When adding a positive integer, count to the _____ on the number line.
b.
When adding a negative integer, count to the __
on the number line.
10. Copy down and complete these statements.
a.
A positive integer added to a positive integer gives a _____ integer.
b.
A negative integer added to a negative integer gives a _____ integer.
EXERCISE 2: (Integer Addition)
WRITTEN EXAMPLES: Find these sums.
1. 4 + - 2
2. 4 + - 3
3.
4+3
5. 4 + 2
6.
7.
4+ 3
9. 4 + 2
10.
11.
4+ -4
4+4
4+ -4
4.
8.
12.
4 + -5
4+5
4 + -5
DS/GRB/JMH: June 17, 2017
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13.
17.
21.
25.
29.
33.
37.
41.
45.
49.
53.
57.
61.
65.
69.
6+ -4
6+ 4
6+ -4
5 + -1
6+ -3
7+ -5
8+ -7
6 +1
8+2
3+ 7
4+9
1+ - 4
4+7
1+ - 3+ 5
5+ -7+9
14.
18.
22.
26.
30.
34.
38.
42.
46.
50.
54.
58.
62.
66.
70.
6+ -5
6+5
6+ -5
5 +1
6+ 3
7+5
8+ 7
6 + -1
8+ -2
3+ - 7
4 + -9
5+ -3
5+ -8
2+ -4 + -6
1+ - 4 + - 7
15.
19.
23.
27.
31.
35.
39.
43.
47.
51.
55.
59.
63.
67.
71.
6+ -6
6+6
6+ -6
5+ -9
6+ -9
7+ -9
8+ -9
6 + -1
8+ -2
3+ - 7
4 + -9
6+ -2
3+ 9
3+ 5 + - 7
2+ 5 + -8
16.
20.
24.
28.
32.
36.
40.
44.
48.
52.
56.
60.
64.
68.
72.
6+ -7
6+ 7
6+ -7
5+9
6+9
7+9
8+9
6 +1
8+2
3+ 7
4+9
8+6
2+ -7
4 + -6+ -8
3+ - 6 + - 9
Copy down and complete these statements.
*73.
*74.
i.
ii.
i.
ii.
Given that 79 + 86 =165 then - 79 + - 86 = ?
In general, if a +b = c then - a + -b = ?
Given that 165 + - 86 = 79 then -165 + 86 = ?
In general, if a + -b = c then - a + b = ?
Copy down and complete these addition tables.
*75
*76
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Page 4 of 11
EXERCISE 3: (Integer Subtraction)
WRITTEN EXAMPLES: Copy down and complete these statements (using number patterns or
the number line to help you).
1. a.
5-1=
b.
5 + -1 =
2.
a.
5-4=
b.
5 + -4 =
5-2=
5 + -2 =
5-3=
5 + -3 =
5-3=
5+ 3=
5-2=
5 + -2 =
5-4=
5 + -4 =
5-1=
5 + -1 =
5-5=
5 + -5 =
5-0=
5+0=
5-6=
5+ 6=
5- 1=
5+1=
5-7=
5+ 7=
5- 2=
5+2=
5-8=
5 + -8 =
5 - -3 =
5+3=
5-9=
5+ 9=
5- 4=
5+4=
Note:
The statements in 1a. and 2a. can be checked by adding back.
Copy down and complete these statements.
3.
4.
The last example in 1a. and 1b. shows that instead of subtracting 9 we can add _____.
The last example in 2a. and 2b. shows that instead of subtracting -4 we can add _____.
Hence, instead of subtracting an integer we can add the _____ integer.
i.e. first number – second number = first number + ______________ second number.
Note:
Be careful to use the symbols for minus and negative correctly.
Minus (-) is an operation on 2 numbers (e.g. 8 - 2)
Negative (-) indicates a number less than zero. (e.g. -3)
Thus 0 - 3 = -3
Find these differences.
5. 4 - 6
6.
9. 9 - 2
10.
13. 7 - 7
14.
17. 1 - 2
18.
21. 3 - 2
22.
25. 1 - 3
26.
29. 4 - 1
30.
33. 4 - 4
34.
37. 2 - 9
38.
41. 8 - 2
42.
45. -4 - -9
46.
49. 6 - 4
50.
53. 5 - 8
54.
57. -1 + 5 - 9
58.
61. 3 + 7 - 11
62.
4 - -6
9 - -2
7 - -7
3 - -4
5 - -4
3 - -5
5 - -2
6 - -6
-1 - -5
6-2
-3 - 8
7 - -3
6 - -9
-1 + -5 - -9
-3 + -7 - -11
7.
11.
15.
19.
23.
27.
31.
35.
39.
43.
47.
51.
55.
59.
63.
-4
-6
9-2
-7 - 7
-5 - 6
-7 - 6
-5 - 7
-6 - 1
-3 - 3
1-8
-9 - -1
4 - -7
-8 - 4
-6 - 8
2 + 6 - -10
4 + 8 - -12
8.
12.
16.
20.
24.
28.
32.
36.
40.
44.
48.
52.
56.
60.
64.
-4 - -6
-9 - -2
-7 - -7
-7 - -8
-9 - -8
-7 - -9
-7 - -2
-5 - -5
-2 - 4
7 - -1
3-6
-9 - -3
-5 - -9
2 + -6 - 10
4 + -8 - 12
Copy down and complete these tables by subtracting the numbers in the top row from those in
the left hand column.
EXERCISE 4: (Integer Multiplication)
DS/GRB/JMH: June 17, 2017
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WRITTEN EXAMPLES:
1. Copy down and complete these statements (using patterns to help you).
a.
b.
4´4 =
4´4 =
-
4´3=
4´2 =
4 ´1=
4´0 =
2.
4´3=
4´2 =
4 ´1 =
4´0 =
4 ´ -1 =
4 ´ -1 =
4 ´ -2 =
4 ´ -2 =
4´ -3=
4´ -3=
4´ 4 =
4´ -4 =
Copy down and complete this multiplication table.
-
3.
Copy down and complete these statements.
a. A positive integer multiplied by a positive integer gives a _____ integer.
b. A positive integer multiplied by a negative integer gives a _____ integer.
4.
Copy down and complete these statements.
a. A negative integer multiplied by a positive integer gives a _____ integer.
b. A negative integer multiplied by a negative integer gives a _____ integer.
Find these products:
5. 2 ´ - 3
9. - 4 ´ 3
13. 6 ´ - 3
17. - 7 ´ 3
21. - 6 ´ - 5
25. - 5 ´ - 7
29. - 9 ´ - 5
33. - 7 ´ - 9
37. 5 ´ - 5
41.
45.
49.
53.
-
1´ - 2 ´ 3
2 ´ - 3´ - 4
1´ 3´ - 5
5´ 4´ 3
6.
10.
14.
18.
22.
26.
30.
34.
38.
42.
46.
50.
54.
-
1´ 4
2´ -6
-
2´ -7
2´ -9
-
7´4
4 ´ -9
6´ -8
9´ -8
6´6
-
1´ 2 ´ 3
2 ´ - 3´ 4
1´ - 4 ´ - 7
4 ´ 5´ -6
7.
11.
15.
19.
23.
27.
31.
35.
39.
43.
47.
51.
55.
-
5´ -2
3´ - 5
5´ 4
4 ´ -6
-
-
9´ 3
8´ -5
9´6
3´ - 3
7´ -7
-
1´ 2 ´ - 3
2 ´ 3´ 4
2´ -4´6
5´ -6´ -7
8.
12.
16.
20.
24.
28.
32.
36.
40.
44.
48.
52.
56.
-
6 ´ -1
4 ´ -2
8´ -2
-
3´ - 8 -
8´ -4
7´6
8´ -7
4´4
2
(-8)
-
1´ - 2 ´ - 3
2 ´ 3´ - 4
2´5´8
5 ´ - 8 ´11
DS/GRB/JMH: June 17, 2017
Page 6 of 11
EXERCISE 5: (Integer Division)
WRITTEN EXAMPLES: Copy down and complete these statements.
1.
2´ 3 = 6 and so 6 ¸ 3 =_____ which shows that a positive integer divided by a
positive integer gives a _____ integer.
2.
2 ´ 3 = - 6 and so - 6 ¸ 3 = _____ which shows that a negative integer divided by
a positive integer gives a _____ integer.
2 ´ - 3 = 6 and so 6 ¸ - 3 = _____ which shows that a positive integer divided by a
3.
negative integer gives a _____ integer.
4.
2 ´ - 3 = - 6 and so - 6 ¸ - 3 = _____ which shows that a negative integer divided
by a negative integer gives a _____ integer.
5.
9.
13.
17.
21.
25.
29.
33.
37.
41.
45.
Find these quotients:
6.
6¸2
12 ¸ 3
10.
16 ¸ 8
14.
-21 ÷ -7
18.
-27 ÷ 9
22.
-35 ÷ -7
26.
45 ÷ 9
30.
-56 ÷ -7
34.
4´ 3
38.
2
8´3
42.
4
2´ -8
46.
4
EXERCISE 6:
WORKED EXAMPLES:
( 4)
( 4)
a.
c.
-
8¸ -4
12 ¸ - 6
18 ¸ - 2
-24 ÷ 3
-28 ÷ -4
36 ÷ -4
-48 ÷ -8
-63 ÷ 9
6´2
3
6´4
8
5´ -6
3
7.
11.
15.
19.
23.
27.
31.
35.
39.
43.
47.
9.
13.
10 ¸ - 2
15 ¸ 5
20 ¸ - 4
25 ÷ -5
-32 ÷ -8
-42 ÷ 6
- 54 ÷ -6
81 ÷ -9
3´ - 6
40.
9
4 ´ -5
44.
10
8´5
48.
4
8.
12.
16.
20.
24.
28.
32.
36.
-
Find the value of the following.
-
2
= - 4 ´ - 4 =16
b.
-
4 2 = -1´ 4 2 = -16
-
3
= - 4 ´ - 4 ´ - 4 =16 ´ - 4 = - 64
d.
-
4 3 = -1´ 4 3 = -1´ 64 = - 64
WRITTEN EXAMPLES: Evaluate
2
3
1. ( -1)
2. ( -1)
5.
9¸ -3
14 ¸ - 2
18 ¸ 3
-24 ÷ -6
30 ÷ -5
-40 ÷ -5
- 49 ÷ 7
-72 ÷ - 8
2´9
6
2 ´ -10
5
4 ´ -9
6
(-2)
2
(-3)
2
(-6)
2
6.
10.
14.
- 2
3.
(-2)
3
(-3)
2
(-7)
3
7.
11.
15.
-
(-1)
4
(-2)
2
(-5)
2
(-8)
4
4.
8.
12.
16.
-
(-1)
5
(-2)
3
(-5)
2
(-9)
5
-
17.
21.
25.
1
- 3
1
2
(-10)
18.
22.
26.
22
- 3
2
3
(-10)
19.
23.
27.
32
- 3
3
4
(-10)
20.
24.
28.
52
- 3
5
5
(-10)
29.
(-20)
30.
(-20)
31.
(-30)
2
32.
(-30)
2
35.
4 ´ - 52
36.
-
4 ´ - 52
3
39.
8 ´ - 53
40.
-
8 ´ - 53
2
( 5)
8 ´ ( 5)
3
33. 4 ´
-
2
34.
-
4 ´ ( - 5)
37.
-
3
38.
-
8 ´ ( - 5)
3
Given that 210 =1024 , find the value of
DS/GRB/JMH: June 17, 2017
Page 7 of 11
41.
45.
( 2)
10
-
-
211
42.
-
46.
29
210
43.
211
47.
( 2)
51.
46
55.
( 4)
-
44.
9
48.
Given that 4 5 =1024 , find the value of
49.
53.
( 4)
-
-
5
46
50.
-
54.
44
45
* EXERCISE 7
WRITTEN EXAMPLES: Find the value of
a. x - y + z
b. x - ( y + z )
e. x - y ´ z
f. ( x - y) ´ z
52.
-
4
x ¸ y´ z
x+y¸z
c.
g.
56.
( 2)
11
-
-
29
( 4)
-
-
6
44
x ¸ ( y ´ z)
( x + y) ¸ z
d.
h.
1. x = 9, y = - 3, z = 3
where
2. x = - 6, y = 2, z = -1
3. x = - 8, y = - 4, z = - 2
REVISION EXERCISE:
1. 6+- 3
2.
5. 6 + 3
6.
9. 6+ 3
10.
2
13. ( 4 )
14.
17. 2 ´ (- 5)
2
18.
-
3.
7.
11.
15.
6- 3
6-3
6- 3
3
( -4)
-
-
2 ´ ( 5)
-
2
-
6´ 3
6´3
6´ 3
- 2
4
4.
8.
12.
16.
19.
2 ´ - 52
20.
24.
-
If 310 = 59049 find the value of
21.
25.
( 3)
-
10
- 11
3
-
2´ 5
11
3
23.
311
26.
39
27.
( 3)
9
28.
-
31.
35.
39.
43.
x´y
y´ z
x´z
z2
32.
36.
40.
44.
x¸y
y¸z
x¸z
5z 2
47.
51.
55.
5z 3
x - y+ z
x ¸ y´ z
48.
52.
56.
- 3
46.
50.
54.
z
x+y-z
x´y¸z
3
)
-
2
( 3)
- 10
(
45. z
49. x + y + z
53. x ´ y ´ z
-
22.
If x = -8,y = 4 and z = -2 find the value of
29. x + y
30. x - y
33. y + z
34. y - z
37. x + z
38. x - z
41. 5z
42. - z = -1´ z
- 2
-
6¸ 3
6¸3
6¸ 3
- 3
4
-
-
39
z
x-y-z
x¸y¸z
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Page 8 of 11
TEST EXERCISE
In questions 1 – 20 simplify
1.
2.
8+- 2 =_____
8-- 2 =_____
(40 marks)
3.
8´- 2 =_____
4.
8¸- 2 =_____
5.
-
8 + 2 =_____
6.
-
8 - 2 =_____
7.
-
8 ´ 2 =_____
8.
-
8 ¸ 2 =_____
9.
-
8+- 2 =_____
10.
-
8-- 2 =_____
11.
-
8´- 2 =_____
12.
-
8¸- 2 =_____
13.
( - 9)2 =_____
14.
- 2
15.
( - 9)3 =_____
16.
- 3
17.
6 ´(- 5)2 =_____ 18.
6´- 52 =_____
20.
-
24.
- 5
9 =_____
-
6 ´(- 5)2 =_____ 19.
9 =_____
6´- 52 =_____
Given that 56 =15 625 find the value of
21. ( - 5)6 =_____
22. - 56 =_____
23.
( - 5)5 =_____
If x=- 4, y=- 2 and z=- 1 find the value of
25. x + y + z =____________________
26.
x + y - z =____________________
=________
27.
x - y + z =____________________
=________
28.
=________
29.
x ´ y ´ z =____________________
x ¸ y ´ z =____________________
30.
32.
a - b ´ c =____________________
34.
a + b ¸ c =____________________
36.
a - b ¸ c =____________________
=________
(a - b) ´ c =____________________
=________
38.
=________
39.
(a + b) ´ c =____________________
=________
=________
37.
x ¸ y ¸ z =____________________
=________
=________
35.
x ´ y ¸ z =____________________
=________
=________
If a=- 6, b = 4 and c=- 2 find the value of
33. a + b ´ c =____________________
x - y - z =____________________
=________
=________
31.
5 =_____
(a + b) ¸ c =____________________
=________
40.
(a - b) ¸ c =____________________
=________
DS/GRB/JMH: June 17, 2017
Page 9 of 11
EXTRA EXERCISE
1. a. Evaluate (if possible).
b.
2. a.
b.
3. a.
(i)
(ii)
(iii)
64
64
For what values of a does a exist?
4. a.
64
(iv)
3 -
(iv)
-
64
Evaluate (if possible).
(i)
(ii)
25 ´ 4
25 ´ 4 (iii)
25 ´ - 4
For what values of a and b does a ´ b = a ´ b ?
25 ´
-
4
Evaluate (if possible).
(i)
b.
3
36
9
(ii)
-
-
36
9
36
9
(iii)
(iv)
Evaluate (if possible).
( 7)
2
(ii)
( 7)
-
b.
For what values of a does
c.
For what values of a does
9
a
a
?
=
b
b
For what values of a and b does
(i)
36
-
2
(iii)
( 16 )
2
(iv)
(
-
16
)
2
a2 = a ?
( a)
2
= a?
DS/GRB/JMH: June 17, 2017
Page 10 of 11
5.
In this question you are to find two numbers given their sum and product. Copy down and
complete each table. (The first four examples have been done for you - check these answers).
Product
4
4
-5
-5
6
6
-6
-6
24
24
6.
a.
Sum
5
-5
4
-4
5
-5
5
-5
10
-10
Numbers
4 and 1
-4 and -1
5 and -1
-5 and 1
Product
-24
-24
1
1
-2
-2
4
4
-4
-9
Sum
10
-10
2
-2
1
-1
4
-4
0
0
Numbers
Copy down and complete the following tables. (The first 2 lines of each table have been
done for you – check these).
x
-1
0
1
2
3
4
5
6
7
6-x
x (6 - x )
6 - -1 = 7 -1 x 7 = -7
6-0=6 0x6=0
x
-5
-4
-3
-2
-1
0
1
2
3
x +4
x -2
(x + 4)(x - 2)
-5 + 4 = -1 -5 - 2 = -7
-1 x -7 = 7
-4 + 4 = 0 -4 - 2 = -6
0 x -6 = 0
Use the tables above to help you copy down and complete the statements in (b), (c) and (d).
b.
c.
d.
i.
ii.
If x = 0 then x (6 - x ) =
If x = 0 then (x + 4)(x - 2) =
.
i.
ii.
If x (6 - x ) = 0 then x =
If (x + 4)(x - 2) = 0 then x =
or
i.
ii.
The maximum value of x (6 - x ) =
The minimum value of (x + 4)(x - 2) =
.
.
or
.
and it occurs when x =
.
and it occurs when x =
.
DS/GRB/JMH: June 17, 2017
Page 11 of 11