1
Whole numbers
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The number system
Rounding and estimation
Operating with numbers
Square numbers and square roots
Order of operations
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Contents:
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10
WHOLE NUMBERS
(Chapter 1)
In this chapter you will study the “Hindu-Arabic” number system which is used extensively
throughout the modern world. However, you may also be interested in other number systems
used by ancient cultures.
THE EGYPTIAN NUMBER SYSTEM
There is archaeological evidence that as long ago as
3600BC the Egyptians were using a detailed number
system. The symbols used to represent numbers were
pictures of everyday things. These symbols are some
of their hieroglyphics or ‘sacred picture writings’.
The Egyptian number system is an example of a base
ten system because each symbol stands for ten of the
previous symbols.
1
10
100
1000
10 000
100 000
staff
hock
scroll
lotus flower
bent stick
burbay fish
This system does not have place
values as our system does.
For example, 214 could be written as
or
1 000 000
astonished man
or
This system needs many symbols for certain numbers, such as 999.
DISCUSSION
²
How would you represent the number 645 or 7013?
²
What number is represented by
?
THE ROMAN NUMBER SYSTEM
The Romans used the following symbols for their number system:
1
5
10
50
was
was
was
was
represented by
represented by
represented by
represented by
100 was represented by C
500 was represented by D
1000 was represented by M
I
V
X
L
Unlike the Egyptian system, this system did have to be written in order as the value would
change if the order changed.
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VI stands for 1 after 5 or 6.
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IV stands for 1 before 5 or 4, whereas
For example,
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WHOLE NUMBERS
The system could be simplified to use less symbols.
For example, 14 is written XIIII or XIV.
The Roman Empire spread to many parts of the world. Roman
numbers are still used today on some watches and clocks, and they
are sometimes used to write dates on buildings and memorials.
11
(Chapter 1)
XI
XII
X
II
IX
VIII
VII
I
III
VI
IV
V
DISCUSSION
²
²
How would you represent the number 645 or 7013?
What number is represented by CXVII?
Any study of Mathematics starts with these questions about numbers. How do we write them
and how do we use them or operate with them?
OPENING PROBLEM
A sixteen storey hotel with floors G, 1, 2, 3, ....., 15 has no accommodation
on the ground floor. On the even numbered floors (2, 4, 6, ......) there are 28
guest rooms and on the odd numbered
floors there are 25 guest rooms. Room
cleaners work for four hours each day, during which
time each cleaner can clean 12 guest rooms. Each
cleaner is paid at a rate of $16 per hour.
Consider the following questions:
² How many floors are odd numbered?
² How many guest rooms in total are on all
the odd numbered floors?
² If each guest room has three chairs, how
many chairs are on an even numbered floor?
² How many guest rooms are in the hotel?
² How many cleaners are required to clean all guest rooms assuming the hotel was
‘full’ the previous night?
² What is the total cost of hiring the cleaners to clean the guest rooms of the hotel?
Problems like the one above require an understanding of numbers, their properties, and the
operations between the numbers.
WORDS WE USE
Many words used in mathematics have special meanings and we should not avoid using them.
It is important to learn what each word (or phrase) means and to use it correctly.
When we write any number, we write some combination of the ten symbols: 1, 2, 3, 4, 5,
6, 7, 8, 9 and 0. These symbols are called digits.
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The numbers we use for counting, such as 1, 2, 3, 10, 100, 550 000 etc, are usually called
the counting numbers.
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12
WHOLE NUMBERS
(Chapter 1)
The mathematical name for the counting numbers is the natural numbers. The set of these
natural numbers is endless. There is no largest natural number. We say the set of all natural
numbers is infinite.
If we add the number zero (0) to the set of natural numbers, then our new set is the set of
whole numbers.
A
THE NUMBER SYSTEM
A method of writing numbers is called a number system. The system we use was developed
in India 2000 years ago and introduced to Europe by Arab traders about 1000 years ago. We
therefore call our system the Hindu-Arabic system.
There are three features which make this system useful and more efficient than other systems
such as those used by the Egyptians or Romans.
²
²
It uses only 10 digits to construct all the natural numbers.
It has a place value system where digits represent different numbers when placed in
different place value columns.
It uses the digit 0 to indicate a void in a place value.
²
The combined digits we use to represent numbers are called numerals. The digits 5 and 3
combine to form both the numeral 53 for the number “fifty three” and the numeral 35 for the
number “thirty five”.
PLACE VALUES
The place (or position) of a digit in a number determines its value.
For example, 5378 is really
5 thousand
5000
3 hundred
300
+
Reminder:
and
+
seventy
70
1
10
100
1000
units
tens
hundreds
thousands
eight
8
+
this is expanded form
ten thousands
hundred thousands
millions
ten millions
Example 1
Write in numeral form the number “three thousand, two hundred and
seven”.
What number is represented by the digit 6 in the numeral 1695?
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b six hundred
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a 3207
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Self Tutor
25
a
10 000
100 000
1 000 000
10 000 000
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WHOLE NUMBERS
(Chapter 1)
13
EXERCISE 1A
1 Write in numeral form:
a forty seven
b six hundred and forty eight
c seven hundred and one
d three thousand, four hundred and forty eight
e six hundred and twenty five thousand, nine hundred and ninety
f three million, six hundred thousand, nine hundred and seventy three.
2 When writing out a cheque to pay a debt, the amount must be written in numbers and
words. Write the following amounts in words:
a $91
b $362
c $4056
d $9807
e $43 670
f $507 800
3 What number is represented by the digit 7 in the following?
a 47
b 67
c 372
e 4709
f 17 000
g 3067
i 175 236
j 5 700 000
k 67 000 000
4 Write the following numbers:
a one less than nine
d 2 more than 3000
5 Put
a
b
c
d
e
b
e
two greater than ten
c
the largest two digit number.
a
b
Express 50 000 + 6000 + 70 + 4 in simplest form.
Write 6807 in expanded form.
a
b
50 000 + 6000 + 70 + 4 = 56 074
6807 = 6000 + 800 + 7
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7 Write in expanded form:
a 730
b
0
one less than 200
Self Tutor
6 Express the following in simplest form:
a 90 + 7
c 8000 + 4
e 70 000 + 60 + 5
5
702
370 000
146 070
the following numbers in order beginning with the smallest:
Kylie 57 kg, Amanda 75 kg, Sarah 49 kg, Lindy 60 kg
Josh 183 cm, Gavin 148 cm, Tony 138 cm, Matt 184 cm
$1100, $1004, one thousand and forty dollars
Barina 708 kg, Laser 880 kg, Excel 808 kg, Corolla 890 kg
forty dollars, forty four dollars, fourteen dollars, fifty four dollars,
forty five dollars.
Example 2
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14
WHOLE NUMBERS
(Chapter 1)
a Use the digits 7, 1, and 9 once only to make the largest number you can.
b Write the largest number you can using the digits 3, 1, 0, 4, 5, and 7 once only.
8
B
ROUNDING AND ESTIMATION
ROUNDING
Often we do not need or cannot find an exact value of a number, but rather we want a
reasonable estimate of it.
For example, we can estimate the distance to the sun, or the population of Earth.
One way of doing this is to round off numbers.
We may round off numbers by making them into, for example, the nearest number of tens.
157 is approximately 16 tens or 160
153 is approximately 15 tens or 150
We say 157 is rounded up to 160 and 153 is rounded down to 150.
We can use the symbol + or ¼ to mean
“is approximately equal to”.
+ and » are both
used to represent the
phrase ‘is approximately
equal to’.
Thus, 157 + 160:
When a number is halfway between tens we always round
up, i.e., 155 + 160.
ROUNDING RULES
The rules for rounding off are:
²
If the digit after the one being rounded off is less than 5 (i.e., 0, 1, 2, 3 or 4)
we round down.
²
If the digit after the one being rounded off is 5 or more (i.e., 5, 6, 7, 8 or 9)
we round up.
Example 3
Self Tutor
6705 to the nearest 100
15 579 to two figures
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3143 + 3000
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100
fto nearest 100g
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6705 + 6700
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fto nearest 10g
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769 + 770
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Round off:
a 769 to the nearest 10
c 3143 to one figure
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WHOLE NUMBERS
(Chapter 1)
15
EXERCISE 1B
1 Round off to the nearest 10:
a 43
b 65
e 199
f 451
c
g
98
797
d
h
147
9995
2 Round off to the nearest 100:
a 87
b 369
e 991
f 1426
c
g
442
11 765
d
h
650
34 037
3 Round off to the nearest 1000:
a 784
b 5500
e 12 324
f 23 497
c
g
7435
53 469
d
h
9987
670 934
4 Round off to one figure:
a 69
b
e 963
f
197
2555
c
g
293
6734
d
h
347
39 500
5 Round off to two figures:
a 891
b
e 561
f
166
5647
c
g
750
9750
d
h
238
23 501
6 Round off to the accuracy given:
$35 246 (to the nearest $1000)
a distance of 3651 km (to the nearest 100 km)
a weekly salary of $375 (to one figure)
last year a company’s profit was $237 629 (to the nearest $10 000)
the population of a town is 16 723 (to the nearest thousand)
the number of people at a football match is 35 381 (to two figures)
a
b
c
d
e
f
PUZZLE
ROUNDING WHOLE NUMBERS
Click on the icon to obtain a printable version of this puzzle.
PUZZLE
Round the numbers to the given amount
1
2
Across
1
4866
4
64
5 10 938
7 27 194
8
85
10 2629
3
4
5
6
7
8
9
10
to
to
to
to
to
to
the
the
the
the
the
the
nearest
nearest
nearest
nearest
nearest
nearest
10
10
100
1000
10
1000
Down
1
44
2 7247
3 751
4 550
5 165
6 8500
7 293
9
45
to
to
to
to
to
to
to
to
the
the
the
the
the
the
the
the
nearest
nearest
nearest
nearest
nearest
nearest
nearest
nearest
10
100
100
100
10
1000
10
10
ESTIMATION
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To avoid errors it is important to make estimates of the answers to problems. An estimate is
not a guess. It is a quick and easy approximation of the correct answer.
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16
WHOLE NUMBERS
(Chapter 1)
By making an estimate we can tell if our answer is reasonable, particularly when we are using
our calculator and may have entered the numbers incorrectly.
When estimating we usually round to the first digit and put zeros in the other places.
Example 4
Self Tutor
Find the approximate value of 7235 £ 591.
We round off to the first digit and put zeros in the other places.
7235 £ 591 + 7000 £ 600
+ 4 200 000
The estimate tells us the correct answer should have 7 places in it.
We expect the answer to be about 4 million.
7 Estimate the following using 1 figure working:
a 389 £ 63
b 4619 £ 22
d 389 £ 2178
e 588 £ 11 642
c
f
Example 5
4062 £ 638
29 £ 675 328
Self Tutor
3946 ¥ 79
+ 4000 ¥ 80
+ 400 ¥ 8
+ 50
Find the approximate value
of 3946 ¥ 79.
fusing 1 figure workingg
fdividing each number by 10g
8 Estimate the following using 1 figure working:
a 641 ¥ 59
b 2038 ¥ 49
d 2780 ¥ 41
e 85 980 ¥ 299
c
f
5899 ¥ 30
36 890 ¥ 786
9 In the following questions, round the given data to one figure to find the approximate
value asked for.
a Tracy delivers 405 papers on a paper round.
She does this every week for a year. Find
an approximation for the number of papers
delivered in the year.
b In an orchard there are 103 orange trees in
each row. There are 58 rows.
Find the approximate number of orange trees
in the orchard.
c If a trip of 1023 km from Adelaide to Canberra took 19 hours, find my approximate
average speed in kilometres per hour.
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d If Joe can type at 52 words per minute, find an approximate time needed for him
to type a document of 3920 words.
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WHOLE NUMBERS
(Chapter 1)
17
e Tim counted 42 jelly beans in the bottom layer
of a jar and thinks that there are 38 layers in
the jar. What is his estimate of the number of
jelly beans in the jar?
f Sally earned $404 per week for 7 months of
the year. Estimate the amount of money she
earned.
LIBRARY RESEARCH
Research the following and round off to the accuracy requested.
Do not forget to record the name of the reference (book/magazine title), the
value given in the reference, and your rounded value.
² The population of your nearest capital city (nearest 10 000).
² The distance of a marathon run (nearest km).
² The distance between Adelaide and Melbourne (nearest 100 km).
² The population of Australia (nearest 100 000).
² The population of the world (nearest billion).
² The distance to the sun (nearest million km).
C
OPERATING WITH NUMBERS
There are four basic operations that are carried out with numbers:
Addition
Subtraction
Multiplication
Division
+
¡
£
¥
to
to
to
to
find
find
find
find
a
a
a
a
sum
difference
product
quotient
Here are some words which are frequently used with these operations:
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numbers being added or subtracted
the result of a multiplication
numbers which divide exactly into another number
the result of a division
the number by which we divide
the number being divided
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< 2 is the quotient
3 is the divisor
in 6 ¥ 3 = 2
:
6 is the dividend
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14 is the product of 2 and 7
2 and 3 are factors of 6
25
in 2 £ 7 = 14
in 2 £ 3 = 6
0
3 and 5 are terms
8, 2 and 6 are terms
100
terms
product
factors
quotient
divisor
dividend
For example,
in 3 + 5
in 8 ¡ 2 + 6
5
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100
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²
²
²
²
²
²
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18
WHOLE NUMBERS
(Chapter 1)
SUMS AND DIFFERENCES
²
To find the sum of two or more numbers, we add them.
For example, the sum of 3 and 16 is 3 + 16 = 19.
²
To find the difference between two numbers, we subtract the smaller from the larger.
For example, the difference between 3 and 16 is 16 ¡ 3 = 13.
²
When adding or subtracting zero (0), the number remains unchanged.
For example, 23 + 0 = 23, 23 ¡ 0 = 23.
²
When adding several numbers, we do not have to carry out the addition in the given
order. Sometimes it is easier to change the order.
Example 6
Self Tutor
Find a the sum of 187, 369 and 13
a
b the difference between 37 and 82
187 + 369 + 13
= 187 + 13 + 369
= 200 + 369
= 569
the difference between 37 and 82
= 82 ¡ 37
= 45
b
EXERCISE 1C
1 Simplify the following:
a 3+0
b
e 1¡0
f
0+3
23 + 47 ¡ 0
5¡0
20 + 0 ¡ 8
c
g
2 Simplify the following, taking easy paths where possible:
a 8 + 259 + 92
b 137 + 269 + 63
d 163 + 979 + 21
e 567 + 167 + 33
g 978 + 777 + 22
h 99 + 899 + 1901
3 Find:
a the sum of 5, 7 and 8
c the sum of the first 10 natural numbers
d
h
c
f
i
423 + 0 + 89
53 ¡ 0 + 47
987 + 241 + 159
364 + 779 + 636
89 + 75 + 25 + 11
the difference between 19 and 56
by how much 639 exceeds 483
b
d
4 Solve the following problems:
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a What number must be increased by 374 to get 832?
b What number must be decreased by 674 to get 3705?
c Mount Cook in New Zealand is 3765 m above sea
level, whereas Mount Kościuszko in New South
Wales is 2231 m high. How much higher is Mount
Cook than Mount Kosciuszko?
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WHOLE NUMBERS
(Chapter 1)
19
d In a golf tournament Aaron Baddeley won the first prize of $163 700 and Tiger
Woods came second with $97 330. What was the difference between the two prizes?
What would they each have won if they had tied?
e My bank account balance was $7667 and
I withdrew amounts of $1379, $2608 and
$937. What is my bank balance now?
f Sally stands on some scales with a 15 kg
dumbbell in each hand. If the scales read
92 kg, what does she weigh?
PRODUCTS AND QUOTIENTS
²
The word product is used to represent the result of a multiplication.
For example, the product of 3 and 5 is 3 £ 5 = 15.
²
The word quotient is used to represent the result of a division.
For example, the quotient of 15 ¥ 3 is 5.
²
When multiplying, changing the order can often be used to simplify the process.
²
Multiplying by one (1) does not change the value of a number.
For example, 17 £ 1 = 17, 1 £ 17 = 17.
²
Multiplying by zero (0) produces zero.
For example, 17 £ 0 = 0.
²
Division by zero (0) is meaningless.
We say it is undefined.
For example, 0 ¥ 4 = 0
but 4 ¥ 0 is undefined.
Neither the Egyptians nor
the Romans had a
symbol to represent
nothing. The symbol 0
was called zephirum in
Arabic. Our word zero
comes from this.
Example 7
Self Tutor
a 7£8
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70 £ 800
70 £ 800
= 7 £ 10 £ 8 £ 100
= 56 £ 1000
= 56 000
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f
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50
0
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100
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c
c
80 £ 9
50 £ 6
7 £ 1300
b
e
h
25
0
5
95
100
50
the product:
8£9
5£6
7 £ 13
75
25
0
5
95
100
50
75
25
0
5
5 Find
a
d
g
7 £ 80
= 7 £ 8 £ 10
= 56 £ 10
= 560
b
75
7£8
= 56
a
b 7 £ 80
25
Find the products:
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80 £ 90
50 £ 600
70 £ 13 000
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WHOLE NUMBERS
6 Find
a
d
g
(Chapter 1)
the quotient:
8¥4
36 ¥ 9
56 ¥ 8
80 ¥ 4
360 ¥ 90
560 ¥ 80
b
e
h
8000 ¥ 40
3600 ¥ 9
56 000 ¥ 800
c
f
i
Example 8
Self Tutor
a 4 £ 37 £ 25
Simplify:
4 £ 37 £ 25
= 4 £ 25 £ 37
= 100 £ 37
= 3700
a
b 17 £ 8 £ 125
17 £ 8 £ 125
= 17 £ 1000
= 17 000
b
7 Simplify the following, taking short cuts where possible:
a 5 £ 41 £ 2
b 25 £ 91 £ 4
d 50 £ 200 £ 19
e 57 £ 125 £ 8
g 4 £ 8 £ 125 £ 250
h 8 £ 2 £ 96 £ 125 £ 50
8 Simplify, if possible:
a 6£0
d 0 ¥ 11
g 0£1
j 0 £ 37
6¥0
11 £ 0
0£0
87 £ 0
b
e
h
k
a 87 £ 15
8
£ 1
43
87
130
fmultiply 87 by 5g
fmultiply 87 by 10g
faddingg
7
5
5
0
5
24
456
38
76
76
0
Check these
results on your
calculator!
b 456 ¥ 19
)
87 £ 15 = 1305
)
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456 ¥ 19 = 24
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100
50
c
f
75
25
0
107 £ 9
507 ¥ 13
5
95
50
75
25
0
b
e
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
20 £ 113 £ 5
789 £ 250 £ 40
5 £ 57 £ 8 £ 125 £ 200
fbring 6 downg
f19 goes into 76 four timesg
9 Simplify the following:
a 39 £ 13
d 98 ¥ 7
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c
f
i
f19 goes into 45 twiceg
100
19
² 4 £ 25 = 100
² 8 £ 125 = 1000
Self Tutor
Simplify the following:
b
Reminder:
0£6
0 £ 11
0¥1
87 ¥ 0
c
f
i
l
Example 9
a
To find the
quotient of two
numbers we
divide them.
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117 £ 17
1311 ¥ 23
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WHOLE NUMBERS
10 Find:
the product of 17 and 32
b
the product of the first 5 natural numbers.
a
c
PUZZLE
(Chapter 1)
the quotient of 437 and 19
OPERATIONS WITH WHOLE NUMBERS
Click on the icon to obtain a printable version of this puzzle.
1
2
6
Across
1 11 £ 12
4 4 £ 1234
6 247 + 366
8 1146 ¥ 6
10 427 £ 4
11 347 ¡ 128
3
4
21
5
7
8
9
10
11
Down
1 445 ¡ 249
2 972 ¥ 4
3 7£8
5 845 ¡ 536
7 129 + 58
8 85 £ 2 + 12
9
10
PUZZLE
1000 ¥ 5 ¡ 1
204 ¥ 12
11 Solve the following problems:
a What must I multiply $25 by to get $1375?
b What answer would I get if I start with 69
and add on 8, 31 times?
c I planted 400 rows of cabbages and each
row contained 250 plants. How many
cabbages were planted altogether?
d Ian swims 4500 m in a training session. If
the pool is 50 m long, how many laps does
he swim?
e A contractor bought 34 loads of soil each weighing 12 tonnes at $13 per tonne.
What was the total cost?
f All rooms of a motel cost $78 per day to rent. The motel has 6 floors and 37
rooms per floor. What is the total rental received per day if the motel is fully
occupied?
g How many 38-passenger buses are needed to transport 646 students to the athletics
stadium?
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12 Revisit the Opening Problem. A sixteen storey hotel with floors G, 1, 2, ....., 15 has no
accommodation on the ground floor. On the even numbered floors (2, 4, 6, ......) there
are 28 guest rooms and on the odd numbered floors there are 25 guest rooms. Room
cleaners work for four hours each day during which time each cleaner can clean 12 guest
rooms. Each cleaner is paid at a rate of $16 per hour.
a How many floors are odd numbered?
b How many guest rooms in total are on all the odd numbered floors?
c If each guest room has three chairs, how many chairs are on an even numbered
floor?
d How many guest rooms are in the hotel?
e How many cleaners are required to clean all guest rooms assuming the hotel was
‘full’ the previous night?
f What is the total cost of hiring the cleaners to clean the guest rooms of the hotel?
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SA_08-6
22
WHOLE NUMBERS
(Chapter 1)
ACTIVITY 1
CALCULATOR USE
Over the next few years you will be performing a lot of
calculations in mathematics and other subjects. You can use
your calculator to help save time with your calculations.
Note: ²
A calculator will not necessarily give you a correct answer unless you understand what to do.
Not all calculators work the same way. You will need to check how your
calculator performs each type of operation.
²
Try these calculations:
1 Press the keys in this order and check that you get the correct answer:
a 8
6
+
b 11
=
-
5
c 3
=
8
×
d 16
=
÷
2
=
2 Press the keys in this order and check that you get the correct answer:
15
×
12
3
×
÷
27
128
×
+
26943
[Answer is 29 503.]
=
3 Test yourself on these problems. To find out if you have the correct answer, turn your
calculator upside down to find the correct word, given in brackets.
a 4378 ¡ 51 095 + 657 £ 1376 ¥ 6 ¥ 3
b 2 + 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 4 + 4 + 773
c 1 £ 2 £ 3 £ 4 £ 5 £ 6 £ 7 £ 8 £ 9 £ 10 ¡ 3 628 462
Answers: a [lose] b [bib] c [bee]
Perhaps you can invent more of these problems?
D SQUARE NUMBERS AND SQUARE ROOTS
If a number can be represented by a
square arrangement of dots it is called a
square number.
For example, 9 is a square number as it
can be represented by the 3 £ 3 square
shown:
Square Geometric Symbolic
number
form
form
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4
3
32
3£3
9
4
42
4£4
16
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100
1
50
1£1
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25
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The table alongside shows the first four
square numbers:
1
5
We say ‘three squared is equal to nine’
and we write 32¡=¡9 using index notation.
Factor
Value
form
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Y:\HAESE\SA_08-6ed\SA08-6_01\022SA08-6_01.CDR Thursday, 26 October 2006 2:00:01 PM PETERDELL
SA_08-6
WHOLE NUMBERS
(Chapter 1)
23
A calculator can help you to work out the value of square numbers.
For example, 152 can be found by pressing 15
×
15
=
by pressing 15 x 2
=
. The answer is 225.
or
EXERCISE 1D
1 For the 5th and 6th square numbers:
a draw a diagram to represent them
b state their values.
2
a Manually calculate the 7th, 8th, 9th and 10th square numbers.
b Use your calculator to write the 17th, 20th and 50th square numbers.
3
a Write down two numbers between 7 and 41 that are both odd and square.
b Write down two numbers between 45 and 105 that are both even and square.
4
a Use a calculator to complete the following:
12
112
1112
11112
=
=
=
=
b Have you noticed a pattern? Complete the following without using your calculator:
i 11 1112 =
ii 111 1112 =
c Investigate other such patterns with square numbers. If you find any, share them
with your class!
a Copy and complete the following pattern:
5
1 = 1 = 12
1 + 3 = 4 = 22
1 + 3 + 5 = 9 = 32
1+3+5+7 = =
1+3+5+7+9 = =
b Use the pattern to find the sum of the first:
i 6 odd numbers
ii 10 odd numbers
6
=
=
=
=
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b Hence predict the following:
i 172 ¡ 162
Check your answers using a calculator.
5
‘n’ odd numbers
a Copy and complete the following pattern:
12 ¡ 02
22 ¡ 12
32 ¡ 22
42 ¡ 32
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ii
892 ¡ 882
SA_08-6
24
WHOLE NUMBERS
(Chapter 1)
SQUARE ROOTS
What number multiplied by itself gives 25? The answer is 5 as 5 £ 5 = 25.
p
p
means the square root of.
We say 5 is the square root of 25 and write 25 = 5, where
p
p
To find 961 using your calculator, press
961 =
or
961 ENTER .
2nd
press
The answer is 31.
7 Find the number which must be multiplied by itself to get:
a 16
b 4
c 9
e 81
f 1
g 2500
8 Evaluate without a calculator:
p
p
a
9
b
49
p
p
e
64
f
1600
d
h
36
1 000 000
d
g
p
144
p
3600
h
p
0
p
10 000
c
p
2704
d
p
169
c
9 Use your calculator to find:
p
p
a
576
b
7921
10 Find two consecutive integers which the following square roots lie between:
p
p
p
p
a
3
b
7
c
28
d
57
p
p
p
p
e
131
f
157
g
230
h
385
p
Hint: As 13 lies between the perfect squares 9 and 16, 13 lies
p
p
between 9 and 16.
E
ORDER OF OPERATIONS
When two or more operations are carried out, different answers can result depending on the
order in which the operations are performed.
to find the value of 16 ¡ 10 ¥ 2, Sonia decided to subtract first and then
divide. Wei decided to divide first and then subtract.
For example,
Sonia’s method:
Subtract first then divide.
16 ¡ 10 ¥ 2
=6¥2
=3
Wei’s method:
Divide first then subtract.
16 ¡ 10 ¥ 2
= 16 ¡ 5
= 11
Which answer is correct, 3 or 11?
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To avoid this problem, a set of rules for the order of performing operations has been agreed
upon by all mathematicians.
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Y:\HAESE\SA_08-6ed\SA08-6_01\024SA08-6_01.CDR Thursday, 26 October 2006 2:00:27 PM PETERDELL
SA_08-6
WHOLE NUMBERS
(Chapter 1)
25
RULES FOR ORDER OF OPERATIONS
²
²
²
²
Perform operations within Brackets first, then
calculate any part involving Exponents, then
starting from the left, perform all Divisions and Multiplications as you come to them.
Finally, working from the left, perform all Additions and Subtractions.
The word BEDMAS may help you remember this order.
Note:
²
If an expression contains more than one set of brackets, work the innermost
brackets first.
The division line of fractions behaves like a ‘grouping symbol’ or set of
brackets. This means that the numerator and denominator must be found
before doing the division.
²
Using these rules, Wei’s method is correct in the above example, and 16 ¡ 10 ¥ 2 = 11.
Example 10
Self Tutor
Evaluate: 35 ¡ 10 ¥ 2 £ 5 + 3
35 ¡ 10 ¥ 2 £ 5 + 3
= 35 ¡ 5 £ 5 + 3
= 35 ¡ 25 + 3
= 10 + 3
= 13
fdivision and multiplication
working from leftg
fsubtraction and addition
working from leftg
EXERCISE 1E
1 Evaluate the following:
a 5+6¡6
d 9¥3+4
g 30 ¥ 3 ¥ 5
j 7£4¡3£5
b
e
h
k
7+8¥2
100 + 6 ¡ 7
18 ¥ 3 + 11 £ 2
8+6¥3£4
Example 11
c 8¥2+7
f 7£9¥3
i 6+3£5£2
l 4+5¡3£2
Self Tutor
Evaluate: 2 £ (3 £ 6 ¡ 4) + 7
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fcomplete bracketsg
fmultiplication nextg
faddition lastg
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2 £ (3 £ 6 ¡ 4) + 7
= 2 £ (18 ¡ 4) + 7
= 2 £ 14 + 7
= 28 + 7
= 35
If you do not
follow the order
rules, you are likely
to get the wrong
answer.
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Y:\HAESE\SA_08-6ed\SA08-6_01\025SA08-6_01.CDR Thursday, 26 October 2006 2:00:48 PM PETERDELL
SA_08-6
26
WHOLE NUMBERS
(Chapter 1)
2 Evaluate the following, remembering to complete the brackets first:
a
(12 + 3) £ 2
b (17 ¡ 8) £ 2
c
d 5 £ (7 + 3)
e 36 ¡ (8 ¡ 6) £ 5
f
g 5 + 4 £ 7 + 27 ¥ 9
h (14 ¡ 8) ¥ 2
i
j
17 ¡ (5 + 3) ¥ 8
k (12 + 6) ¥ (8 ¡ 5)
l
m 36 ¡ (12 ¡ 4)
n 52 ¡ (10 + 2)
o
3 Evaluate the following:
a 6 £ 8 ¡ 18 ¥ (2 + 4)
c 5 + (2 £ 10 ¡ 5) ¡ 6
e (2 £ 3 ¡ 4) + (33 ¥ 11 + 5)
g (50 ¥ 5 + 6) ¡ (8 £ 2 ¡ 4)
i (7 ¡ 3 £ 2) ¥ (8 ¥ 4 ¡ 1)
(3 + 7) ¥ 10
18 ¥ 6 + 5 £ 3
6 £ (7 ¡ 2)
5 £ (4 ¡ 2) + 3
25 ¡ (10 ¡ 3)
10 ¥ 5 + 20 ¥ (4 + 1)
18 ¡ (15 ¥ 3 + 4) + 1
(18 ¥ 3 + 3) ¥ (4 £ 4 ¡ 7)
(10 £ 3 ¡ 20) + 3 £ (9 ¥ 3 + 2)
(5 + 3) £ 2 + 10 ¥ (8 ¡ 3)
b
d
f
h
j
Example 12
Self Tutor
Simplify: 5 + [13 ¡ (8 ¥ 4)]
5 + [13 ¡ (8 ¥ 4)]
= 5 + [13 ¡ 2]
= 5 + 11
= 16
finnermost brackets firstg
fremaining bracket nextg
faddition lastg
4 Simplify:
a [3 £ (4 + 2)] £ 5
c [4 £ (16 ¡ 1)] ¡ 6
e 5 + [6 + (7 £ 2)] ¥ 5
g [(2 £ 3) + (11 ¡ 5)] ¥ 3
b
d
f
h
[(3 £ 4) ¡ 5] £ 4
[(3 + 4) £ 6] ¡ 11
4 £ [(4 £ 3) ¥ 2] £ 7
19 ¡ [f3 £ 7g ¡ f9 ¥ 3g] + 14
Example 13
Self Tutor
16 ¡ (4 ¡ 2)
14 ¥ (3 + 4)
Simplify:
16 ¡ (4 ¡ 2)
14 ¥ (3 + 4)
16 ¡ 2
14 ¥ 7
fbrackets firstg
14
2
=7
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Y:\HAESE\SA_08-6ed\SA08-6_01\026SA08-6_01.CDR Tuesday, 31 October 2006 12:31:15 PM PETERDELL
SA_08-6
WHOLE NUMBERS
(Chapter 1)
27
5 Simplify:
a
21
16 ¡ 9
b
18 ¥ 3
14 ¡ 11
c
(8 £ 7) ¡ 5
17
d
12 + 3 £ 4
5+7
e
3£7¡5
2
f
3 £ (7 ¡ 5)
2
g
2£8¡1
8¡6¥2
h
56 ¥ 8 ¡ 7
56 ¥ (8 ¡ 7)
i
25 ¡ (16 ¡ 11)
12 ¥ 4 + 2
Example 14
Self Tutor
Simplify: 3 £ (6 ¡ 2)2
3 £ (6 ¡ 2)2
= 3 £ 42
= 3 £ 16
= 48
fbrackets firstg
fexponent nextg
fmultiplication lastg
PUZZLE
Click on the icon to obtain a printable version of this puzzle.
1
2
3
5
7
Across
1 40 £ 5 ¡ 17
3 100 ¡ (7 ¡ 1)
5 (1 + 5 £ 50) £ 25
7 3 £ (3 + 20)
9 8 £ 11 ¡ 7
10 100 ¡ 9 £ 2
12 5 £ (6 + 7)
14 153 ¥ 3 + 3 £ 1000
16 90 ¡ 4 £ 4
17 9 £ 100 + 8 £ 5
4
6
8
9
10 11
12 13
14
15
16
17
6 Simplify:
a 3 £ 42
d (5 ¡ 2)2 ¡ 6
b
e
PUZZLE
Down
1 100 + 24 ¥ 4
2 10 £ 4 ¡ 20 ¥ 5
3 10 000 ¡ 3 £ 100 + 2 £ 8
4 7£7¡2£2
6 (7 ¡ 3) £ (6 + 1)
8 100 £ 100 ¡ 14 £ 14
11 625 ¥ (20 + 5)
13 10 £ (9 £ 6)
14 70 ¡ 3 £ 11
15 2 £ 5 + 3 £ 3
2 £ 33
3 £ 4 + 52
c
f
32 + 23
4 £ 32 ¡ (3 + 2)2
7 Replace the ¤ with either +, ¡, £ or ¥ to make a true statement:
a 3 + 15 ¤ 3 = 8
b 10 ¤ 7 + 15 = 18
c 8 ¤ 4 ¡ 10 = 22
d (18 ¤ 2) ¥ 10 = 2
e (10 ¤ 3) ¤ 7 = 1
f 15 ¤ 3 + 2 ¤ 5 = 15
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8 Put brackets into the following to make them true:
a 18 ¡ 6 £ 3 + 2 = 38
b 48 ¡ 6 £ 3 + 4 = 6
d 8+4¥2+2=3
e 5 + 3 £ 6 ¡ 10 = 38
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c
f
32 ¥ 8 ¥ 2 = 8
13 + 5 ¥ 5 + 4 = 2
SA_08-6
28
WHOLE NUMBERS
(Chapter 1)
9 In these number puzzles each letter stands for a different one of the digits 0, 1, 2, 3,
:::::: to 9: There are several solutions to each puzzle. Can you find one of them? Can
you find all of them?
a
D O G
b
S U R F
c
Y O U
+
C A T
¡ S A N D
£
M E
H A T E
S E A
L O V E
ACTIVITY 2
BRACKETS AND MEMORY KEYS
Memory keys are used to store and retrieve numbers in and
from the memory. We store a number in the memory so that
we can recall it for later use.
Alternatively, brackets keys ( and ) can be used.
² M+
Adds the number in the display to what is already in the memory.
² STO or Min
Replaces what is in the memory by the number in the display.
² RCL or MR
Displays the number which is stored in the memory.
Note: Calculators which have multiple memory locations require you to specify which
memory location you wish to use to store or retrieve numbers after pressing
STO or RCL .
These locations are usually labelled A, B, C, etc., and are accessed by pressing
the ALPHA button.
Example 15
Self Tutor
Use your calculator to find: 15 £ 60 ¥ (8 + 7)
Method 1:
15
=
×
do (8 + 7) first and put result in the memory
STO
60
gives the result of 15 £ 60
=
÷ RCL =
15
Method 2:
×
completes the calculation
60
÷
( 8
+
7
)
( and ) enable us to work from left to
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The second method
is clearly better for
this example.
right without using the memory keys.
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Y:\HAESE\SA_08-6ed\SA08-6_01\028SA08-6_01.CDR Thursday, 26 October 2006 2:01:14 PM PETERDELL
SA_08-6
WHOLE NUMBERS
Example 16
(Chapter 1)
29
Self Tutor
Find: (3 £ 37 + 9) ¥ 1, (3 £ 37 + 9) ¥ 2, (3 £ 37 + 9) ¥ 3,
(3 £ 37 + 9) ¥ 4, (3 £ 37 + 9) ¥ 5, (3 £ 37 + 9) ¥ 6
3
37
×
+
9
=
gives us the dividend for each expression
puts this result into the memory
STO
Answer:
RCL
÷
1
=
calculates the first quotient
RCL
÷
2
=
calculates the second quotient
60
RCL
÷
3
=
calculates the third quotient
40
RCL
÷
4
=
calculates the fourth quotient
30
RCL
÷
5
=
calculates the fifth quotient
24
RCL
÷
6
=
calculates the sixth quotient
20
(Use both memory and bracket keys.)
Try these calculations:
1
(47 £ 63) + (57 £ 28) = 4557
360 ¡ (14 £ 17) + (8 £ 11) = 210
120 ¥ (17 + 23) = 3
a
c
e
120
INVESTIGATION
b 47 £ (63 + 57) £ 28 = 157 920
d 15 £ (13 ¡ 6) £ 19 ¡ 143 = 1852
f 12 648 ¥ (17 £ 24) = 31
NUMBER CRUNCHING MACHINE
Any positive integer can be fed into a number crunching machine which
produces one of two results:
input, x
² If the integer fed in is even, the machine
divides the number by 2.
What to do:
² If the integer fed in is odd, the machine
subtracts one from the number.
number crunch
1 Find the result if the following numbers are fed into the machine:
a 26
b 15
c 42
d
output
117
2 What was the input to the machine if the output is:
a 8
b 13?
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It is possible to feed the output from the machine back into the input, and continue to do
so until the output reaches zero.
For example, with an initial input of 11, the following would occur:
11
10
5
4
2
1
0:
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Y:\HAESE\SA_08-6ed\SA08-6_01\029SA08-6_01.CDR Friday, 27 October 2006 3:35:37 PM PETERDELL
SA_08-6
30
WHOLE NUMBERS
(Chapter 1)
We see that 6 steps or “crunches” are necessary to reach zero.
3 Give the number of “crunches” required to reach zero if you start with:
a 7
b 24
c 32
There are three 4-step numbers.
The method of finding them is to work in
reverse.
4
0
1
Hence the only 4-step numbers are: 5, 6 and 8.
Can you determine all the 5-step numbers?
3
6
4
5
2
8
5 By changing the rules for the number crunching machine, different outputs can be
obtained. Try some different possibilities for yourself.
For example:
² If the number is divisible by 3, divide it by 3.
² If the number is not divisible by 3, subtract 1.
REVIEW SET 1A
1 Write 4738 in word form.
2 What number is represented by the digit 4 in 86 482?
3 What is the difference between 895 and 2718?
4 Round off 48 526 to the nearest 100.
5 Find an approximate value of 106 £ 295.
6 Calculate 123 £ 36.
7 How many buses would be required to transport 329 students if each bus holds a
maximum of 47 students?
8 Simplify the following:
a 20 £ 33 £ 5
b
125 £ 7 £ 8
9 A class contains 14 boys and 15 girls. If each student is given 13 pencils, how many
pencils are given out altogether?
10 If a baker sells 24 dozen rolls at 15 cents each, how much money does he receive?
11 Find 1834 ¡ 712 + 78.
12 A woman has $255 in her purse. She gives $35
to each of her five children.
How much money does she have left?
13 Find the 9th square number.
p
14 Find 121.
15 Find the sum of the first five square numbers.
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25
0
b
5
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16 Simplify the following:
a 7 £ 4 ¡ 18 ¥ 2
31 ¡ 11
c
12 ¥ 6
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Y:\HAESE\SA_08-6ed\SA08-6_01\030SA08-6_01.CDR Tuesday, 31 October 2006 12:33:54 PM PETERDELL
SA_08-6
WHOLE NUMBERS
(Chapter 1)
31
17 Put brackets into the following to make a true statement:
a 4 £ 2 + 3 ¡ 5 = 15
b 20 ¥ 4 + 1 + 5 = 2
REVIEW SET 1B
1 Simplify 14 £ 0.
2 What number is represented by the digit 3 in 43 209?
3 Round off 46 804 to two figures.
4 Determine 2763 + 427.
5 By how much does 738 exceed 572?
6 Write 4000 + 20 in simplest form.
7 Use the digits 7, 5, 9, 0, 2, once only to make the largest number you can.
a
b
c
d
e
f
g
h
8
Find the sum of 8, 15 and 9.
Find the difference between 108 and 19.
What number must be increased by 293 to get 648?
Determine 119 + 19 + 81:
Determine 5 £ 39 £ 20:
Find the product of 14 and 38.
Find the quotient of 437 and 19.
Use 1 figure working to find an approximate answer to 5032 £ 295:
9 How many times larger than 7 £ 8 is 700 £ 80?
10 The Year 8 students at a school are split into 4 equal classes of 27 students. The
school decides to increase the number of classes to 6. How many students will there
be in each of the new classes if the students are divided equally between them?
11 If 73 students have a total mass of 4161 kg, what is their average mass?
12 My bank account contains $3621 and I make monthly withdrawals of $78 for 12
months. What is my new bank balance?
13 How many square numbers are there between 60 and 130?
p
14 Between which two integers does 29 lie?
15 Simplify the following:
a 5£6¡4£3
c 5 £ 8 ¡ 18 ¥ (2 + 4)
3 + 18 £ 22
(5 + 4) £ 2 + 20 ¥ (8 ¡ 3)
b
d
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16 Put brackets into the following to make a true statement:
a 9¡8¥2+2 =7
b 2+8¥4¡2=5
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Y:\HAESE\SA_08-6ed\SA08-6_01\031SA08-6_01.CDR Thursday, 26 October 2006 2:02:09 PM PETERDELL
SA_08-6
32
WHOLE NUMBERS
(Chapter 1)
PUZZLE 1
MATCHSTICK PUZZLES
In playing with matches a number of interesting puzzles have been
developed.¡ It is impossible to state any general rules for solving puzzles
with matches but the fun and the challenge remain.
Investigate the following puzzles:
1 In the configuration of 12 matches given:
a remove 4 matches to leave 1 square
b remove 2 matches to leave
i 3 squares
ii 2 squares
c shift 4 matches to obtain 3 squares.
2 Using 17 matches we can obtain the rectangle shown:
a Remove 5 matches to leave 3 congruent squares.
b Remove 2 matches to leave 6 squares.
3 Using the 24 match rectangle shown:
a remove 4 matches to make 5 squares
b make 2 squares by removing 8 matches
c remove 8 matches to leave
i 2 squares ii 3 squares iii 4 squares
d remove 12 matches to leave 3 squares
e shift 8 matches to make 3 squares.
4 The triangle alongside has area 6 units2 .
a Move two matches to make the area 5 units2 .
b See what other areas you can make by moving
matches 2 at a time.
PUZZLE 2
HOW MANY SQUARES
consists of 1 square
consists of 4 + 1 = 5 squares, 4 with sides one unit and 1 with
sides 2 units.
What to do:
Investigate the total number of squares in a
a 3 by 3 square,
b 4 by 4 square,
c 5 by 5 square.
cyan
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Can you determine, without drawing figures,
the total number of squares in a
d 6 by 6 square,
e 20 by 20 square?
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Y:\HAESE\SA_08-6ed\SA08-6_01\032SA08-6_01.CDR Wednesday, 8 November 2006 5:02:25 PM DAVID3
SA_08-6