maths
Matters
for CSEC® Examinations
Macmillan Mathematics for the Caribbean
y
3
0
2
x
Geoff Buckwell and Robert Solomon
CSEC® is a registered trade mark(s) of the Caribbean Examinations Council
(CXC). Maths Matters for CSEC® Examinations is an independent publication
and has not been authorized, sponsored, or otherwise approved by CXC.
CMSB_00_prelims(i-xii).indd 1
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Macmillan Education
Between Towns Road, Oxford, OX4 3PP
A division of Macmillan Publishers Limited
Companies and representatives throughout the world
www.macmillan-caribbean.com
ISBN 978-0-230-40032-0
Text © Geoff Buckwell and Robert Solomon 2011
Design and illustration © Macmillan Publishers Limited 2011
First published in 2011
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Illustrated by EXPO Holdings, Malaysia
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Printed and bound in Malaysia
2015 2014 2013 2012 2011
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Contents
Introduction
Scope and sequence
1 Numbers theory
1.1 Classifiying numbers
1.2 Operations, identity and inverse
1.3 Closure
1.4 Distributive law and associative law in number problems
1.5 Other bases (revision)
vii
viii
1
1
4
5
6
7
2 Rounding and errors
2.1 Significant figures, decimal places and standard form
2.2 Errors in measurement
2.3 Rounding effects when substituting in formulae
11
11
14
15
3 Consumer arithmetic
3.1 Income
3.2 Taxes
3.3 Utilities
3.4 Currency
3.5 Paying by instalments
3.6 Interest
20
20
27
31
33
35
38
4 Sets and sequences
4.1 Definitions
4.2 Sequences
4.3 Finding the rule of a sequence
47
47
50
54
5 Algebra: expressions
5.1 Algebraic operations
5.2 Indices
5.3 Binary operations
5.4 Algebraic fractions
60
60
63
64
65
6 Algebra: equations
6.1 Linear equations
6.2 Simultaneous equations
6.3 Inequalities
6.4 Changing the subject
74
74
76
79
81
7 Coordinate geometry of the straight line
7.1 Straight-line graphs
7.2 Gradients
7.3 Gradients and intercepts of straight lines
7.4 Finding the equation of a straight line
7.5 Parallel and perpendicular lines
7.6 Distance between points
7.7 Midpoints
7.8 Graphical solution of simultaneous equations
7.9 Inequalities in two dimensions
87
87
92
95
98
100
104
106
108
110
iii
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8 Quadratics
8.1 Expanding a bracket (revision)
8.2 Expanding pairs of brackets
8.3 Identities
8.4 Factorising two terms (revision)
8.5 Factorising a quadratic
8.6 The general quadratic ax2 + bx + c
8.7 Quadratic equations
8.8 The quadratic formula
8.9 Quadratic equation problems
8.10 The quadratic curve, symmetry, equations and gradient
115
115
116
117
118
119
122
123
125
127
129
9 Ratio and proportion
9.1 Ratio
9.2 Rate
9.3 Direct proportion
9.4 Inverse proportion
9.5 Proportion to a power
134
134
137
138
142
144
10Geometry 1
10.1 Lines, angles, polygons and solids
10.2 Basic drawing and constructions
10.3 Symmetry
150
150
153
157
11Geometry problems
11.1 Lines, angles and polygons
11.2 Congruent triangles
11.3 Similar figures
165
165
168
174
12Transformations
12.1 Translations using vectors
12.2 Transformations (revision)
12.3 Combining transformations
12.4 Glide reflections
180
180
183
185
191
13Surface area and volume
13.1 Revision of units
13.2 Approximate area
13.3 Area of regular shapes (revision)
13.4 Surface area
13.5 Volume
13.6 The cylinder
13.7 Pyramids and cones
13.8 The sphere
13.9 Compound shapes
198
198
200
201
203
204
207
209
212
213
14Trigonometry
14.1 Pythagoras’ theorem
14.2 The trigonometric ratios
14.3 Bearings
14.4 The sine rule
14.5 Using the sine rule to find angles
14.6 The cosine rule
219
219
221
225
227
234
231
iv
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14.7 Using the cosine rule to find angles
14.8 Solving triangles
234
235
15Circle theorems
15.1 Definitions
15.2 Angle at the centre of a circle
15.3 Angle in a semicircle
15.4 Angles in the same segment
15.5 Cyclic quadrilaterals
15.6 Mixed examples
15.7 Tangents
15.8 Alternate segment theorem
15.9 Midpoint of a chord
242
242
243
245
247
249
251
252
256
258
16Functions
16.1 Domain and range
16.2 Composite functions
16.3 Formulae for composite functions
16.4 Inverse functions
264
264
267
268
269
17Statistics
17.1 Data and frequency tables
17.2 Statistical diagrams
17.3 Measures of central tendency
17.4 Range and interquartile range
17.5 Median and quartiles from a frequency table
17.6 Cumulative frequency
17.7 Probability
277
277
279
284
287
289
290
293
18Vectors
18.1 The arithmetic of vectors and geometry
18.2 Position vectors
18.3 Magnitude of a vector, unit vectors
18.4 Midpoints
299
299
302
303
305
19Matrix algebra
19.1 Arrays of numbers
19.2 Adding and subtracting matrices
19.3 Multiplication by a scalar
19.4 Matrix multiplication
19.5 The identity matrix
19.6 Inverse matrices
308
308
309
311
311
314
315
Optional special objectives
20Further quadratics
20.1 Completing the square
20.2 Greatest or least value and line of symmetry
20.3 Solving equations by completing the square
20.4 Line of symmetry and turning points by sketching
20.5 Simultaneous equations: one linear, one quadratic
319
319
321
323
324
326
21Regions and linear programming
21.1 Inequalities revision
329
329
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21.2 Revision of y = mx + c
21.3 Regions
21.4 Regions with sides not parallel to the axes
21.5 Situations which give rise to inequalities
21.6 Listing the possibilities
21.7 Linear programming
330
331
333
336
337
338
22Further graphs
22.1 Graphs of powers of x
22.2 Distance–time graphs
22.3 Acceleration
22.4 Velocity–time graphs
344
344
346
348
349
23Measurement, geometry and trigonometry
23.1 Area of a triangle
23.2 Area of a segment
23.3 Lines and planes
23.4 Lengths and angles in three dimensions
23.5 Lengths within solids
23.6 Angle between lines
355
355
357
358
359
361
362
24 Vectors and matrices
24.1 Equivalent (equal) and null vectors
24.2 Collinear points
24.3 Transformations using matrices
24.4 Enlargements
24.5 The unit square
24.6 Common matrices
24.7 Successive transformations using matrices
24.8 Simultaneous equations
368
368
369
370
371
373
375
375
376
Model examination papers
380
Glossary
397
Answers
403
Index
442
vi
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Preface
Maths Matters for CSEC® Examinations is a course of four books for secondary schools in the Caribbean. The
first three books cover Grades 7, 8 and 9. The fourth book prepares students for the CSEC examination.
The books provide comprehensive coverage of the syllabus. For each topic there is a thorough and clear
explanation, followed by worked examples to show how the content can be used. There are many
exercises to provide practice for students. These exercises are carefully graded to cater for all abilities, and
some exercises can be used for revision. As well as the material in the textbooks, there are Workbooks for
extra practice. The fourth book ends with examination-style test papers, to provide practice for the CSEC
examinations. The textbooks are suitable for use either in the classroom or at home.
Each unit contains the following features:
Key skills you will learn in this unit
This lists the learning objectives of the unit.
What you need to know
This lists the previous knowledge required for the unit. Where relevant, revision material is provided.
Summary
This sums up what has been achieved in the unit.
Progress exercise
This is an exercise that covers all the material of the unit.
Multiple choice test
This provides practice in answering the type of questions set in examination papers.
The material is made relevant to the daily experiences of students, and to modern life in the
Caribbean. It is consistent with modern technology, while retaining knowledge of traditional methods.
The authors are confident that this series provides all that is needed for the student to achieve success
in the CSEC examination.
Mathematics is becoming increasingly important, through the spread of information technology. To
understand and succeed in the modern world it is necessary to appreciate the power and significance
of mathematics.
Currencies of the Caribbean
The different countries of the Caribbean have different currencies, such as the Jamaican dollar (J$) the
Barbados dollar (Bds$) and so on. Below is a list of the currencies, with the exchange rates with the
United States dollar (US$).
Country
Currency
Symbol
Approximate value of US$1
Jamaica
Jamaican dollar
J$
J$85
Trinidad and Tobago
Trinidad and Tobago dollar
TT$
TT$6.3
Barbados
Barbados dollar
Bds$
Bds$2
Antigua, St Lucia etc.
East Caribbean dollar
EC$
EC$2.7
Belize
Belize dollar
BZ$
BZ$1.95
Guyana
Guyana dollar
GY$
GY$200
Many people travel abroad during their lives, for work or for holidays. They meet different traditions
and institutions, and in many cases the money systems are different. These different currencies are used
throughout this book, which will provide experience in dealing with money from other countries.
vii
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Scope and sequence
Syllabus heading
Section in book
Computation
1.1
Computation
1.2
1.2
Conversion between fractions, decimals, percentages
1.1
1.3
Metric units
13.1, 3.4
1.4
Significant figures, decimal places
2.1
1.5
Standard form
2.1
1.6
Fraction and percentage of a whole
1.1
1.7
One quantity as a fraction or percentage of another
1.1
1.8
Compare by ratio
9.1
1.9
Divide in ratio
9.1
1.10
Problems involving fractions, decimals, percentages, ratio, rates, proportion
Arithmetic mean
9.1→9.5
17.3
Number theory
2.1
Sets of numbers
1.1, 4.1
2.2
Order numbers
1.1
2.3
Sequence from rule
4.2
2.4
Find rule of sequence
4.3
2.5
Subsets of numbers
1.1, 4.1
2.6
Sets of factors and multiples
4.1
2.7
HCF and LCM
5.4
2.8
Place value in base n (≤ 10)
1.5
2.9
Properties of numbers
1.2→1.4
2.10
Solve problems
1.2→1.4
Consumer arithmetic
3.1
Discount, sales tax, profit, loss
3.1, 3.2
3.2
Profit etc. as a percentage
3.1
3.3
Problems involving marked price etc.
3.1
3.4
HP, mortgages
3.5
3.5
Simple interest
3.6
3.6
Compound interest appreciation
3.6
3.7
Measures, money
3.4
3.8
Rates, taxes, utilities, income etc.
3.1→3.3
Sets
4.1
Concepts relating to sets
4.1
4.2
Representing sets
4.1
4.3
Relationships among sets
4.1
4.4
List subsets
4.1
4.5
Elements in ∪ etc.
4.1
4.6
Venn diagrams
4.1
4.6
Solve problems involving Venn diagrams
4.1
4.7
Problems involving number theory etc. using set theory
1.1
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Section in book
Measurement
5.1
Perimeter of polygon etc.
5.2
Length of arc
5.3
Area of polygon etc.
13.3
5.4
Area of sector
23.2
*5.5
1
2
23.1
*5.6
Area of segment
23.2
5.7
Estimate area
13.2
5.8
Surface area
13.4→13.9
5.9
Volume of solids
13.5→13.9
5.10
Convert between units
13.1
5.11
SI units
13.1
5.12
Time, distance, speed
9.2, 13.1
5.13
Estimate margin of error
2.1, 2.2, 2.3
5.14
Maps and scale drawings
9.1
5.15
Problems involving measurement
13.4→13.9
ab sin C
• Scope and sequence
Syllabus heading
Statistics
6.1
Types of data
17.1
6.2
Frequency table
17.1
6.3
Features for set of data
17.1→17.6
6.4
Statistical diagrams
17.2
6.5
Interpret diagrams
17.2
6.6
Measures of central tendency
17.3
6.7
Which of mean, median and mode is appropriate?
17.3
6.8
Measures of spread
17.4→17.6
6.9
Cumulative frequency
17.6
6.10
Ogive
17.6
6.11
Use statistical diagrams
17.2, 17.6
6.12
Find percentage from cumulative frequency curve
17.6
6.13
Sample space
17.7
6.14
Experimental, theoretical probabilities
17.6
6.15
Inferences from statistics
17.1→17.7
Algebra
7.1
Use of symbols
5.1
7.2
Translate algebra to words
5.1
7.3
Arithmetic with directed numbers
1.1→1.4
7.4
Four operations with algebraic symbols
5.1
7.5
Substitution
5.1
7.6
Binary operations
5.3
7.7
Distributive law
5.1
7.8
Algebraic fractions
5.4
7.9
Laws of indices
5.2
7.10
Linear equations one unknown
6.1
7.11
Simultaneous equations
6.2
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Syllabus heading
Section in book
7.12
Linear inequality
6.3
7.13
Change subject
6.4
7.14
Factorisation
5.1, 8.4→8.6
7.15
Quadratics
8.6→8.9
7.16
Word problems
6.1, 6.2
7.17
Simultaneous equations, one non-linear equation
20.5
7.18
Prove identities
8.3
7.19
Represent direct and indirect variation
9.3→9.5
7.20
Solve problems involving direct and inverse variation
9.3→9.5
Relations functions graphs
8.1
Concept of relation
16.1
8.2
Represent relations
16.1
8.3
Concept of function
16.1
8.4
Function notation
16.1
8.5
Distinguish relation and function
16.1
8.6
Graphs of linear functions
7.1
8.7
Intercepts of linear
7.3
8.8
Gradient of straight line
7.3
8.9
Equation of straight line
7.4
8.10
Parallel and perpendicular lines
7.5
8.11
Length and midpoint of segments
7.6, 7.7
8.12
Simultaneous equations solved graphically
7.8
8.13
Illustrate inequality
21.3→21.7
8.14
Graph of linear equations in two variables
7.1→7.8
*8.15
Linear programming
21.1→21.7
8.16
Composite functions
16.2, 16.3
8.17
Concept of inverse
16.4
8.18
Find inverse
16.4
8.19
Evaluate inverse
16.4
8.20
Inverse of composite
16.4
8.21
Quadratic features
8.5→8.10
*8.22
Axis of symmetry, turning point in completing square formula
20.1→20.4
*8.23
Sketch from completing square formula
20.2, 20.4
*8.24
Non-linear function graphs
22.1
*8.25
Travel graphs
22.2, 22.4
Geometry trigonometry
9.1
Concepts of geometry
10.1
9.2
Draw and measure angles
10.2
9.3
Construct lines etc.
10.2
9.4
Symmetry
10.3
9.5
Geometrical problems
11.1→11.3
9.6
Translations with vectors
12.1
9.7
Image under transformation
12.1→12.4
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Syllabus heading
Section in book
9.8
Object and image
12.1→12.4
9.9
Describe transformations
12.1→12.4
9.10
Combination of transformations
12.3
9.11
Relation between object and image
12.1→12.4
9.12
Pythagoras
14.1
9.13
Find trigonometric ratios
14.2
9.14
Use trigonometric ratios
14.2
9.15
Practical use of trigonometric ratios
14.2, 14.3
9.16
Sine and cosine rules
14.4→14.8
9.17
Find position given bearing
14.3
9.18
Determine bearing
14.3
9.19
Problems with bearings
14.3
*9.20
3-dimension problems
23.3→23.5
9.21
Circle theorems
15.1→15.9
Vectors matrices
10.1
Concept of vectors
12.1, 18.1
10.2
Combine vectors
18.1
10.3
Position vectors
18.2
10.4
Magnitude of vectors
18.3
10.5
Vectors in geometry
18.1→18.4
24.1, 24.2
10.6
Concepts of matrices
19.1
10.7
Combining matrices
19.2→19.4
10.8
Determinant of 2 by 2
19.6
10.9
Singular matrices
19.6
10.10
Inverse of 2 by 2
19.6
10.11
Matrix from transformation
24.3→24.6
10.12
Matrix of combinations
24.7
10.13
Matrices to solve problems
24.8
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Unit 1 Number theory
Key skills developed in this unit
What you need to know
▶ How to classify numbers
▶ Set notation
▶ How to relate different sets of numbers
▶ Venn diagrams
▶ How to find an inverse
▶ Understand closure
▶ Use the distributive and associative laws
1.1 Classifying numbers
(i) Different numbers
Different types of numbers can be put into various sets. These sets are defined as follows:
The set of counting numbers is N = {1, 2, 3, 4, 5, …}
The set of whole numbers is W = {0, 1, 2, 3, 4, 5, …}
Whole numbers are also called integers.
The set of all integers is Z = {… , –3, –2, –1, 0, 1, 2, 3, …}
The set of positive integers is Z+ = {1, 2, 3, 4, 5, …}
The set of negative numbers is Z– = {–1, –2, –3, –4, –5, …}
A number that can be written as a fraction is called a rational number. Rational numbers are denoted
by Q.
1
4
For example, 2 ∈ Q, and –4 ∈ Q (because –4 = – 1 )
A number that cannot be written
_ as a fraction is called an irrational number. Irrational numbers can
be denoted by either Q′ or I or Q.
For example, π ∈ Q′ and √3 ∈ Q′.
Note:
(i) All exact decimals are fractions, and so they are rational.
(ii) All recurring decimals are fractions, and so they are rational.
..
Hence, 3.45 ∈Q and 2.624 ∈Q.
The complete union of all rational and irrational numbers is called the set of real numbers, and is denoted by R.
Example 1
Copy and complete each statement. Write the appropriate symbol to make each statement true.
3
(a) 23 □ Z+ (b) 4 □ W (c) –4 □ Z
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Number theory •
Unit 1
Solution
(a) 23 is a positive integer, and so 23 ∈ Z+
3
3
(b) 4 is not a whole number, and so 4 ∉ W
(c) –4 is an integer, and so –4 ∈ Z
Example 2
Draw a Venn diagram to show the relationship between the sets Z+,
Q and R.
R
Q
Solution
Z+
Since both Z+ and Q are subsets of R, we can use R as the universal set.
Also, Z+ ⊂ Q.
Exercise 1.1A
In questions 1 to 15, copy each statement. Write the appropriate symbol to make each statement true.
3
1 2 □ Q 2 – 4 □ Q
3 –5 □ Z+
.
5 0.4 □ Q 6 π □ W 7 Q □ Z– 4 1.5 □ Q
9 N □ W
12 N □ Z+
10 0 □ Z+
11 √3 □ Q
8 W □ Z+
2
13 1000 □ R
14 3 □ N
15 Q □ R
16 Draw a Venn diagram to show the relationship between the sets N, Q and Z–.
(i) Ordering real numbers
All real numbers (whether they are rational or irrational) can be represented on the number line by a
point. They can be put into order of size.
3
The numbers 8 , 0.342, √0.84 and 0.3 in increasing order are:
3
3
0.3, 0.342, 8 , √0.84 (√0.84 = 0.92, 8 = 0.375)
2 2
The numbers 0, – 3 , 3 , –0.72, π in decreasing order are:
2
2
π, 3 , 0, – 3 , –0.72
2
(π = 3.142, 3 = 0.67)
Exercise 1.1B
Arrange the numbers in each set in order of size, smallest number first.
1 5 , √0.2 , 8π , 0.39
1
5
2 –2.3, 2π
, –2 3 , 2 17
3
3
3
3 – 7 , –0.3, (0.234)2, 0.1
4 15.9, 3.22, √200 , 4π
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Unit 1
We can convert between fractions, decimals and percentages as shown in the following.
3
• To convert 8 to a decimal, divide 8 into 3. The result is 0.375. To convert this to a percentage,
multiply by 100. The result is 37.5%.
3
So 8 , 0.375 and 37.5% are equivalent.
42
• Number theory
(ii) Using fractions, decimals and percentages
21
• To convert 0.42 to a fraction, write it as 100 , then simplify to 50 . To convert 0.42 to a percentage,
multiply by 100, obtaining 42%.
21
9
So 50 , 0.42 and 42% are equivalent.
• To convert 45% to a decimal, divide by 100, obtaining 0.45. To convert this to a decimal, divide by
45
9
100, obtaining 100, and then simplify to 20 .
So 20 , 0.45 and 45% are equivalent.
We can find a fraction or percentage of a quantity by multiplying.
4
4
For example: 9 of 4.5 m is 9 × 4.5 = 2 m.
60
60% of 30 kg is 100 × 30 = 18 kg.
We can express one quantity as a fraction or percentage of another by dividing.
80
1
For example: 80 cm as a fraction of 4 m is 400 = 5 .
80
As a percentage, it is 400 × 100 = 20%.
Exercise 1.1C
(Simplify answers if possible. Where relevant give answers correct to 3 significant figures.)
1 Convert the following.
13
2
a 20 to a decimal and a percentage
c 0.9 to a fraction and a percentage
b 7 to a decimal and percentage
d 0.815 to a fraction and a percentage
e 48% to a decimal and a fraction
f 87.5% to a fraction and a decimal
2 In a class of 50 pupils, 22 are girls.
a What fraction are girls?
b What percentage are girls?
3 In a herd of 90 cattle, 25 are bulls.
a What fraction are bulls?
b What percentage are bulls?
4 There are 1800 books in a school library. 450 of the books are fiction.
a What percentage are fiction?
b What fraction are non-fiction?
5 Find the following.
3
a 5 of J$900
d 15% of 80 kg
5
b 8 of 640 litres
e 82% of TT$500
3
c Half of 2 4 metres
1
f 33 3 % of 36 hours
6 Dwight received EC$480 to spend during his holiday. He spent five-eighths of it in the first week.
How much did he have left?
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Number theory •
Unit 1
1.2 Operations, identity and inverse
The basic operations on numbers are
Addition +
Subtraction –
Multiplication ×
Division ÷
(i) Addition and subtraction
For any two numbers, a + b = b + a
This is the commutative law of addition.
Note that because a – b ≠ b – a, subtraction is not commutative
The identity under an operation leaves a number unchanged under that operation.
Because a + 0 = 0 + a = a, then 0 (zero) is the identity under addition.
(ii) Multiplication and division
Since a × b = b × a, multiplication is commutative.
However, a ÷ b ≠ b ÷ a , and so division is not commutative.
a×1=1×a
Hence, 1 is the identity under multiplication.
(iii) Inverse
Addition
If the result of combining two numbers under an operation is the identity, then one number is the
inverse of the other.
Here is an example:
4 + –4 = 0
We say that –4 is the additive inverse of 4 under addition. We also say that 4 is the additive inverse
of –4 under addition.
The additive inverse of x under addition is –x.
Multiplication
Look at this multiplication:
3
4
4
×3=1
4
3
3
This means that 3 is the multiplicative inverse of 4 under multiplication, and also that 4 is the
4
multiplicative inverse of 3 under multiplication.
1
The multiplicative inverse of x under multiplication is x . This is also called the reciprocal.
There may be a button on your calculator for this, either x –1 or
1
x
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Unit 1
• Number theory
Example 3
(a) Find the additive inverse of each of these.
1
(i)7
(ii)– 4 (iii)–2.4
(b) Find the multiplicative inverse of each of these.
3
(i)2
(ii)– 4 (iii)2.5
Solution
1
(a) (i)–7
1
(b) (i) 2
1
(ii)– – 4 = 4 3
(iii)– –2.4 = 2.4
4
(ii)1 ÷ – 4 = – 3 1
(iii) 2.5 = 0.4
Exercise 1.2
In questions 1 to 10, find the additive inverse of each number.
1 0.5
2 7
3
–2
1
3
4 –2.8 5 4
2
9 – 3 10 2y
6 –1 2 7 –5.6
8 1000
In questions 11 to 20, find the multiplicative inverse of each number.
11 3
5
12 6 13 0.2
14 4x
15 –4
3
16 25 17 – 4 18 t2 19 100 20 –0.01
1.3 Closure
Since 4 + 8 = 12, we can see that if you add two positive integers, the result is another positive integer.
We say that the set of positive integers Z+ is closed under addition.
However, 4 – 8 = –4, and so Z+ is not closed under subtraction.
Example 4
Which of the following are closed?
(a) Q under multiplication
(b) Z– under addition
(c) N under division
Solution
(a) A fraction × a fraction = a fraction
It is closed.
(b) –4 + –7 = –11 for example, and so adding two negative integers gives a negative integer. It is
closed.
(c) 3 ÷ 4 = 0.75, which is not an integer.
It is not closed.
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Number theory •
Unit 1
Exercise 1.3
Which of these is closed?
1 N under subtraction
2 Q under addition
3 R under division
4 Z+ under multiplication
5 W under multiplication
6 Q under division
7 Z– under subtraction
8 Z– under multiplication
1.4 Distributive law and associative law in number problems
We know that x(a + b) = x × (a + b) = x × a + x × b = xa + xb.
We also know that x(a – b) = x × (a – b) = x × a – x × b = xa – xb.
This is known as the distributive law of multiplication over addition or subtraction.
To work out a + b + c, you can bracket it as (a + b) + c or a + (b + c).
In other words, (a + b) + c = a + (b + c).
This is known as the associative law of addition.
Also a × b × c = (a × b) × c = a × (b × c)
This is the associative law of multiplication.
By using the distributive, associative and commutative laws, we can often make calculations easier.
Example 5
Work out the following in the simplest way.
(a) 23 × 47 + 23 × 53
(b) 193 + 76 + 107
(c) 25 × 113 × 0.4
Solution
(a) The distributive law means we can factorise this as:
23 × (47 + 53) = 23 × 100 = 2300
(b) The associative law means we can write this as:
193 + 76 + 107 = (193 + 107) + 76 = 300 + 76 = 376
(c) The commutative law means we can write this as (25 × 0.4) × 113 = 10 × 113 = 1130
Exercise 1.4
Use the distributive or associative laws to work out the following calculations. Do not use a calculator.
1 19 × 38 + 19 × 62
2 25 × 39 × 8
3 4.9 + 11.8 + 23.2
4 2.5 × 11.8 × 4
5 123 × 56 + 123 × 44
6 4 4 + 11 11 + 2 4
3
7
1
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Unit 1
In all of the calculations we have done so far, base 10 (denary) has been used.
Remember that a number such as 6742 is really:
(6 × 103) + (7 × 102) + (4 × 101) + 2
• Number theory
1.5 Other bases (revision)
We are using powers of 10.
However, any number can be used for the base.
(i) Binary
Base 2 (binary) is the base used by a computer to represent a number. It uses only the digits 0 and 1,
and works using the powers of 2 (1, 2, 4, 8, 16, 32, …) as opposed to base 10, which uses the powers
of 10 (1, 10, 100, 1000, …)
A number such as 11012 is a binary number. The subscript 2 indicates that 2 is the base.
Example 6
Convert the binary number 11012 into base 10.
Solution
11012 = 1 × 23 + 1 × 22 + 0 × 21 + 1
=1×8+1×4+1
= 13
Hence: 11012 = 13 in base 10
Example 7
Convert the following numbers into binary.
(a) 53 (b)1043
Solution
(a) Keep dividing by 2 and write down the remainders.
(b)
2 ) 53
2 ) 26 r 1
2 ) 13 r 0
Now read upwards.
2 ) 6 r 1
2 ) 3 r 0
2 ) 1 r 1
2 ) 0 r 1
Hence: 53 = 110 1012
2 ) 1043
2 ) 521 r 1
2 ) 260 r 1
2 ) 130 r 0
2 ) 65 r 0
2 ) 32 r 1
Read upwards.
2 ) 16 r 0
2 ) 8 r 0
2 ) 4 r 0
2 ) 2 r 0
2 ) 1 r 0
2 ) 0 r 1
So: 1043 = 10 000 010 0112
Notice that considerably more digits are needed for a number expressed in binary.
Calculations in binary arithmetic are very simple, because only 0 and 1 are used.
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Number theory •
Unit 1
Example 8
Evaluate in binary arithmetic.
(a) 1011 + 111 + 1
(b) 10 101 – 1110
(c) 110 × 11
Solution
(a)     1011
     111
+      1
+ 10011
1 1 1
So: 1011 + 111 + 1 =
(b)
10 011
1 2
2 0 2
  10101
–   1110
     111
So: 10 101 – 1110 = 111
(c)     110
×     11
    1100
     110
  10010
1
So: 110 × 11 = 10 010
(ii) Other bases
Example 9 illustrates how we work in other bases.
Example 9
(a) Change 123 into base 3.
(b) Find 3115 + 4245 + 435.
(c) Work out 248 × 358.
Solution
(a)
3 ) 123
2 ) 41 r 0
2 ) 13 r 2
Read upwards.
2 ) 4 r 1
2 ) 1 r 1
2 ) 0 r 1
123 = 11 1203
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Unit 1
• Number theory
(b)    311
    424 1 + 4 + 3 = 8 = 5 + 3 = 135 (base 5)
+    43
  1333
   1 1 1
The answer is 13335.
(c)     24
×    35
    740
    144
  1104
The answer is 11048.
Exercise 1.5
1 Convert these numbers into binary.
a 27
b 56
c 235
d 59
e 102
f 78
g 106
h 305
i 244
j 98
e 11 111
2 Convert these binary numbers into base 10.
a 1100
b 10 101
c 10 111
d 111
f 100 001
g 1 101 101
h 110 111
i 101 010 101
3 Evaluate in binary arithmetic, and then convert your answer to denary.
a 111 + 1111
b 111 × 101
c 11 × (101 + 11)
d 1111 – 1011
e (111 – 11) × (11 011 – 1101)
4 Convert these numbers into base 5.
a 46
b 67
c 109
d 348
e 244
5 Evaluate in base 8.
a 34 × 22
b 45 – 17
c 567 + 123 + 456
Summary
▶ Numbers can be put into a category, integer, rational or irrational.
▶ Addition and multiplication are commutative.
▶ Addition and multiplication have identities of 0 and 1 respectively.
▶ An inverse combines with a number to produce the identity.
▶ If the result of combining two numbers from a set gives an answer in the same set, we say that the
operation is closed.
▶ Any whole number can be used as a base for writing a number. Binary arithmetic is very important
in computing.
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Number theory •
Unit 1
Progress exercise 1
1 Write down the first eight prime numbers. 2 Find two irrational numbers between 1 and 2.
3 Is the set Q closed under addition? 4 What is the additive inverse of –1.6?
5 What is the multiplicative inverse of 0.5? 6 What is the additive inverse of 4x?
3
7 Work out without a calculator: 8 × 33 × 12.5 8 What is the multiplicative inverse of 1 4 ?
9 Is the set Z+ closed under division?
10 Evaluate 1942 – 1062 without a calculator.
11 Change these numbers to binary.
a 89
b 432
c 128
d 100
12 Evaluate 111 × 1011 × 11 in binary.
13 Evaluate 2322 + 1223 + 333 in base 4.
Multiple choice test 1
1 The number of prime numbers between 20 and 50 is
A 6
B 7
C 8
D 9
2 The sum of two rational numbers is
I always rational II sometimes an integer III sometimes irrational
A I and II
B I and III
C I only
D II and III
B 13 × 19
C 14 × 13
D none of these
C –x2 + 2x
D 0
3 14 × 19 – 19 =
A 14
4 The additive inverse of x2 – 2x is
A –2x + x2
5 If
p
q
B x2 + 2x
is a recurring decimal, then
A q must be a multiple of 3
B q must be odd
C q must be prime
D q is not a power of 2
6 4 × 104 + 2 × 103 =
I 6 × 107
II 42 000
A II only
B III only
7 The multiplicative inverse of
4
A – 11 B
4
3
III 6 × 104
3
24
C II and III only D I and III only
is
C
11
4
4
D 11
8 In base 5, the value of the digit 4 in 24335 is
A 400
B 40
C 100
D 25
9 What are the next two numbers in the binary sequence?
1, 100, 111, 1010, 1101, 10 000, … A 10 011, 10 100
B 10 011, 10 110
C 11 011, 11 110
D 10 101, 10 111
10
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