# giáo trình mathematics fundamentals

DR RAMZAN RAZIM KHAN & DR DES HILL

MATHEMATICS
FUNDAMENTALS

1

Mathematics Fundamentals
1st edition
© 2017 Dr Ramzan Razim Khan & Dr Des Hill & bookboon.com
ISBN 978-87-403-1864-7
Peer review by Dr Miccal Matthews, The University of Western Australia

2

CONTENTS

MATHEMATICS FUNDAMENTALS

CONTENTS
1

Numbers and operations

5

2

Algebra

17

3

Simultaneous equations

25

4

31

5

Indices

41

6

Re-arranging formulae

53

7

Exponentials and logarithms

57

8

Equations involving logarithms and/or exponentials

65

9

77

10

Functions and graphs

87

11

103

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MATHEMATICS FUNDAMENTALS

CONTENTS

12

Calculus

111

123

Appendix 2: Solutions to problem sets

141

Index

197

4

NUMBERS AND OPERATIONS

MATHEMATICS FUNDAMENTALS

1. Numbers and operations
Types of Numbers
The natural numbers are the usual numbers we use for counting:
1, 2, 3, 4, · · ·
We also have the very special number zero
0
and the negative numbers

· · · − 4, −3, −2, −1
which are used to indicate deﬁcits (that is, debts) or to indicate
opposite directions. These numbers collectively form the integers

· · · − 4, −3, −2, −1, 0, 1, 2, 3, 4, · · ·

The basic operations of arithmetic
Hopefully, the everyday operations +, −, × and ÷ are familiar:
(i) 3 + 4 = 7

(iii) 4 × 9 = 36

(ii) 11 − 6 = 5

(iv) 24 ÷ 6 = 4

Arithmetic involving negative numbers
When negative numbers are involved we need to be careful, and
must use brackets if necessary.
Multiplication examples:
(i) 4 × (−7) = −28

(iii) (−3) × (−6) = 18

(ii) (−5) × 6 = −30

(iv) 3 × 5 = 15

Other examples:
(i) 13 + (−8) = 5

(iv) −16 − 13 = −29

(ii) 21 − (−3) = 24

(v) (−12) ÷ 4 = −3

(iii) −16 + 13 = −3

(vi)

−12
=3
−4

5

positive × negative = negative
negative × positive = negative
negative × negative = positive
positive × positive = positive

NUMBERS AND OPERATIONS

MATHEMATICS FUNDAMENTALS

BIDMAS. The order of operations
Consider the simple question:
What is 2 + 3 × 4?
Which operation (+ or ×) do we perform ﬁrst?
Mathematical expressions are not simply calculated from left to
right. They are performed in the following order:
B:

Expressions within Brackets (. . .)

I:

Indices (powers, square roots, etc)

DM:
AS:

We will consider Indices in detail later

Divisions (÷) and Multiplications (×)

A couple of things to note:
• Division and multiplication have the same precedence.
• Addition and subtraction have the same precedence.
• Operations with the same precedence are performed left to right.
Some BIDMAS examples
(i) 2 + 3 × 4 = 2 + 12 = 14
(ii) (4 + 6) × (12 − 4) = 10 × 8 = 80
(iii) 4 + 7 × (6 − 2) = 4 + 7 × 4 = 4 + 28 = 32
(iv) 3 × (4 + 8) − 2 × (10 − 6) = 3 × 12 − 2 × 4 = 36 − 8 = 28

When brackets are involved we can
omit the multiplication sign. So
instead of 3 × (4 + 8) − 2 × (10 − 6) we
could write 3(4 + 8) − 2(10 − 6)

More examples

4÷2×3÷3

= 2×3÷3
= 6÷3
=2

− [2 − 3(4 − 6 ÷ 2 − (−3))]
= −[2 − 3(4 − 3 + 3)]
= −[2 − 3 × (1 + 3)]
= −(2 − 3 × 4)

When brackets are nested we
usually use different sorts, e.g.
(· · · )
[· · · ]
{· · · }

= −(2 − 12)
= −(−10)
= 10

6

NUMBERS AND OPERATIONS

MATHEMATICS FUNDAMENTALS

Word problems
We are sometimes asked quesions in written or spoken conversation that we will be able to use mathematics to help us solve.
Our ﬁrst task is to turn the ‘word problem’ into a mathematics
problem.
Example: What is three times the difference between 12 and 4?
3 × (12 − 4)

= 3×8
= 24

Another example: Joy takes the product of the sum of two and
six, and the difference between nineteen and nine. She then result
the answer by four. What number does she arrive at?
{(2 + 6) × (19 − 9)} ÷ 4

= {8 × 10} ÷ 4
= 80 ÷ 4
= 20

7

NUMBERS AND OPERATIONS

MATHEMATICS FUNDAMENTALS

The number line
The integers sit nicely on a number line:

-5

-4

-3

-2

-1

0

1

2

3

4

5

Fractions
These are numbers that sit between the integers. For example, the
number “one-half”, which we write “ 12 ”, is exactly halfway between
0 and 1.
1
2

-5

s
-4

-3

-2

-1

0

1

2

3

4

5

Fractions are ratios of integers, that is,
p
q

Note that p or q or both can be negative

where the numerator p and denominator q are integers. When the
numerator (top line) is smaller than the denominator (bottom line)
we call this a proper fraction.
The number “four and two-ﬁfths” is written “4 52 ” and is between
4 and 5. To pinpoint this number, imagine splitting up the interval
between 4 and 5 into ﬁfths (ﬁve equal pieces). Now, starting at 4,
move right along two of these ﬁfths.

-5

s
-4

-3

-2

-1

0

1

2

3

4

5

We call 4 25 a mixed numeral, as it is the sum of an integer and a
proper fraction.

Converting mixed numerals to improper fractions
An improper fraction is one in which the numerator is larger than
the denominator.
Example: We wish to convert the mixed numeral 3 21 into an
improper fraction. We have three wholes and one half:

or three lots of two halves (making six) plus one half is
seven halves:

so

3

7
1
=
2
2

8

NUMBERS AND OPERATIONS

MATHEMATICS FUNDAMENTALS

Arithmetically, we write
3

3×2+1
6+1
7
1
=
=
=
2
2
2
2

Some examples
4×3+1
12 + 1
13
1
=
=
=
3
3
3
3

4

2
1

1
2×4+1
8+1
9
=
=
=
4
4
4
4

5
1×6+5
6+5
11
=
=
=
6
6
6
6

3
8×5+3
40 + 3
43
=
=
=
5
5
5
5
Note: Improper fractions turn out to be used more often in
maths, mainly because mixed numbers are confusing in algebra.
8

Multiplication of fractions
To multiply two fractions simply multiply their numerators and
denominators separately. For example,
1×5
5
1 5
× =
=
2 6
2×6
12

Think of this as ﬁnding “one-half of
ﬁve-sixths”

Another example is
3 2
×
5 7

=

3×2
5×7

=

6
35

Division of fractions
The notation 2 ÷ 3 means the same as 32 , that is “two-thirds”. Now
think of 2 and 3 as fractions, that is
2÷3 =

2 3
÷
1 1

We can also use fraction multiplication to say that
2 1
2
= ×
3
1 3
Comparing these last two we have
2 1
2 3
÷ = ×
1 1
1 3
That is, dividing by a fraction is equivalent to multiplying by its
reciprocal, that is, the fraction having been inverted. For example,
2 9
18
2 5
÷ = × =
7 9
7 5
35

9

NUMBERS AND OPERATIONS

MATHEMATICS FUNDAMENTALS

More examples
1 2 5
× ÷
3 7 6
1
4

×
3
5

7
8

7
32
3
5

=

5 2
1
2 ÷ ×3
4 3
7

=

=

2
5
÷
21 6

=

=

7
3
÷
32 5

13 2 22
÷ ×
4
3
7

=

2
6
×
21 5

=

=

7
5
×
32 3

13 3 22
× ×
4
2
7

12
105

=

=

35
96
858
56

Factorization of numbers
This is the process of expressing a given number as a product of
two (or more) smaller numbers. For example,
28 = 4 × 7
108

64

= 2 × 54
= 2×6×9

= 2 × 32
= 2 × 2 × 16
···
= 2×2×2×2×2×2

Later we’ll write this as 64 = 26

360°
thinking

.

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10

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NUMBERS AND OPERATIONS

MATHEMATICS FUNDAMENTALS

Cancellation of numbers
The process of simplifying fractions by looking for factors which
are common to both the numerator and the denominator and then
eliminating (cancelling) them. For example,
18
20

=

✁2 × 9
✁2 × 10

=

9
10

100
250

=

10 × ✚
10

25 × ✚
10

=

10
25

=

5
✁×2
5×5

=

2
5

24
108

=

✁4 × 6
✁4 × 27

=

6
27

=

3×2

3×9

=

2
9

Consider the sum

1 2
+
3 5
We can’t add these fractions yet because thirds are different to
ﬁfths. However, we can make use of reverse cancellation to make
the denominators the same:
1 2
+
3 5

=
=

1×5
2×3
+
3×5
5×3
6
5
+
15
15

Now we just have a total of 5 + 6 = 11 ﬁfteenths, that is
1 2
+
3 5

=
=

5+6
15
11
15

9
− 52 . Following
The same applies to subtraction. For example, 20
the above pattern the denominator would be 20 × 5 = 100, but we
can simplify the numbers a little as follows. Since 5 divides into 20
we could just do this instead:

9
2

20 5

=
=
=
=

9
2×4

20 5 × 4
8
9

20 20
9−8
20
1
20

Two examples
1 4
1×7 4×4
7
16
23
+
=
+
=
+
=
4 7
4×7 7×4
28 28
28
5
12
9
5
21
5
16
2 1
+ −
=
+

=

=
9 6 54
54 54 54
54 54
54

11

which if required can be simpliﬁed:
2×8
8
= ✁
=
27
✁2 × 27

NUMBERS AND OPERATIONS

MATHEMATICS FUNDAMENTALS

Decimals
Decimals are a convenient and useful way of writing fractions with
denominators 10, 100, 1000, etc.
2
is written as 0.2
10
35
is written as 0.35
100
The number 237.46 is shorthand for

4
is written as 0.04
100
612
is written as 0.612
1000

4
6
+
10 100
The decimal point separates the whole numbers from the fractions.
To the right of the decimal point, we read the names of the digits individually. For example, 237.46 is read as ‘two hundred and
thirty-seven point four six’.
The places after the decimal point are called the decimal places.
We say that 25.617 has 3 decimal places.
Some fractions expressed as decimals
We can write
2 × 100 + 3 × 10 + 7 × 1 +

3
10

as

0.3

4
− 10

47
100

as

0.47

12
10

23
− 100

as

−0.23

327
100

931
1000

as

0.931

6
100

as

0.06

as

−0.4

as

1.2

as

3.27

28
1000

as

0.028

5
1000

as

0.005

= 1+

2
10

= 3+

27
100

Common fractions expressed as decimals
All fractions can be written as decimals. However, the decimal
sequence may go on forever.
Examples
1
= 0.1
10

1
= 0.01
100

1
= 0.5
2
3
= 0.75
4
1
= 0.3333 · · ·
3
1
= 0.142857142857 · · ·
7

1
= 0.25
4
1
= 0.2
5
2
= 0.6666 · · ·
3

1
= 0.001
1000
1
= 0.125
8
2
= 0.4
5
2
− = −0.4
5
22
1
= 3 + = 3.1428 · · ·
7
7

12

The · · · indicate that the sequence goes
on forever

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