**Unformatted text preview: **Unit 1 Algebra Welcome to MST124 Welcome to MST124
In this module you’ll learn the essential ideas and techniques that underpin
university-level study in mathematics and mathematical subjects such as
physics, engineering and economics. You’ll also develop your skills in
communicating mathematics.
Here are some of the topics that you’ll meet.
Vectors are quantities that have both a size and a direction. For example,
a ship on the ocean moves not only with a particular speed, but also in a
particular direction. Speed in a particular direction is a vector quantity
known as velocity.
Calculus is a fundamental topic in mathematics that’s concerned with
quantities that change continuously. If you know that an object is moving
at a constant speed, then it’s straightforward to work out how much
distance it covers in any given period of time. It’s not so easy to do this if
the object’s speed is changing – for example, if it’s accelerating, as a
falling object does. Calculus can be used to deal with situations like this. The motion of a ship consists
of speed and direction Matrices are rectangular arrays of numbers – for example, any
rectangular table of numbers forms a matrix. Matrices have many
applications, which involve performing operations on them that are similar
to the operations that you perform on individual numbers. For example,
you can add, subtract and multiply matrices.
Sequences are lists of numbers. Sequences whose numbers have a
connecting mathematical relationship arise in many diﬀerent contexts. For
example, if you invest £100 at a 5% rate of interest paid annually, then the
value in pounds of your investment at the beginning of each year forms the
sequence 100, 105, 110.25, 115.76, 121.55, . . ..
The complex numbers include all the real numbers that you know about
already, and also many ‘imaginary’ numbers, such as the square root
of −1. Amazingly, they provide a simple way to deal with some types of
complicated mathematics that arise in practical problems.
You’ll see that not only do the topics above have important practical
applications, but they’re also intriguing areas of study in their own right.
One of the main aims of the ﬁrst few units of MST124 is to make sure that
you’re conﬁdent with the basic skills in algebra, graphs, trigonometry,
indices and logarithms that you’ll need. The mathematics in the later
units of the module depends heavily on these basic skills, and you’ll ﬁnd it
much easier and much quicker to study and understand if you can work
with all the basic skills ﬂuently and correctly.
To help you attain conﬁdence with these skills, the ﬁrst few units of the
module include many revision topics, as well as some new ones. Which
parts, and how much, of the revision material you’ll need to study will
depend on your mathematical background – diﬀerent students start
MST124 with widely diﬀering previous mathematical experiences. When
you’re deciding which revision topics you need to study, remember that The speed of a falling ball
increases as it falls 3 Unit 1 Algebra
even though you’ll have met most of them before, you won’t necessarily
have acquired the ‘at your ﬁngertips’ ﬂuency in working with them that
you’ll need. Where that’s the case, you’ll beneﬁt signiﬁcantly from working
carefully through the revision material. Information for joint MST124 and MST125 students
If you are studying Essential mathematics 2 (MST125) with the same
start date as MST124, then you should not study the MST124 units
on the dates shown on the main MST124 study planner. Instead, you
should follow the MST124 and MST125 joint study planner, which is
available from the MST124 and MST125 websites. This is important
because you will not be prepared to study many of the topics in
MST125 if you have not already studied the related topics in MST124.
The MST124 and MST125 joint study planner ensures that you study
the units of the two modules interleaved in the correct order.
The MST124 assignment cut-oﬀ dates shown in the MST124 and
MST125 joint study planner are the same as those shown on the
MST124 study planner. Introduction
The main topic of this ﬁrst unit is basic algebra, the most important of the
essential mathematical skills that you’ll need. You’ll ﬁnd it diﬃcult to
work through many of the units in the module, particularly the calculus
units, if you’re not able to manipulate algebraic expressions and equations
ﬂuently and accurately. So it’s worth spending some time now practising
your algebra skills. This unit gives you the opportunity to do that.
The unit covers a lot of topics quite rapidly, in the expectation that you’ll
be fairly familiar with much of the material. You should use it as a
resource to help you make sure that your algebra skills are as good as they
can be. You may not need to study all the topics – you should concentrate
on those in which you need practice. For many students these will be the
topics in Sections 3 to 6. A good strategy might be to read through the
whole unit, doing the activities on the topics in which you know you need
practice. For the topics in which you think you don’t need practice, try one
or two of the later parts of each activity to make sure – there may be gaps
and rustiness in your algebra skills of which you’re unaware. Remember to
check all your answers against the correct answers provided (these are at
the end of the unit in the print book, and can be obtained by pressing the
‘show solution’ buttons in some screen versions).
As with all the units in this module, further practice questions are available
in both the online practice quiz and the exercise booklet for the unit. 4 1 Numbers Working through the revision material in this unit should also help you to
clarify your thinking about algebra. For example, you might know what
to do with a particular type of algebraic expression or equation, but you
might not know, or might have forgotten, why this is a valid thing to do. If
you can clearly understand the ‘why’, then you’ll be in a much better
position to decide whether you can apply the same sort of technique to a
slightly diﬀerent situation, which is the sort of thing that you’ll need to do
as you study more mathematics.
Some of the topics in the unit may seem very easy – basic algebra is
revised starting from the simplest ideas. Others may seem quite
challenging – some of the algebraic expressions and equations that you’re
asked to manipulate may be more complicated than those that you’ve dealt
with before, particularly the ones involving algebraic fractions and indices.
The ﬁnal section of the unit, Section 6, describes some basic principles of
communicating mathematics in writing. This will be important
throughout your study of this module and in any further mathematical
modules that you study.
If you ﬁnd that much of the content of this unit (and/or Unit 2) is
unfamiliar to you, then contact your tutor and/or Learner Support Team
as soon as possible, to discuss what to do.
The word ‘algebra’ is derived from the title of the treatise al-Kit¯
ab
al-mukhtas.ar f¯i h.is¯
ab al-jabr wa’l-muq¯
abala (Compendium on
calculation by completion and reduction), written by the Islamic
mathematician Muh.ammad ibn M¯
us¯a al-Khw¯
arizm¯i in around 825.
This treatise deals with solving linear and quadratic equations, but it
doesn’t use algebra in the modern sense, as no letters or other
symbols are used to represent numbers. Modern, symbolic algebra
emerged in the 1500s and 1600s. 1 Numbers
In this section you’ll revise diﬀerent types of numbers, and some basic
skills associated with working with numbers. 5 Unit 1 Algebra 1.1 Types of numbers
We’ll make a start by brieﬂy reviewing some diﬀerent types of numbers.
Remember that all the deﬁnitions given here, and all the other deﬁnitions
and important facts and techniques given in the module, are also set out in
the Handbook, so you can refer to them easily.
The natural numbers, also known as the positive integers, are the
counting numbers,
1, 2, 3, . . . .
(The symbol ‘. . . ’ here is called an ellipsis and is used when something has
been left out. You can read it as ‘dot, dot, dot’. In some texts the natural
numbers are deﬁned to be 0, 1, 2, 3, . . . .)
The natural numbers, together with their negatives and zero, form the
integers:
. . . , −3, −2, −1, 0, 1, 2, 3, . . . .
The Latin word integer consists of the preﬁx in, meaning ‘not’,
attached to the root of tangere, meaning ‘to touch’. So it literally
means ‘untouched’, in the sense of ‘whole’. The rational numbers are the numbers that can be written in the form
integer
;
integer
that is, as an integer divided by an integer.
For example, all the following numbers are rational numbers:
3
4, 2 13 , 4, −4, − 78 , 0.16, 7.374. You can check this by writing them in the form above, as follows:
3
7
4
−4
−8
16
7374
,
,
,
,
,
,
.
4
3
1
1
7
100
1000
The real numbers include all the rational numbers, and many other
numbers as well. A useful way to think of the real numbers is to envisage
them as lying along a straight line that extends inﬁnitely far in each
direction, called the number line or the real line. Every point on the
number line corresponds to a real number, and every real number
corresponds to a point on the line. Some points on the line correspond to
rational numbers, while others correspond to numbers that are not
rational, which are known as irrational numbers. Figure 1 shows some
numbers on the number line. 6 1 −
−4 −3 −2 √ 2 − 25 −1 √
2 1
2 0 1 9
2 e π
2 3 Numbers 4 5 Figure 1 Some numbers on the number line √ √
Four of the numbers√marked in Figure 1 are irrational, namely − 2, 2, e
and π. The number 2 is the positive square root of 2, that is, the positive
number that when multiplied by itself
√ gives the answer 2. Its value is
approximately 1.41. The number − 2 is the negative of this number. The
numbers e and π are two important constants that occur frequently in
mathematics. You probably know that π is the number obtained by
dividing the circumference of any circle by its diameter (see Figure 2). Its
value is approximately 3.14. (The symbol π is a lower-case Greek letter,
pronounced ‘pie’.) The constant e has value approximately 2.72, and you’ll
learn more about it in this module, starting in Unit 3.
√ √
To check that the numbers − 2, 2, e and π are irrational, you have to
prove that they can’t be written as an integer
√ divided by an integer. If
you’d like to √
see how this can be done for 2, then look at the document
A proof that 2 is irrational on the module website. Proving that e and π
can’t be written as an integer divided by an integer is more diﬃcult, and
outside the scope of this module. circumference diameter Figure 2 The circumference
and diameter of a circle Every rational number can be written as a decimal number. To do this,
integer
you divide the top number of the fraction of the form
by the
integer
bottom number. For example,
1
8 = 0.125,
2
3 = 0.666 666 666 . . . ,
83
74 = 1.1 216 216 216 216 . . . . As you can see, the decimal form of 18 is terminating: it has only a ﬁnite
number of digits after the decimal point. The decimal forms of both 23 and
83
74 are recurring: each of them has a block of one or more digits after the
decimal point that repeats indeﬁnitely. There are two alternative notations
for indicating a recurring decimal: you can either put a dot above the ﬁrst
and last digit of the repeating block, or you can put a line above the whole
repeating block. For example,
2
˙
and
3 = 0.666 666 666 . . . = 0.6 = 0.6,
83
˙ ˙
74 = 1.1 216 216 216 216 . . . = 1.1216 = 1.1216 . In fact, the decimal form of every rational number is either terminating or
recurring. Also, every terminating or recurring decimal can be written as
an integer divided by an integer and is therefore a rational number. If
you’d like to know why these facts hold, then look at the document
Decimal forms of rational numbers on the module website. 7 Unit 1 Algebra
The decimal numbers that are neither terminating nor recurring – that is,
those that are inﬁnitely long but have no block of digits that repeats
indeﬁnitely – are the irrational numbers. This gives you another way to
distinguish between the rational and irrational numbers, summarised
below. Decimal forms of rational and irrational numbers
The rational numbers are the decimal numbers that terminate or
recur.
The irrational numbers are the decimal numbers with an inﬁnite
number of digits after the decimal point but with no block of digits
that repeats indeﬁnitely. So, for example, the irrational number π has a decimal expansion that is
inﬁnitely long and has no block of digits that repeats indeﬁnitely. Here are
its ﬁrst 40 digits:
π = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 . . . .
You might like to watch the one-minute video clip entitled The decimal
expansion of π, available on the module website.
In 2006, a Japanese retired engineer and mental health counsellor,
Akira Haraguchi, recited the ﬁrst 100 000 digits of π from memory. It
took him 16 hours. Natural
numbers
Integers
Rational
numbers
Real
numbers Figure 3 Types of numbers 8 Figure 3 is a summary of the types of numbers mentioned in this
subsection. It illustrates that all the natural numbers are also integers, all
the integers are also rational numbers, and all the rational numbers are
also real numbers.
In Unit 12 you’ll learn about yet another type of number. The complex
numbers include all the numbers in Figure 3, and also many ‘imaginary’
numbers, such as the square root of −1. The idea of imaginary numbers
might seem strange, but these numbers are the foundation of a great deal
of interesting and useful mathematics. They provide a natural, elegant way
to work with seemingly complicated mathematics, and have many practical
applications. 1 Numbers 1.2 Working with numbers
In this subsection, you’ll revise some basic skills associated with working
with numbers. It’s easy to make mistakes with these particular skills, and
people often do! So you should ﬁnd it helpful to review and practise them.
Before doing so, notice the label ‘(1)’ on the right of the next paragraph.
It’s used later in the text to refer back to the contents of the line in which
it appears. Labels like this are used throughout the module. The BIDMAS rules
When you evaluate (ﬁnd the value of) an expression such as
200 − 3 × (1 + 5 × 23 ) + 7, (1) it’s important to remember the following convention for the order of the
operations, so that you get the right answer. Order of operations: BIDMAS
Carry out mathematical operations in the following order.
B
I
D
M
A
S brackets
indices (powers and roots)
0
divisions
same precedence
multiplications
0
additions
same precedence
subtractions Where operations have the same precedence, work from left to right. As you can see, the I in the BIDMAS rules refers to ‘indices (powers and
roots)’. Remember that raising a number to a power means multiplying it
by itself a speciﬁed number of times. For example, 23 (2 to the power 3)
means three 2s multiplied together:
23 = 2 × 2 × 2.
In particular, squaring and cubing a number mean raising it to the
powers 2 and 3, respectively. When you write an expression such as 23 ,
you’re using index notation. Taking a root of a number means taking its
square root, for example, or another type of root. Roots are revised in
Subsection 4.1.
If you type an expression like expression (1) into a calculator of the type
recommended in the MST124 guide, then it will be evaluated according to
the BIDMAS rules. However, it’s essential that you understand and
remember the rules yourself. For example, you’ll need to use them when
you work with algebra.
Example 1 reminds you how to use the BIDMAS rules. It also illustrates
another feature that you’ll see throughout the module. Some of the worked
examples include lines of blue text, marked with the following icons
. 9 Unit 1 Algebra
This text tells you what someone doing the mathematics might be
thinking, but wouldn’t write down. It should help you understand how you
might do a similar calculation yourself. Example 1 Using the BIDMAS rules Evaluate the expression
200 − 3 × (1 + 5 × 23 ) + 7
without using your calculator. Solution
The brackets have the highest precedence, so start by evaluating
what’s inside them. Within the brackets, ﬁrst deal with the power,
then do the multiplication, then the addition.
200 − 3 × (1 + 5 × 23 ) + 7 = 200 − 3 × (1 + 5 × 8) + 7
= 200 − 3 × (1 + 40) + 7
= 200 − 3 × 41 + 7
Now do the multiplication, then the addition and subtraction from
left to right.
= 200 − 123 + 7
= 77 + 7
= 84 You can practise using the BIDMAS rules in the next activity. Remember
that where division is indicated using fraction notation, the horizontal line
not only indicates division but also acts as brackets for the expressions
above and below the line. For example,
1+2
1 + 32 means (1 + 2)
,
(1 + 32 ) that is, (1 + 2) ÷ (1 + 32 ). In a line of text, this expression would normally be written as
(1 + 2)/(1 + 32 ), with a slash replacing the horizontal line. The brackets
are needed here because 1 + 2/1 + 32 would be interpreted as
1 + (2/1) + 32 .
Part (b) of the activity involves algebraic expressions. Remember that
multiplication signs are usually omitted when doing algebra – quantities
that are multiplied are usually just written next to each other instead
(though, for example, 3 × 4 can’t be written as 34). 10 1 Activity 1 Numbers Using the BIDMAS rules (a) Evaluate the following expressions without using your calculator.
(i) 23 − 2 × 3 + (4 − 2) (ii) 2 − 1
2 ×4 (iii) 4 × 32 1+2
(vi) 1 − 2/32
1 + 32
(b) Evaluate the following expressions when a = 3 and b = 5, without
using your calculator.
* +
b
2
(i) 3(b − a)
(ii) a + b(2a + b)
(iii) a + 9
(iv) 30/(ab)
a
(iv) 2 + 22 (v) Rounding
When you use your calculator to carry out a calculation, you often need to
round the result. There are various ways to round a number. Sometimes
it’s appropriate to round to a particular number of decimal places (often
abbreviated to ‘d.p.’). The decimal places of a number are the positions of
the digits to the right of the decimal point. You can also round to the
nearest whole number, or to the nearest 10, or to the nearest 100, for
example. More often, it’s appropriate to round to a particular number of
signiﬁcant ﬁgures (often abbreviated to ‘s.f.’ or ‘sig. ﬁgs.’). The ﬁrst
signiﬁcant ﬁgure of a number is the ﬁrst non-zero digit (from the left), the
next signiﬁcant ﬁgure is the next digit along (whether zero or not), and so
on.
Once you’ve decided where to round a number, you need to look at the
digit immediately after where you want to round. You round up if this
digit is 5 or more, and round down otherwise. When you round a number,
you should state how it’s been rounded, in brackets after the rounded
number, as illustrated in the next example.
Notice the ‘play button’ icon next to the following example. It indicates
that the example has an associated tutorial clip – a short video in which a
tutor works through the example and explains it. You can watch the clip,
which is available on the module website, instead of reading through the
worked example. Many other examples in the module have tutorial clips,
indicated by the same icon. 11 Unit 1 Algebra Example 2 Rounding numbers Round the following numbers as indicated.
(a) 0.0238 to three decimal places
(b) 50 629 to three signiﬁcant ﬁgures
(c) 0.002 958 2 to two signiﬁcant ﬁgures Solution
(a) Look at the digit after the ﬁrst three decimal places: 0. 023 8.
It’s 8, which is 5 or more, so round up.
0.0238 = 0.024 (to 3 d.p.) (b) Look at the digit after the ﬁrst three signiﬁcant ﬁgures:
506 29. It’s 2, which is less than 5, so round down.
50 629 = 50 600 (to 3 s.f.) (c) Look at the digit after the ﬁrst two signiﬁcant ﬁgures:
0.00 29 582. It’s 5, which is 5 or more, so round up.
0.002 958 2 = 0.0030 (to 2 s.f.) Notice that in Example 2(c), a 0 is included after the 3 to make it clear
that the number is rounded to two signiﬁcant ﬁgures. You shoul...

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