Get Your Head Around: Basic Algebra I

146 Pages·2019·0.38 MB·English

by  Austin Hartnell-Jones

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Get Your Head Around:
Basic Algebra I
by
Austin Hartnell-Jones
© Copyright 2019 Austin Hartnell-Jones.
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Welcome to ‘Get Your Head Around: Basic
Algebra I’. You will probably be aware that
mathematics is considered to be the
language of science and technology. It is
also often used in many other important
areas of life, including economics and
finance.
If we think of mathematics as a ‘language’,
then we can think of algebra as being the
‘grammar’, making up the basic rules and
building blocks of that language.
Whilst mathematics is hugely important in
scientific applications, I probably won’t
use too many scientific examples, at least
not in the early stages of the book.
Instead, I will try to use more ‘everyday’
examples that can more easily be
visualised. This is to encourage you to get
a genuinely instinctive ‘feel’ for algebra.
Ideally, it should seem like a lot of
common sense. If it doesn’t, you might be
over-thinking it, or you might have missed
something earlier on.
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It is very important not to move on to a
new section until you have thoroughly
grasped the earlier sections at a
fundamental ‘common-sense’ level.
The only other thing I’ll say before we get
started is that the book begins with some
very simple stuff. Don’t worry if this seems
a little rudimentary, it will get more
interesting later on.
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1. Equations
Let us begin. First of all, we’ll introduce
the idea of ‘equations’, as well as some of
the symbols and notations used.
As you progress with your mathematical
education, you will pick up on the fact that
much of what is written down is basically
a short-hand way of writing statements
that could otherwise be written in plain-
English (but would take up a lot more
space).
Lets start with a plain-English statement.
Imagine we go to a grocer’s store and we
want to buy some eggs (four eggs to be
precise). We see that the grocer is selling
eggs at 25p (or £0.25) each. So, one egg
costs £0.25.
We wanted four eggs. As the grocer
unfortunately doesn’t give any discount for
multiple purchases, the cost for four eggs
is going to be four times the cost for one
egg. So, the total cost will be £1.00.
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This all seems very straightforward. Lets
write down our initial statement as an
equation:
1 egg = £ 0.25
Here, the equals symbol ‘=’ is basically
saying ‘has a cost equal to’. We can think
about the whole equation as being
equivalent to saying ‘the cost of one egg
equals £0.25’, or in more common English
‘the cost of one egg is £0.25’. As we are
talking about money and cost, it can be
assumed that this is what the equation is
about. All equations have two sides, the
left hand side and the right hand side. The
left hand side is what is on the left of the
equals sign ‘=’, and the right hand side is
what is on the right hand side of the
equals sign ‘=’. The equation ‘equates’ the
two sides. It says that they are equal to
each other, that they are the same. In this
case, as we are talking about cost and
money, the equation is about the cost of
the eggs. If we were talking about the
weight of the eggs, then the equation
would be about weight. In that case, we
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would obviously have a weight value on
the right hand side instead of a money
value. The context of an equation will
generally be known to us.
It is useful to note that there is nothing
particularly important about which side of
the equation things go on. We could also
have written the previous equation as:
£ 0.25 = 1 egg
This isn’t perhaps the way we would say it
if we were talking, but it is perfectly fine
as an equation. We might say it this way if
we said ‘£0.25 will buy me one egg’.
Now, lets write down the equation for ‘the
cost of four eggs is £1.00’:
4 eggs = £ 1.00
So far, so good. Nothing very interesting,
but we need to make sure we are building
our understanding on very solid
foundations. Let’s slow down a little. How
did we get from the first equation to the
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second equation? We basically increased
the number of eggs from 1 to 4 (four times
as much) and also increased the cost by
the same amount (four times as much).
Lets see that in steps:
1 egg = £ 0.25
multiply both sides by 4:
4 × 1 egg = 4 × £0.25
rewrite:
4 eggs = £ 1.00
At this point, its probably worth pointing
out something fairly obvious. If I write:
4 eggs = 4 × £ 0.25

and also that:
4 × £ 0.25 = £ 1.00
Then it makes absolute sense that:
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4 eggs = £ 1.00
In plain English, this is like saying ‘4 eggs
cost 4 times £0.25, which is £1.00, so that
means 4 eggs cost £1.00’. In more general
terms, we can say ‘if A equals B and B
equals C, then A equals C’. Or in
equations:
if:
A = B
and:
B = C
then:
A = C
Sometimes in mathematics, you will see
two equations written on one line, like:
A = B = C
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This just means ‘A equals B and B equals
C’. Its just a shorter way of writing the two
separate equations for A = B and B = C.
For our equation for 4 eggs, we could have
written:
4 × 1 egg = 4 × £ 0.25 = £ 1.00
Now, all of this might seem so intuitively
obvious that we probably didn’t even need
to think about it. The bit about multiplying
both sides by the same number doesn’t
just apply to this equation about eggs, we
can do the same thing with any equation.
If the left hand side and the right hand
side are equal, then 4 times the left hand
side will also be equal to 4 times the right
hand side. In fact, any number times the
left hand side will be equal to any number
times the right hand side. For instance:
8 × 1 egg = 8 × £ 0.25

which we would rewrite as:
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