Download Revision notes 1 - University of Warwick

Department of Economics | Maths Revision Notes
These Maths revision notes have been developed for incoming students to the Department
of Economics because past experience has shown that many new Economics students have
forgotten much of the algebra they learned at school. This lack of basic proficiency leads
to their struggling with certain modules throughout the year. Some even fail the
examination at the end of these modules. Therefore if your basic algebra is weak, it is
essential that you revise thoroughly now.
These revision notes are intended to help with you with your revision. You may also find it
helpful to obtain a GCSE textbook, such as the Letts publication, Revise GCSE Mathematics
(only the sections on arithmetic and algebra are relevant, though the statistics section is a
useful preparation for Statistical Techniques A which you will take in term 2). If you find
yourself struggling it may also a good idea to ask more mathematically competent friends to
coach you!
None of the maths in the Economics modules is "maths for the sake of maths" understanding it will make you better at economics!!
Revision notes 1
1. Addition and subtraction with positive and negative numbers
The key point to understand here is that the + and - signs serve two distinct purposes:
(i) We place a + or - sign in front of a number to indicate whether that number is positive or
negative. Thus (+5) is a positive number and (-3) is a negative number. The brackets are not
strictly necessary but have been added to make it absolutely clear that the + and - signs are
attached to the numbers.
(ii) We place a + or - sign between two numbers to indicate whether the operation of addition
or subtraction is to be performed.
This results in four possibilities:
(a)
(b)
(c)
(d)
Adding a positive number, e.g. (+5) + (+3). The result is (+8)
Subtracting a positive number, e.g. (+5) - (+3). The result is (+2)
Adding a negative number, e.g. (+5) + (-3). The result is (+2)
Subtracting a negative number, e.g. (+5) - (-3). The result is (+8)
To understand these rules, we could think of Anne's bank balance being (+5) dollars and
John's bank balance being (+3) dollars. Then in case (a) Anne and John are perhaps planning
an evening out together, so are adding their bank balances together to see how much they
have to spend. They arrive at their combined bank balance, (+8) dollars. In case (b) Anne
and John wish to see which of them is the richer, by subtracting John's balance from Anne's
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in order to calculate the difference between them, which is (+2) dollars. In other words,
Anne has two dollars more than John.
Cases (c) and (d) are a little more difficult. In case (c), Anne's bank balance is (+5) dollars
as before, but John's account is in the red; his balance is (-3) dollars. When they add their
balances, therefore, they find that their combined balance is only (+2) dollars.
In case (d), Anne's bank balance is (+5) dollars and John's is (-3), as in case (c). This time,
though, they are subtracting John's balance from Anne's in order to calculate the difference
between them, which is (+8) dollars. In other words, Anne has eight dollars more than John.
John would need to pay eight dollars into his account in order to have the same balance as
Anne.
In summary, adding a positive number [case (a)] and subtracting a negative number [case (d)]
both result in addition of the two numbers.
Subtracting a positive number [case (b)] and adding a negative number [case(c)] both result
in subtraction of the second number from the first.
In practice of course we don't usually bother to place a + sign in front of positive numbers,
nor bother with the brackets. So cases (a) - (c) above would actually be written as:
(a)
(b)
(c)
(d)
5 + 3. The result is 8 (Anne and John's combined bank balance).
5 - 3. The result is 2 (the difference between Anne' balance and John's).
5 + (-3). The result is 2 (their combined balances, when John is in the red).
5 - (-3). The result is 8 (the difference between Anne's balance and John's, when John
is in the red).
It makes no difference to the application of the rules if the first number (5, in our examples)
is negative. Of course, it changes the answers, because we are now assuming that Anne's
bank balance is (-5) instead of 5. [Check for yourself that the answers to (a), (b), (c) and (d)
in this case are -2, -8, -8 and -2 respectively.]
2. Multiplication and division with negative numbers
One approach to understanding the rules for multiplication of positive and negative numbers
is as follows: If a number is multiplied by +1, the number is left unchanged. If a number is
multiplied by -1, the number is left unchanged in absolute magnitude but its sign is reversed;
that is, if it was previously positive it becomes negative and vice versa. Thus:
(+5) x (+1) = (+5)
(+5) x (-1)
= (-5)
(-5) x (+1) = (-5)
(-5)
x (-1) = (+5)
However, consistency clearly requires that these rules must hold for multiplication by any
number, not just -1. This implies that:
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(1)
(+5) x (+3) = + 15
(2)
(+5) x (-3) =
-15
(3)
(-5) x (+3) =
-15
(4)
(-5) x (-3) = +15
The way to remember these rules is that if the two numbers have the same sign (cases 1 and
4), the result is positive, while if they have different signs (cases 2 and 3), the result is
negative.
Division
Since division simply reverses multiplication, consistency requires that division should obey
the same sign rules as apply to multiplication. Thus:
(1)
(+5) ÷ (+3) = +5 = + 5
+3
3
(2)
(+5) ÷ (−3) = +5 = - 5
-3
3
(3)
(-5) ÷ (+3) = -5 = - 5
3
+3
(4)
(-5) ÷ (−3) = -5 = + 5
−3
3
Notice that the division operation can be written in two ways: either using the ÷ sign, or by
writing a fraction in which the first number appears in the top (called the numerator) and the
second in the bottom (the denominator). This second way is more convenient, hence is more
common than the first.
Again, the way to remember these rules is that if the two numbers have the same sign (cases
1 and 4), the result is positive, while if they have different signs (cases 2 and 3), the result is
negative.
From now on we shall follow the normal convention and omit the + sign and the brackets in
front of positive numbers.
3. Brackets and when we need them
When addition and subtraction are mixed with multiplication and division, you get different
answers depending on which part of the calculation you do first. For example,
6 + 8 ÷ 2
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is ambiguous. If you do the addition first you get 14 ÷ 2 = 7, but if you do the division
first you get 6 + 4 = 10.
Mathematicians have set up certain conventions in an attempt to avoid ambiguity in such
cases. Specifically, the convention is that you work from left to right in an expression,
carrying out any division operations first, then any multiplication, then addition and finally
subtraction. Applying these conventions to the expression above tells us to do the division
first, so the “correct” answer is 10.
But, these conventions are not followed universally, and so the only way to be sure of
avoiding ambiguity is to use brackets to indicate which of any alternatives is correct in the
particular case. The rule is that whatever is in the brackets must be done first. Thus
(6 + 8) ÷ 2
= 14 ÷ 2
=7
but
6 + (8 ÷ 2)
= 6 + 4
= 10
However, because as noted earlier the division sign ( ÷ ) is not often used, these two
expressions would not be written in this way. Instead we would write
6 +8
to denote (6 + 8) ÷ 2
2
When we write
6 +8
2
it is understood that the whole of the numerator (that is, 6 + 8) must
be divided by 2. That is why
In contrast, we write 6 +
6 +8
2
has the same meaning as (6 + 8) ÷ 2.
8
to denote 6 + (8 ÷ 2).
2
6 +8
has the same meaning as (6 + 8) ÷ 2. In both expressions this
2
means that the 6 and the 8 must both be divided by 2. Overlooking this is one of the most
common mistakes made by students taking Maths A. This rule will be considered
further below, when we look at fractions.
To summarise,
4. Fractions
In arithmetic the idea of fractions such as 14 , 12 , 83 is familiar.
As noted earlier, we often write, for example, 2 to denote 2 ÷ 3
3
Thus a fractions (better described as a ratio or quotient) results whenever we divide one
number by another. So we need to be clear about the rules for manipulating these fractions.
Note that in any expression such as 2 (also often written as 2/3), 2 is called the numerator
3
and 3 the denominator.
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"Cancelling" and Fractions
It is easy to see that 63 = 12 .
What we have done here (almost without thinking) is to divide both the numerator and the
denominator by 3; that is:
3
6
=6
3
3
3
=21
We divided numerator and denominator by 3 because we saw that both numerator and
denominator were divisible by 3 without remainder; in other words we saw that 3 was a
factor of both the numerator and the denominator. This process is called cancelling and can
be done whenever you can find something which is a factor of both the numerator and the
denominator. Before cancelling, though, you must make sure that the whole of the numerator
and the whole of the denominator are divisible by the factor, without remainder.
The process of cancelling can also be reversed; that is, we can multiply numerator and
denominator by any number. For example,
3 3 x 4 12
=
=
8 8 x 4 32
“Cancelling” and its reverse are permissible because multiplying or dividing both numerator
and denominator by the same thing leave a fraction unchanged in its value.
4.1 Addition and Subtraction of fractions
Addition
1 1
+
2 4
The way we tackle this is to see, intuitively, that a half is the same thing as two quarters, so
the answer is
1 1
2 1 3
+
=
+ =
2 4
4 4 4
Consider:
2 1
+
5 2
4 and 1 = 5 , so
Again we see, almost without thinking, that 25 = 10
2
10
Another example is:
2
1
4
5
9
+
=
+
=
5
2
10
10
10
What we need to do now is to make this intuitive process systematic, as follows. In the
previous example, 10 is called the common denominator. A common denominator is any
number which is divisible (without remainder) by the denominators of the two fractions to be
added. Thus 10 is a common denominator because both 5 and 2 divide into it without
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remainder. (This follows from the fact that 10 = 5 x 2). However, 20, 30, 40 etc are also
common denominators because they too can be divided by both five and two without
remainder. But ten is the lowest common denominator. It is the smallest number which is
divisible by both 5 and 2 without remainder. (Using the lowest common denominator when
adding fractions is not essential but simplifies the calculation.)
How do we find a common denominator? Consider the example:
1 1
+
7 11
We have to find a number which is divisible by both 7 and 11 without remainder. It occurs to
us that if we take the product of 7 and 11, that is 7 X 11 or 77, this is certain to be divisible
by both 7 and 11 without remainder.
1 1 x 11 11
=
=
7 7 x 11 77
1x7
7
1
=
=
11 11 x 7 77
11
7
18
1 1
=
+
=
+
7 11
77
77
77
So we can then write:
and
so
The pattern here is that we took each fraction in turn and multiplied both its numerator and its
denominator by the denominator of the other fraction.
Thus in the example above where we took
2
2x2 4
becomes
=
5
5 x 2 10
so
2 1
+
5 2
and
1
1x5 5
becomes
=
2
2 x 5 10
4
5 9
2 1
=
+
=
+
5 2
10 10 10
Things are a little more tricky when there are three or more fractions to be added. The easiest
method is to add them in pairs. For example, given
1 1 1
+ +
8 4 5
we first add the 1/8 and the 1/4, then add the result of this to 1/5, as follows
1 1 1
1 2 1
3 1
3x5 1x8
15 8
23
+ + =
+ + = + =
+
=
+
=
8 4 5
8 8 5
8 5
8x5 5x8
40 40
40
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Alternatively we can multiply numerator and denominator of each fraction by the product of
the other two (or more) denominators. This gives:
1 1 1
1x 4x5 1x8x5 1x8x4
+ + =
+
+
8 4 5
8x4x5 4x8x5 5x8x4
=
20
40 32
92 23
+
+
=
=
160 160 160
160 40
This second method does not always result in the lowest common denominator being found;
in our example the common denominator is 160, whereas the lowest common denominator is
40. Consequently may lead to some unnecessary calculation, but this method does have the
merit of being reliable and reasonably easy to remember.
Subtraction
The method is exactly the same as the method for addition. For example
1 2
2 5
=
5
4
5-4
1
=
=
10 10
10
10
4.2 Multiplication and Division of fractions
Multiplication
The rule for multiplying one fraction by another is very simple. You simply multiply the two
numerators together to obtain the numerator of the answer, and multiply the two
denominators together to obtain the denominator of the answer. Thus in general:
3 2
3x 2 6
3
X =
=
=
8 5
8 x 5 40 20
This generalises to any number of terms, e.g.:
3 2 3 3 x 2 x 3 18
9
X X =
=
=
8 5 4 8 x 5 x 4 160 80
Reciprocals
From the multiplication rule, we can get
1
4 1
4 X1
4
4X = X =
= =1
4
1 4
1X 4
4
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Here, 4 and 1 are said to be reciprocals. By definition, their product is unity (in other
4
words, when you multiply one by the other the result is 1).
"of" Sometimes we meet a question such as: what is half of 18? Obviously the answer is
9, but we need a procedure for answering this kind of question so that we don't have to rely
on intuition (which can lead us astray)
The key insight is to see that "half of 18" is found by multiplying 18 by 1 . That is,
2
18
"half of 18" = 18 x 1 = = 9 .
2 2
So "of" just means " multiplied by". For example, 3 of 1 = 3 x 1 = 3x1 = 3
4
2 4
2 4x2 8
Division of Fractions
From the multiplication rule above, we can get
1
10
1
10 X 1
=
X
=
= 5
10 X
1
2
1 X 2
2
But we also know that:
10 ÷ 2 = 5
Combining these, we see that
10 ÷ 2 = 10 X 1
2
1
So dividing by 2 is equivalent to multiplying by its reciprocal, 2
This gives us the fundamental rule for dividing by a fraction: you simply invert the fraction
and multiply instead of dividing.
For example,
2
3
The rule is that we first invert the fraction ( i. e. 2 becomes 3 )
3
2
it instead of dividing. Thus:
Consider: 1 ÷
1 ÷
and then multiply by
2
3
1
1
3
3
= 1 X
=
= 1
=
X
3
2
2
1
2
2
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This answer may seem puzzling, but it may help to think in words. The question is: "How
many two-thirds are there in 1?" And the answer is: "One and a half".
This rule holds equally when one fraction is divided by another.
4
1
1
1
1
4
÷
=
X
=
=
8
4
8
1
8
2
Thus:
Here the question is: "How many quarters are there in an eighth?" And the answer is: "A
half." (In other words, an eighth is only half as large as a quarter.)
WARNING: A COMMON MISTAKE: Cancelling fractions when addition or
subtraction is involved
Students often go wrong when confronted with something like
3 + 2
5
=
8
8
If you try to cancel the 2 with the 8, you get
3 +
2
2
3 + 1
=
=
4
8
2
The way to understand why
4
= 1 , which is wrong
4
5
8
is correct while 1 is wrong is to recall from our earlier
is another way of writing (3 + 2) ÷ 8. This means that the addition
discusssion that 3+2
8
must be performed first. So whenever addition or subtraction appear in the numerator or
denominator of a fraction, the addition or subtraction must be done first before any cancelling
takes place.
Cancelling fractions when multiplication or division is involved
In contrast to the above, it is quite OK to cancel between numerator and denominator when
only multiplication or division signs appear in numerator and denominator. For example:
3 x 2
6
3
=
=
8
8
4
(This is the same as the previous case, except that we now have a multiplication sign instead
of a plus sign in the numerator.)
In this case you can cancel the 2 with the 8 and get the right answer. For doing so gives:
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3 x
8
2
2
2
=
3 x 1
4
=
3
4
Summarising, whenever you have something like 3+2
; that is, something with addition or
8
subtraction in the numerator or denominator, you must do the addition or subtraction first
before doing any cancelling.
But when you have something like 3 x8 2 ; that is, something with only multiplication (or
division) in the numerator and denominator, you can cancel individual elements in the
numerator and denominator, if you wish. But you will also get the right answer if you
choose instead to perform the multiplication or division in numerator or denominator first,
and then do the cancelling.
5. Powers and roots
Squares and square roots
The product 3 X 3 is often written as 32. We read this as “three squared” and means 2
threes multiplied together. Thus 3 X 3 = 32 = 9. (It is called a square because 32 is the
area of a square with sides of length a.)
Similarly 23 = 2 X 2 X 2,
25 = 2 X 2 X 2 X 2 X 2
and so on.
The process which is the reverse of squaring is called taking (or finding) the square root.
Thus 3 is the square root of 9 because 32 = 9. We write the square root of 9 as √9. Thus
√9 = 3 because 32 = 9.
Because squaring and taking the square root are mutually reversing processes, it follows that
if you square a number and then take the square root, you arrive back where you started; that
is: √(32) = 3. Similarly (√3)2 = 3.
Negative square roots. Every positive number has 2 square roots, one positive and the other
negative. For example, 9 has two square roots. One, as we have just seen, is +3. This is
because (+3)2 = (+3) X (+3) = 9.
But (-3) is also a square root of 9, because (-3)2 = (-3) X (-3) = 9. (This is because when
two negative numbers are multiplied together, the result is a positive number - see section 2.2
above).
So the two square roots of 9 are +3 and -3 (sometimes written more compactly as ± 3). In
symbols, we write: √9 = ± 3. We normally consider only the positive root unless the
context of the problem requires us to consider both.
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Square roots of negative numbers. Suppose we are seeking the square root of (-9). Let us
denote the number we are looking for by the symbol y (this is just a label). By definition, y is
a number such that y2 = (-9). Thus y2 is negative.
But, from section XX above we know that if y is a positive number then y2 = y X y is
positive. So y cannot be positive if y2 is negative.
Similarly we know from section XX above that if y is a negative number then y2 = y X y
is positive. So y cannot be negative if y2 is negative.
Combining these two, we can say that y can be neither positive nor negative if y2 is negative.
Therefore the square root of a negative number does not exist. (This is not quite true. In
more advanced maths the concept of an imaginary number is introduced, and this number has
the property that its square is negative. Thus to be completely precise we have to say that the
square root of a negative number does exist, but is not a real number (but rather, is an
imaginary number).)
6. Decimals, fractions and percentages
6.1 Converting decimal numbers to fractions and vice versa
To quickly refresh our memories as to the meaning of decimal numbers, recall that by
definition
1
,
10
1
,
0.01 =
100
1
,
0.001 =
1000
125
1
0.125 = 1000 (which =
after cancelling between top and bottom)
8
and so on.
0.1 =
The pattern is that the number of zeros in the denominator of the fraction is always equal to
the number of digits after the decimal point. (Except that a zero or zeros at the right hand end
1
55
; 0.5500 = 0.55 =
don't count. E.g. 0.10 = 0.1 =
and so on.
10
100
Some conversions from fractions to decimals and vice versa are fairly obvious, e.g.
1
1
1
= 0.5 ;
= 0.1
;
= 0.2
2
10
5
The underlying rule for converting a fraction into a decimal is that you divide the numerator
by the denominator, using decimal long division (being careful to get the decimal point in the
1
we first divide 2 into 1. This won’t go, so we add a
right place!). For example, given
2
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zero and simultaneously enter a decimal point (see figure 1). We then divide 2 into 10, which
goes 5 times, so the answer is 0.5.
Figure 1:
0·5
2 )1 0
1
as another example. To convert this to a decimal number, we divide 8 into 1.
8
This won’t go, so we add a zero and enter a decimal point (see figure 2). We then divide 8
into 10, which goes once with remainder 2. We add another zero and divide 8 into 20, which
goes twice with remainder 4. We add another zero and divide 8 into 40, which goes 5 times
1
= 0.125
without remainder, giving us our answer:
8
Consider
Figure 2
0·125
8 )10
8
20
16
40
40
In practice you will probably use your calculator for this, but it's worth seeing how it is done
without a calculator.
To convert a decimal number into a fraction, simply apply the definitions above. For
example,
5
1
0.5 =
, which after cancelling , =
10
2
125
1
Similarly
0.125 =
, which after cancelling , =
1000
8
A strange feature of decimal numbers is that some fractions ahve no exact decimal
1
equivalent. For example, = (approximately) 0.6666 with the 6's recurring indefinitely.
3
6.2 Percentages
(i) To convert a decimal fraction into a percentage, simply multiply by 100. For example in
the case of 0.5 we get:
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0.5 =
5
500
x 100 =
= 50 (per cent)
10
10
(ii) To convert any fraction into a percentage, simply multiply by 100. For example,
1
.
2
1
100
1
X 100 =
= 50 (per cent). So,
= 50%.
2
2
2
1
1
100
Similarly as a percentage is:
X 100 =
= 11.11% (approx.)
9
9
9
(iii) To convert any percentage into a fraction, simply reverse the above, i.e. divide by 100.
50 1
1
=
=
Thus 50% becomes 50 ÷ 100 = 50 X
100
100 2
(iii) To find, say 5% of 27, recall that 5% simply means “five-hundredths”. So we have to
divide 27 into 100 parts and then take 5 of them. To do this, we divide 27 by 100, then
multiply the result by 5. This gives:
5
135
27 X
=
= 1.35 (per cent). (With some practice, you’ll find you can amaze
100
100
your friends by calculating, say, 2% of 61 (= 1.22%) in your head).
We get:
7. Some Additional Symbols
To conclude these notes, here are some symbols which we will be using later and which may
be new to you:
≠
means "is NOT equal to"
≈
means "is approximately equal to"
≡
means "is identically equal to". This conveys a stronger meaning than
the = sign. The difference between ≡
and = is important in
Economics and we shall return to it later.
∞
means "infinity". (Note that infinity is NOT "a very large number".
Any number, however large, is finite; while the essence of infinity is
that it is not finite)
-∞
means "minus infinity"
x
>
<
means the absolute value of x, i.e. ignoring its sign. For example,
−4 = 4
means "greater than" (for example, 4 > 3)
means "less than" (for example, 3 < 4)
The symbols > and < are called inequalities and can be used in a variety of ways. For
example if we want to say that some number called z is positive (i.e. greater than zero), we
can simply write: z > 0. If on the other hand we want to say the some number k is
negative, we can write: k < 0. A third possibility is that we might want to say that some
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number y has a value which lies between 4 and 5. We can write this as: 4 < y < 5, or
as: 5 > y > 4. Both of these algebraic statements tells us that y is greater than 4 but less
than 5. We shall return to inequality statements later.
Finally, some symbols which you will find useful in setting out your work clearly are:
∴
⇒
means "therefore"
means "implies" or "leads to"
(For example: " x = 3 ⇒ x + 1 = 4")
A note on use of calculators
Many of you will already own calculators and may be in the habit of using them for routine
calculations. My view on calculators is that they are fine for speeding up long and laborious
calculations where a lot of data are involved, such as in an applied economics exercise. But
if used routinely for trivial calculations (such as 100 ÷ 15), they are positively harmful
because they lead to mental laziness and, more important, they prevent you from developing
any "feel" for numbers. (Incidentally, it is worth noting that a calculator cannot in fact
calculate 100 ÷ 15, because it runs out of space in its display! The correct answer is 6 2 ,
3
but the calculator cannot supply this answer, only 6.6666666.........which is only an
approximation.)
However, we shall find later that a calculator is essential for certain kinds of mathematical
computation (specifically, those involving logarithms and exponents); and also for statistics
in the second term. So you will definitely need a calculator, but it should be of the
"scientific" type, with keys marked: " log ", " ln " , and " yx ", and also with some
statistical function keys. It need not cost more than £8-£10. If in addition you already own,
or wish to buy, one of the more expensive "graphics" calculators, with a larger display screen
and which can draw graphs (priced from around £35 up), this is perfectly OK as long as you
don't become too dependent on it.
University rules allow you to use a calculator in examinations, provided it can only
display numbers (i.e. not text or graphics).
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