Department of Economics | Maths Revision Notes 2: Algebra
1.The Meaning of Algebra
Algebra is a form of shorthand which is used to express complex statements clearly and
concisely. Every algebraic expression can therefore be translated into words. For example,
a 2 + b2 = c translates as: "there exist two numbers such that the square root of the sum of
their squares is equal to a third number". Similarly the statement "total sales revenue is equal to
the product of price and quantity sold" can be expressed more compactly as:
R=P X q
(where R = total sales revenue, P = price, q = quantity sold and X denotes multiplication).
As these two examples show, in algebra we use symbols to denote unspecified or variable
numbers. The symbols are either letters of the alphabet (a, b, c, ....... x, y, z); or letters of the
Greek alphabet: α, β, γ etc. (see appendix to these notes for a full listing of the Greek alphabet).
In economics these numbers may be measured in terms of some natural unit; for example if p
denotes the price of a commodity, this must be measured in units of money (£s or $s, say);
while if q denotes the quantity of the goods purchased, this too must be measured in some unit
such as litres (petrol), pints (beer) or kilos (sugar). Some quantities have a time dimension; for
example if we say that someone's income is 2,000 we not only need to specify whether this is £s
or $s but also whether this is a weekly, monthly or annual income. In economics, quantities
with a time dimension are called flows. In contrast, when we say that someone's wealth is
£100,000, it is a quantity which exists at a moment in time; it is a stock, not a flow. The
distinction between stocks and flows is very important in economics.
Finally, some quantities have no dimension; they are pure numbers. For example, if Mr A's
income is £2,000 per annum and Mr B's is £1,000 per annum then the ratio of Mr A's income to
Mr B's is 2 (i.e, Mr A's income is twice that of Mr B). Here, the number 2 is a pure number; it is
not measured in £s or any other unit of measurement.
In both algebra and arithmetic, the symbols + , - , x, ÷ mean "add", "subtract", "multiply" and
"divide" respectively. In algebra, though, some conventions are slightly different:
In arithmetic, we write 4 x 3 to denote "4 multiplied by 3". In the same way, in algebra we can
write a x b to denote "a multiplied by b". But because the multiplication sign x can easily
be mistaken for a number called x , it is not often used in algebra. Instead we just write " ab"
or "a.b" to denote " a multiplied by b". This also has the attraction of being a more compact
notation.1
Similarly, the division sign (÷) is seldom used in algebra. Instead we write a or a b to
b
denote "a divided by b". Notice that an expression of the form a or a b is called an algebraic
b
fraction, or a ratio, or sometimes a quotient. The part of it above the line (a, in this case) is
called the numerator while the part below the line (b, in this case) is called the denominator.
1
In some computer programs, such as M-S Excel, the symbol * is used to denote multiplication.
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2. Rules of Algebra
It is important to realise that the rules of algebra are exactly the same as the rules of arithmetic
which we revised in Revision Notes 1. This means that if ever you are in doubt whether some
operation you have performed is correct, you can always check by inventing a simple numerical
example.
2.1 Adding and subtracting of algebraic expressions
When adding or subtracting it is necessary to obey the rules of signs (see Revision Notes 1), i.e.
taking into account whether the numbers being added or subtracted are themselves positive or
negative. The rules are:
(1)
x + (-y ) = x - y
(Think of adding a debit to your bank balance)
(2)
x - (+y) = x - y
(Think of subtracting a credit from your bank balance)
(3)
x + (+y ) = x + y
(Think of adding a credit to your bank balance)
(4)
x - ( -y ) = x + y
(Think of subtracting a debit from your bank balance)
Thus the rule is: If the signs differ, the result is subtraction (cases 1 and 2). If the signs are the
same, the result is addition (cases 3 and 4)
2.2 Multiplication and Division of Positive and Negative Numbers
Multiplication
The rules are (see Revision Notes 1):
(1)
(+a) x (+b) = + ab
(2)
(+a) x (-b) = -ab
(3)
(-a) x (+b) =
(4)
(-a) x (-b) = +ab
-ab
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 the rules are (see Revision Notes 1):
:
= + ab
(1)
( +a ) ÷
(+b) = +a
+b
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(2)
( +a )
÷
(3)
(-a )
÷
-a
(+b) = +b
= -a
b
(4)
(-a )
÷
a
(-b) = -a
-b = + b
(-b) = +a
-b
= -a
b
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 in front of positive
algebraic 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.
To avoid ambiguity we use brackets. The rule is that whatever mathematical operation is in the
brackets must be done first. Thus
(a + b) ÷ c is not the same as a + (b ÷ c)
However, because as noted earlier the division sign ( ÷ ) is not used in algebra, these two
expressions would not be written in this way. Instead we would write
a +b
to denote (a + b) ÷ c
c
When we write
a +b
c
it is understood that the whole of the numerator (that is, a + b) must be
a +b
c
divided by c. That is why
In contrast, we write a +
b
has the same meaning as (a + b) ÷ c
to denote a + (b ÷ c).
c
Given therefore that
a +b
denotes (a + b) ÷ c, it is clear that the a and the b must be
c
added together before division by c takes place.
Alternatively, a and b must both be divided by c before they are added together. That is,
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a b
a +b
= +
(Try a numerical example, e.g. a = 2, b = 3, c= 4 if you doubt this)
c
c c
Overlooking this is one of the most common mistakes made by students taking QT1.
4. “Expanding” (Multiplying out) Brackets
Consider an expression such as
a X (b + c)
As noted earlier, in algebra we avoid the X sign to denote multiplication, as it can easily get
confused with some number called x. So instead of writing: a X (b + c) we write
a(b + c)
Now, because of the rule on brackets noted above, this tells us that the whole of what is in the
brackets (that is, a + b) has to be multiplied by a. But multiplying the whole of the bracket by a
is equivalent to multiplying each element separately. Thus
a(b + c) = ab + ac
(If you doubt this, check by numerical example, e.g. a =2, b=3, c=4)
and this gives us the basic rule for "multiplying out" or "expanding" expressions which contain
brackets: you multiplying each term inside the brackets by whatever is outside the brackets.
The next question is: what happens if a is negative? We know from section 2.2 above that
multiplying by a negative number reverses the sign of the number being multiplied. So it
follows that
-a(b + c)
= (-a) x (+b) + (-a) x (+c)
=
(-ab)
=
-ab - ac
+
(-ac)
So, a minus sign outside the brackets means that, when brackets are removed by multiplying
out, the sign of each term previously inside the brackets is reversed.
5. Factorisation
This rather forbidding term simply means reversing the process of multiplying out or expanding
an expression. For example, suppose we are given
ab + ac
We see that both terms have a common element or factor, a. This common factor can be taken
outside of a pair of brackets, to give:
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ab + ac = a(b + c)
This process is called factorisation. We can easily check we have performed the process of
factorisation correctly, by multiplying out the result and confirming that we get back to where
we started.
Note that when we are factorising, we must remember the rule of signs. Thus
-ab - ac = -a(b + c)
Forgetting the necessity for a plus sign inside the brackets in cases such as this is another very
common mistake which students make.
6.
Operations with Fractions
Operations with algebraic fractions, such as
a
, obey the same rules as arithmetical fractions
b
(See Revision Notes 1).
In algebra a fraction (better described as a ratio or quotient) appears frequently because we avoid
using the ÷ and therefore write
a or a/b rather than a ÷ b.
b
Thus a fractions results whenever we divide one number by another. So we need to be clear
about the rules for manipulating these algebraic fractions. Recall that in any expression of the
form a or a/b, a is called the numerator and b the denominator.
b
"Cancelling" and Fractions
As in arithmetic, we can multiply or divide the numerator and denominator of a fraction
3
by the same thing without affecting the value of the expression. For example, 3 = 3 = 1 .
6
Here
6
2
3
we have divided numerator and denominator by 2 and thus converted 3 into 1 , which is a
6
2
simpler expression.
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. Thus in algebra for example
ac ac a c
=
=
ab ab a b
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The process of cancelling can also be reversed; that is, we can multiply numerator and
denominator by any number. For example,
a a x c ac
3 3 x 4 12
=
=
=
. Similarly in algebra, =
8 8 x 4 32
b b x c bc
“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.
6.1 Addition and Subtraction of fractions
Addition
In adding two fractions we have to find a common denominator, i..e. something that is divisible
by both numerators without remainder (see Revision Notes 1). The way we do this is to multiply
numerator and denominator of each fraction by the denominator of the other farction. For
example, given
a
b
+
c
d
First we multiply top and bottom of the first fraction by d. Thus
a
a
d
ad
becomes X
=
b
b
d
bd
c
c b
cb
Similarly,
becomes X =
d
d b
bd
Then we add the results, to get:
ad
cb
ad +cb
+
=
bd
bd
bd
which is as far as we can take the calculation without knowing the values of a, b, c and d.
Things are a little more tricky when there are three or more fractions to be added. The easiest
way of tackling this is to add the fractions two at a time. For example, given
a c e
+ +
b d f
We first add the first two fractions together. From the previous example, we know that
ad +cb
a
c
=
+
bd
b
d
So our given problem becomes
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ad +cb
e
+
bd
f
We then multiply the top and bottom of the first fraction by f and the top and bottom of the
second fraction by bd. This gives
f ( ad +cb )
bde
+
fbd
bdf
Notice the use of brackets to indicate that the whole of the top of the first fraction must be
multiplied by f.
So our final answer is
f ( ad + cb ) + bde
fbd
The above method does not always result in the lowest common denominator being found, and
consequently may lead to some unnecessary calculation - as you will find when you do the
examples. But this method does have the merit of being reliable and reasonably straightforward.
Subtraction
The method is exactly the same as the method for addition. For example
a
c
ad cb
−
=
−
b
d
bd bd
=
ad −cb
bd
6.2 Multiplication and Division of fractions
Multiplication
The rule for multiplying algebraic fractions is exactly the same as for arithmetica fractions (see
Revisions Notes 1). We 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:
a c
ac
X =
b d
bd
This generalises to any number of terms, e.g.:
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a c
e
ace
X X
=
b d
f
bdf
Reciprocals. As in arithmetic,
1
b 1
b X1 b
bX = X =
= =1
b
1 b
1X b
b
1
where b and
are said to be reciprocals.
b
Division of Fractions
As in arithmetic, we invert the second fraction and multiply. Thus
a
b
÷
c
d
=
a
b
d
X
c
=
ad
bc
7. Some special cases
Before leaving fractions, let us look at some special cases
(1) For any number x (provided x is not equal to zero (which we write as x ≠ 0), then:
x
=1
x
In words, any number divided by itself is equal to 1.
(2)
If x is equal to zero, we have
x 0
=
x 0
Now, it is very tempting to say that 00 = 0 , but this is not the case.
In fact, 00 is said to be undefined; that is, it has no meaning. If this is puzzling, remember that
it is not necessary in logic that everything you can write down must have a meaning.
0
(3)
Provided x ≠ 0, then
=0
x
Here the logic is that since division is repeated subtraction, the question here is: "How many
times can you subtract x from 0?" And the answer is: "Zero times".
(4)
Provided x is a positive number, then:
x
(Note that the symbol " ∞ " denotes "infinity")
=∞
0
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Again, since division is really nothing more than repeated subtraction, the logic is that we are
asking: "How many times can we subtract zero from a positive number?" To which the answer
is: "An infinite number of times."
By the same reasoning, if x is a negative number then:
x
(Note that the symbol "- ∞ " denotes "minus infinity")
= −∞
0
(5)
1
x
X x = =1
x
x
(Provided x is not zero)
As noted earlier, here x and 1 are said to be reciprocals of one another (that is, each is the
x
inverse of the other). If any number is multiplied by its own reciprocal, the resulting product is
1 (also called "unity").
-------------------------------------8. Powers and roots
8.1 Squares and square roots.
“Squaring” any number, a, means multiplying a by itself, once.
a2 = a x a
Thus
and the number, 2, is called the power, index or exponent of a.
Reversing this process is known as taking the square root, denoted by the symbol
say
Thus we
(because 22 = 4)
4 = 2
and, in general
(if a2 = b)
b = a
Negative square roots.
The number +9 has two square roots: +3 or -3. This is because
(+3) x (+3) = 9
and
We can write this fact as:
(-3) x (-3) = 9
9 = ±3
What is true of the number 9 is true of any positive number: there are two square roots, equal in
absolute magnitude but of opposite sign.
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The square root of a negative number does not exist. This is because any number, when squared,
becomes positive.
Caution: Don't confuse two things: (i) a positive number has two square roots, one of which is
negative; (ii) a negative number has no square roots. (Actually, this last statement is not quite
true - see footnote below) 1
8.2 Cubes and cube roots
23 ≡ 2 x 2 x 2 ≡ 8
and in general
a3 ≡ a x a x a
Reversing this process is known as taking the cube root, denoted by the symbol
say
3
8 = 2
3
. Thus we
(because 23 = 8)
and, in general
3
b = a
(if a3 = b)
Negative cube roots.
Consider the number 64, for example. This has two square roots, +8 and -8.
But it has only one cube root, +4.
The number -4 is not a cube root, because (-4)3 = (-4) x (-4) x (-4) = -64
And in general, every positive number has only one cube root, and that root is positive. (But see
footnote at bottom of this page)
The cube root of a negative number
Consider the number -64, for example. This has no square roots, because (+8) x (+8) = +64 and
(-8) x (-8) = 64.
But it does have one cube root, -4, because
(-4)3 = (-4) x (-4) x (-4) = 16 x (-4) = -64
and in general, every negative number has only one cube root, and that root is negative.
1
Footnote: We said above that a negative number had no square roots. To be precise, a negative number has no
square roots that are real numbers. Mathematicians have invented another class of numbers, called imaginary
numbers, and using this concept it is possible to find the square roots of a negative number. We also said above that
positive numbers had only one cube root. To be precise, a positive number has only one cube root that is a real
number. It is possible to find two more roots for any positive number by using the concept of imaginary numbers.
We shall not develop further the concept of an imaginary number in this module (EC121), but it becomes important
in certain more advanced math techniques for economists.
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9.
Extending the idea of powers
(also known as indexes, indices, or exponents)
In the previous section we considered simple powers such as squares and cubes and their
corresponding square roots and cube roots. Now we want to extend these ideas.
In general an means n a's multiplied together. Notice that it follows immediately from this
that, when n = 1,
a1 ≡ a
Caution:
9.1
Be careful not to confuse a2 (≡ a x a) and 2a (≡ a + a).
Also don't confuse -a2 ≡ -(a x a) and (-a)2 = (-a x -a)
Rules for manipulating powers
(Here a and n and m are any numbers)
an x am ≡ an + m
Rule 1:
Example: if n = 2 and m = 5, we have:
≡ (a x a) x (a x a x a)
≡ a x a x a x a x a
≡
------------------------------------------------am
Rule 2:
≡
am - n
n
a
Example: if m = 5 and n = 3, we have:
a x a x a x a x a
a5
=
= a x a = a2
3
a
x
a
x
a
a
(Notice the a's cancel between top and bottom)
------------------------------------------------Rule 3:
(ab)n ≡ anbn
a2 x a3
a5
Example: if n = 2 we have
(ab)2
≡ (ab) x (ab) ≡ a x b x a x b ≡ a2 x b2 ≡ a2b2
Caution: Don't confuse (ab)n and (a + b)n
------------------------------------------------Rule 4:
(an)m = an x m = anm
Example: if n = 2, m = 3, we have
(a2)3
= (a x a) x (a x a) x (a x a) = a x a x a x a x a x a = a6
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------------------------------------------------Rule 5:
a0 ≡ 1
an
≡ 1 (since any number divided by itself equals 1
Proof:
an
a n ≡ an-n ≡ a0
But, by Rule 2 above,
an
an
Combining these two equations, a 0 ≡ n ≡ 1
a
------------------------------------------------9.2 Negative and fractional powers
Negative Powers. A negative power is equivalent to a reciprocal, i.e.
1
an
1
a0
Proof: using Rule 5, n =
a
an
a0
And using Rule 2, n ≡ a0− n ≡ a−n
a
1
1
-3
=
, and so on.
Thus a - 2 =
,
a
a2
a3
-----------------------------------------------------------------------------------------------------------------------Fractional Powers. A fractional power is equivalent to a root, i.e.
Rule 6:
a-n ≡
Rule 7:
1
an
≡ na
1
2
Thus for example a 2 ≡ a ; in words, a raised to the power one-half is the same thing as the
square root of a.
⎛ 1⎞
Proof: using Rule 4 above, ⎜ a 2 ⎜
⎜ ⎜
⎝ ⎠
2
≡
1
x2
a2
≡ a1 ≡ a
( )
2
But, from the definition of a square root, 2 a ≡ a (i.e. squaring and taking the square root
cancel each other out).
2
⎛ 1⎞
So, a 2
and
⎜ ⎜
⎜
⎜
⎝
⎠
( a)
2
1
are both equal to a. Clearly this can only be true if a ≡
2
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a
2
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-----------------------------------------------------------------------------------------------------------------------Rule 7a:
show that
This extends Rule 7 to cover a case such as
2
a3
≡
3
2
a3
. We can
a2
2
a3
So
is evaluated in any particular case by squaring a, then finding the
cube root of this. For example, if a = 9, a2 = 81 and
3 81 = 4.33 (approximately). (See below for how to find this number)
In general therefore
n
m
a
≡
m n
a
We won't trouble to prove this here.
-----------------------------------------------------------------------------------------------------------------------10
Evaluating powers and roots on your calculator
Evaluating powers. On most calculators the relevant key is labelled
yx
So to evaluate, for example, 26, key in 2, then press the yx key, then press the 6 key. The answer
(64) should then appear on the display.
Evaluating roots. To do this, you need to reverse the above process. This is usually done by
pressing a key marked “2nd F”, “ALT” or “INV” (this varies from one calculator to another)
before pressing the yx key.
To evaluate, for example, 3 81 [which by Rule 7 above is the same thing as (81) 3 ; that is the
cube root of 81), you key in the number 64, then press the “2nd F”, “ALT” or “INV”key, then
the yx key, then key in 0.333333, the decimal equivalent to 1 3 . You can check your answer by
1
raising it to the power 3, which should take you back to 81. Practise this until you are confident.
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The Greek Alphabet
Upper case
Lower case
α
α
alpha
β
β
beta
γ
γ
gamma
δ
δ
delta
ε
ε
epsilon
ζ
ζ
zeta
η
η
eta
Θ
θ
theta
ι
ι
iota
κ
κ
kappa
λ
λ
lamda
µ
µ
mu
Ν
ν
nu
ξ
ξ
xi
ο
ο
omicron
π
π
pi
ρ
ρ
rho
σ
σ
sigma
τ
τ
tau
Υ
υ
upsilon
φ
φ
phi
χ
χ
chi
ψ
ψ
psi
ω
ω
omega
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