Math For Economics Formula Sheet - The Economics Network Page 2
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1
2Arithmetic
Algebra
Sigma Notation
When multiplying or dividing positive and negative
The Greek capital letter sigma ∑ is used as an abbreviation
numbers, the sign of the result is given by:
for a sum. Suppose we have n values: x
, x
, ... x
and we
Removing brackets
a(b + c) = ab + ac
a(b – c) = ab – ac
x
+
x
. . .
x
1
2
n
1
2
n
wish to add them together. The sum
+ and + gives +
e.g. 6 x 3 = 18;
21 ÷ 7 = 3
(a + b)(c + d) = ac + ad + bc + bd
x
+
x
. . .
x
– and + gives –
e.g. (–6) x 3 = –18
(–21) ÷ 7 = –3
n
x
+
x
. . .
x
1
2
x
(a + b)
= a
+ b
+ 2ab;
(a - b)
= a
+ b
– 2ab
n
is written
2
2
2
2
2
2
x
+
x
. . .
x
1
2
n
+ and – gives –
e.g. 6 x (–3) = –18
21 ÷ (–7) = –3
i
1
2
n
∑
1
i
=
(a + b)(a – b) = a
– b
n
2
2
– and – gives +
e.g. (–6) x (–3) = 18
(–21) ÷ (–7) = 3
x
n
x
+
x
. . .
x
Note that i runs through all integers (whole numbers) from
x
n
i
1
2
n
x
+
x
x
. . .
x
∑
i
1
3
i
=
x
means
x
+
x
+
x
1 to n. So, for instance
1
i
2
n
∑
1
i
=
i
∑
=
1
1
2
3
n
i
x
i
∑
=
1
Formula for solving a quadratic equation
4
3
n
–
b
±
b
–
ac
2
means
i
x
x
+
x
+
x
x
3
First:
brackets
If ax
+ bx + c = 0, then
x
=
Order of calculation
means
x
+
∑
x
1
. . .
x
x
i
=
x
+
x
+
x
2
3
1
2
3
i
i
2
means
a
x
x
+
x
+
x
1
2
5
1
n
i
∑
=
i
1
2
3
∑
1
i
=
Second:
x and ÷
means
1
+
2
+
3
+
4
+
5
i
2
2
2
2
2
2
i
1
2
3
1
i
∑
=
1
i
∑
=
3
Third:
+ and –
means
n
x
x
+
x
+
x
= i
∑
1
5
x
3
a
means
1
+
2
+
3
+
4
+
5
m
i
i
1
2
3
5
2
2
means
2
2
2
2
Laws of indices
x
x
+
x
+
x
i
(
)
a
=
a
Example
means
5
1
1
+
2
+
3
+
4
+
5
a
a
=
a
=
a
m
n
mn
i
2
i
∑
=
2
2
2
2
2
m
n
m
+
n
m
–
n
i
1
2
3
means
i
∑
=
1
1
+
2
+
3
+
4
+
5
i
2
2
2
2
2
2
= i
∑
1
x
x
+
x
+
. . .
+
x
1
n
a
i
∑
=
n
= i
∑
1
x
=
=
1
i
1
2
i
=
n
= i
5
∑
1
n
n
∑
numerator
means
1
+
2
+
3
+
4
+
5
i
2
2
2
2
2
2
1
3
Fractions
5
x
means
x
x
+
x
x
+
+
. . .
x
+
+
x
x
This notation is often used in statistical calculations. The
n
Fraction =
=
1
means
1
+
2
+
3
+
4
+
5
a
0
a
=
a
/ 1
=
a
a
=
a
i
2
2
2
2
2
2
–
m
n
n
m
/
n
n
x
m
x
=
n
x
=
+
x
+
. . .
1
+
x
2
3
i
=
1
i
i
1
2
n
denominator
= i
∑
1
x
n
x
+
x
+
. . .
+
x
x
=
∑
i
=
i
n
1
=
n
a
mean of the n quantities, x
i
=
1
∑
1
2
n
, x
, ... and x
is
m
x
=
i
=
n
= i
∑
1
i
=
1
n
1
2
n
∑
(
x
–
x
)
2
x
n
1
2
n
2
n
n
n
∑
var(
x
)
=
i
=
i
–
x
2
i
=
1
i
=
1
x
n
x
+
x
+
. . .
+
x
5
∑
∑
n
n
x
=
means
i
=
1
+
2
+
3
+
4
+
5
i
2
i
=
1
2
1
2
2
2
2
n
2
x
n
x
+
x
+
. . .
+
x
(
x
–
x
)
2
x
n
n
2
∑
n
n
x
=
i
=
i
=
1
1
2
n
var(
x
)
(
x
–
x
)
2
x
2
=
n
1
i
=
n
1
i
–
x
2
= x and b is called the base
i
=
i
=
y = log
x means b
Laws of logarithms
= i
∑
1
n
n
∑
(
x
–
x
)
2
x
2
To add or subtract two fractions, first rewrite each fraction
y
n
n
Adding and subtracting fractions
var(
x
)
=
=
–
x
∑
1
i
∑
1
i
2
i
=
n
i
=
n
b
var(
x
)
=
=
–
x
1
i
1
i
2
i
=
i
=
∑
∑
n
n
so that they have the same denominator. Then, the
e.g. log
2 = 0.3010
means 10
= 2.000 to 4 sig figures
∑
∑
n
n
0.3010
sd
(
x =
)
var(
x
)
x
(
x
–
x
x
)
2
+
x
+
. . .
+
x
x
10
n
n
n
2
numerators are added or subtracted as appropriate and
var(
x
)
x
=
=
=
=
–
x
1
1
i
i
1
2
1
i
2
The variance is
i
i
=
=
i
=
n
Logarithms to base e, denoted log
, or alternatively ln,
(
x
–
x
)
2
x
2
n
n
n
n
∑
∑
∑
n
n
var(
x
)
=
=
–
x
1
i
1
i
2
e
i
=
i
=
the result is divided by the common denominator: e.g.
sd
(
x =
)
var(
x
)
are called natural logarithms. The letter e stands for the
∑
∑
n
n
sd
(
x =
)
var(
x
)
sd
(
x =
)
var(
x
)
exponential constant, which is approximately 2.7183.
4
3
16
15
31
i.e. the mean of the squares minus the square of the mean
(
x
–
x
)
x
+
=
+
=
n
2
n
2
var(
x
)
5
4
20
20
20
sd
(
=
x =
)
var(
x
)
=
–
x
2
i
=
1
i
i
=
1
i
A
∑
∑
n
n
ln
=
ln
+
ln
AB
A
B
;
ln
=
ln
–
ln
;
ln
=
ln
sd
(
x =
)
var(
x
)
A
B
A
n
A
n
The standard deviation (sd) is the square root of the
B
3
5
15
variance:
=
4
3
7
16
11
15
77
31
To multiply two fractions, multiply their numerators and
Multiplying fractions
+
=
+
=
sd
(
x =
)
var(
x
)
5
4
20
20
20
then multiply their denominators: e.g.
Proportion and Percentage
3
2
3
3
9
Note that the standard deviation is measured in the same
To convert a fraction into a percentage, multiply by 100
÷
=
=
3
5
15
5
5
5
3
5
2
10
=
units as x.
4
3
16
15
31
and express the result as a percentage. An example is:
as
a
percentage
is
100
=
62
5 .
%
7
11
77
+
=
+
=
8
8
5
4
20
20
20
5
5
as
a
percentage
is
100
=
62
5 .
%
8
8
3
2
3
3
9
3
5
15
÷
=
=
=
The Greek Alphabet
To divide two fractions, invert the second and then
Dividing fractions
5
3
5
2
10
7
11
77
multiply: e.g.
Α α alpha
iota
Ρ ρ rho
1
1
1
3
Some common conversions are
Ι
ι
=
10
%
=
25
%
=
50
%
=
75
%
Β β beta
Κ κ kappa
Σ σ sigma
3
2
3
3
9
10
4
2
4
÷
=
=
1
1
1
3
5
3
5
2
10
gamma
Λ λ lambda
tau
=
=
=
=
10
%
25
%
50
%
75
%
Ratios are simply an alternative way of expressing
10
4
2
4
Γ γ
Τ τ
delta
Μ µ mu
Υ υ upsilon
fractions. Consider dividing £200 between two people in
∆ δ
epsilon
nu
phi
the ratio of 3:2. This means that for every £3 the first person
1 + x + x
+ x
+ x
+ …
= 1/(1–x)
Series (e.g. for discounting)
2
3
4
gets, the second person gets £2. So the first gets
of the
Ε ε
Ν ν
Φ φ
zeta
xi
Χ χ chi
3 /
1 + x + x
+ x
+ … + x
= (1–x
) /(1–x)
5
2
3
k
k+1
total (i.e. £120) and the second gets
(i.e. £80).
Ζ ζ
Ξ ξ
2 /
Η η eta
Ο ο omicron
Ψ ψ psi
5
(where 0 < x < 1 )
Generally, to split a quantity in the ratio m : n, the quantity
theta
Π π pi
Ω ω omega
is divided into m/(m + n) and n/(m + n) of the total.
Θ θ
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