Mathematics for Economics and Business: Lecture 1
1.1 Introduction
Mathematics is of great importance in business and economics, and has been
for many thousands of years. In fact many of the earliest advances in
mathematics has been motivated by business applications, such as recording
the size of herds of cattle and flocks of sheep, or measuring the sizes of fields
for growing crops. Ever since money was invented, the calculation of interest
rates has been a great source of problems. At the advanced end of
mathematical finance, “rocket scientists" work out the newest combinations of
“financial derivatives" to squeeze a little more profit from the market. But many
of the day-to-day calculations just require some basic arithmetic and algebra
as covered in this course. Much of the material we cover is also of interest in
managing your own personal finances, not a trivial task in these days of
student loans, etc.
Developing skills in mathematics is all about practicing new skills and having
confidence to apply the skills you have. You need to keep up with the material
and not get behind, since each lecture depends on work developed in the
previous lectures.
The module web page, which is accessed through the link
http://www.macs.hw.ac.uk/~jphillips, contains copies of the syllabus and
lecture notes, tutorials, tutorial solutions, etc. You will also find there important
information such as exam details and use of the University Calculator in
exams. It is important to work though the tutorial problems you are given,
which are designed to give you practice in the methods you have learned in
class. There are also supplementary problems sheets if you need extra
practice in any particular area.
1.2 Arithmetic
Arithmetic is to do with calculations involving numbers and the basic
operations of addition, subtraction, multiplication, and division. The simplest
sort of numbers is the integers (whole numbers), such as 3, 89, or -23.
Slightly more complicated are fractions formed by dividing one integer by
1 4
9
another, such as, , and . The number on the top is called the numerator,
3 5
8
whilst the number on the bottom is called the denominator. If the numerator is
bigger than the denominator, it is sometime useful to split the fraction into a
whole number plus a fraction, so for example:
17 16 1
1
1
= + = 2+ = 2
8
8 8
8
8
1
Be careful, however, not to confuse this with 2 × which is something
8
different.
The most general type of number is a decimal number such as 12.43 which
43
.
100
You can convert fractions to decimals using your calculator, but note that
some calculations give a decimal number which to full accuracy have an
infinite number of digits after the decimal place, for example,
1
= 0.3333333.....
3
When working with decimal numbers you should always use at least six digits
in the calculation.
means 12 +
Addition and subtraction
This is carried out in the usual way. Be careful adding and subtracting
fractions (see below). We also need to be careful adding and subtracting
negative numbers. Adding a negative number to a positive one is the same as
subtraction, and vice versa:
5 + (-3) = 5 - 3 = 2
5 - (-3) = 5 + 3 = 8
Multiplication and division
Multiplication of negative numbers follows a rule that if the number of negative
signs is even (2, 4, . . . ) then the result is positive, if the number of negative
signs is odd then the result is negative
-(-3) × (-4) = -12; (-3) × (-4) ×2 = 24
Division follows the same rules as multiplication.
1.3 Precedence
In what order do you do a calculation (what precedes what)?
Normally the rule is multiplication and division first, then addition and
subtraction:
2 - 3 + 4 × 5 = 2 - 3 + 20 = 19
Use brackets to force a different order of precedence
(2 - 3 + 4) × 5 = 3 × 5 = 15
If we choose to multiply out the bracket before adding and subtracting, we
must be careful that the term outside multiplies every term inside the bracket.
(2 - 3 + 4) × 5 = 2 × 5 -3 × 5 + 4× 5 = 10 - 15 + 20 = 15
Factorizing is the opposite rule to multiplying out a bracket
10 - 15 + 20 = 2× 5 - 3× 5 + 4 ×5 = (2 - 3 + 4) × 5