Download Numbers - Queen Mary University of London

Queen Mary University of London
SEFP POM
Numbers
The numbers you will meet on this module are:
The integers or whole numbers, going from minus infinity (−∞) to plus infinity (+∞); it is convenient to distinguish those which are non-negative and those which are positive, as shown
here:
a
The rational numbers or fractions, which are numbers which can be written in the form for some
b
choice of integers a and b. This form consists of a numerator (or ‘top’) and a denominator
(or ‘bottom’). We also write a/b because it takes up less room on the page (or a ÷ b, but this is
non-standard). We can also write the number as a decimal expansion (or an expansion in any
other number base) and we shall say more about that in a minute.
5
Thus the fractions and 5/4 and 5÷4 and 1 14 and 1.25 all mean the same number. And because,
4
for any fraction,
we can change the top as long as we change the bottom in the same way
5
10
(multiplying the top and bottom of by 2).
8
4
The value 1.25 is given by your calculator when you divide 5 by 4 unless your calculator is
in MATH mode (or similar) giving output in fraction format. You should familiarise yourself
with the mode key on your calculator because you will need it later to choose between angle
calculations given in degrees and radians.
this is the same number as
Notice that if a is an integer then it is automatically a fraction a/b as well, because b is allowed
3
x
to be 1. It is sometimes useful in calculations to write 3 as or x as .
1
1
Arithmetic for fractions works according to the following rules:
a c ac
a c
a d
a c da + bc
× =
= ×
+ =
.
b d bd
b d
b c
b d
bd
The real numbers comprise of all the integers, all the fractions which lie between them, and all the
numbers which lie between the fractions (and cannot be written in the form a/b, for integers a
and b). Altogether, this gives us the real line:
The real line is often chosen as the x-axis when we plot graphs.
1
The real numbers which cannot be written as fractions (rational numbers) are called the irrational numbers. They cannot be specified exactly, although sometimes we use a single letter
to denote their exact value. The most famous example is π, the (exact) ratio of circular circumference to diameter which, to 50 decimal places1 , is approximated by:
π ≈ 3.14159265358979323846264338327950288419716939937511.
We can make this approximation more accurate by extending the sequence of decimal places
but this sequence never repeats itself, so we can never declare that we have written down π
exactly.
In contrast, the fraction 22/7 is sometimes used to approximate π. The decimal expansion of
22/7 to 50 decimal places is:
22
≈ 3.14285714285714285714285714285714285714285714285714.
7
This obviously is repeating! Indeed, we can write 22/7 exactly as a decimal expansion by using
some notation such as
22
= 3.142857
7
with the bar indicating a repeating sequence of decimal digits. Note that this only approximates
π to 2 decimal places, because if we round it to 3 decimal places it gives 3.143 (the 8 rounds
the 2 up to 3), whereas π to 3 decimal places is the familiar 3.142.
We can represent 5/4 exactly as a decimal in the same way: 5/4 = 1.250. If you know that you
are representing 5/4 then you will say it is simpler to just write 1.25. But the use of decimal
notation nearly always implies a physical application, in which case we are nearly always
approximating. So we should think of 1.25 as a real number being given to 2 decimal places
of accuracy, which is not the same thing as writing 5/4.
Because decimal expansions imply approximation we must be careful about decimal arithmetic.
For example, your calculator will give the value 1.25 × 1.25 = 1.5625. Unless you know you
are calculating 5/4 × 5/4 this equality, as written, is incorrect! Because if 1.25 is a data value
given to 2 decimal places then we are allowing for an inaccuracy which might make the actual
value, say, 1.249. Now 1.249 × 1.249 = 1.56001, making our first answer inaccurate in the 3rd
decimal place. So the correct calculation should be given as:
1.25 × 1.25 = 1.56 to 2 decimal places.
If we have a sequence of calculations we may have to reduce our decimal places of accuracy
still further. For example,
1.254 = 1.25 × 1.25 × 1.25 × 1.25 = 2.44140625 according to your calculator
1.2494 = 1.249 × 1.249 × 1.249 × 1.249 = 2.43360312,
so if 1.25 is a physical quantity given to two decimal places then we should say that its fourth
power is 2.4 to one decimal place.
1
The 50th digit of π is 0 but the 51st is 5, which rounds the 0 up to 1. If you memorise these 1st 50 decimal places you
will be able to impress people with your ‘mathematical’ skill!
2