Understanding Engineering Mathematics

are calledrational numbersorfractions. All measurements of a physical nature (length,
time, voltage, etc.) can only be expressed in terms of such numbers. Numbers which are
not rational, and cannot be expressed as ratios of integers, are calledirrational numbers.
Examples are

√

2andπ. We will prove that

√
2 is irrational in Chapter 14.
The set of all numbers: integers, rational and irrationals is called the set ofreal numbers.
It can be shown that together these numbers can be used to ‘label’ every point on a
continuous infinite line – thereal line. So called ‘complex numbers’ are really equivalent
to pairs of real numbers. They are studied in Chapter 12, and an introduction is provided
in the Applications section of Chapter 2.
Note that zero, 0, is an exceptional number in that one cannotdivideby it. It is not that
1/0 is ‘infinity’, but simply thatit does not exist at all.Infinity, denoted∞, is not really
a number. It is a concept that indicates that no matter what positive (negative) number
you choose, you can always find another positive (negative) number greater (less) than it.
Crudely,∞denotes a ‘number’ that is as large as we wish.

Solution to review question 1.1.

A.All numbers here are real, so d applies to them all.

(i)−1 is a negative natural number, i.e. an integer; (a, b, c, d)

(ii)^12 is a ratio of integers and is therefore rational; (c, d)

(iii) 0 is an integer – the only one that is its own negative; (a, c, d)

(iv) 7 is a natural number and an integer. It is in fact also a prime

number – that is, only divisible by itself or 1 (Section 1.2.3). It is

also anoddnumber (cannot be exactly divided by 2); (a, c, d, g)

(v)

23

5

is a rational number – actually an improper fraction

(Section 1.2.5); (c, d)

(vi)−^34 is a rational number – a proper fraction (Section 1.2.5); (b,

c, d)

(vii) 0.73 is actually a decimal representation of a rational number

0. 73 =

73

100

sometimes called a decimal fraction, but usually simply a decimal

(Section 1.2.8); (c, d, f)

(viii) 11 is a natural number and an integer – like 7 it is also prime,

and is also odd as any prime greater than 2 must be; (a, c, d, g)

(ix) 8 is another natural number and integer – but it is not prime,

since it can be written as 2× 2 × 2 = 23 (Section 1.2.7). It is

also an even number; (a, c, d)

(x) The square root of 2,

√

2, is not a rational number. This can be

shown by assuming that

√

2 =

m

n