Understanding Engineering Mathematics

(viii) − 2 −( 4 − 5 )=− 2 −(− 1 )

=− 2 + 1

=− 1

(ix)( 4 ÷(− 2 ))× 3 − 4 =

(

4

− 2

)

× 3 − 4

=(− 2 )× 3 − 4

=− 6 − 4

=− 10

(x)( 3 + 7 )÷ 5 +( 7 − 3 )×( 2 − 4 )

= 10 ÷ 5 +( 4 )×(− 2 )

= 2 − 8

=− 6

Notice the care taken in these examples, spelling out each step.

This may seem to be a bit laboured, but I would encourage you

to take similar care, particularly when we come to algebra. Slips

with brackets and signs crop up frequently in most people’s calcula-

tions (mine included!). Whereas this may only lose you one or two

marks in an exam, in real life, an error in sign can convert a stable

control system into an unstable one, or a healthy bank balance into

an overdraft.

1.2.5 Handling fractions

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328 ➤

Afractionorrational numberis any quantity of the form

m

n

n= 0

wherem,nare integers butnis not equal to 0. It is of course essential thatn=0, because
as noted in Section 1.2.1division by zero is not defined.

mis called thenumerator

nis thedenominator

Ifm≥nthe fraction is said to beimproper,andifm<nit isproper.
A number expressed in the form 2^12 (meaning 2+^12 ) is called amixed fraction.In
mathematical expressions it is best to avoid this form altogether and write it as avulgar
fraction,^52 , instead, otherwise it might be mistaken for ‘2×^12 =1’, and it is also more
difficult to do calculations such as multiplication and division using mixed fractions.
The numerator and denominator of a fraction may have common factors. These may be
cancelled to reduce the fraction to its simplest or ‘lowest’ form:

6

12

=

2 × 3

3 × 4

=

2

4

=

1 × 2

2 × 2

=

1

2

Each of these forms areequivalent fractions, but clearly the last one is the simplest.
However, sometimes one of the other forms may be convenient for particular purposes,
such as adding fractions. A very common fraction where we tendnotto cancel down in