Understanding Engineering Mathematics

calculator, or we need to get a rough order of magnitude check on a messy calculation. In
such situations the engineer’s most powerful tool has always been an ability to mentally
estimate quantities and perform quick ‘back of the envelope’ (we still have them, despite
email!) calculations. The trick is to approximate the numbers you are dealing with so
that the calculations become simple, yet some sort of rough accuracy is retained. It is a
matter of judgement and practice. Absolute values of numbers are less important than their
relative values – for example 1021 is significant in

3 × 1021 + 40 × 234

but is relatively insignificant in

1021

10

− 103372415

So, inspect all the numbers occurring in an expression and approximate them each to
an appropriate order of magnitude, rounding as necessary, then perform the (hopefully)
simplified calculation with the results.

Solution to review question 1.1.9

(i)

4. 5 × 105 × 2. 0012

8. 892 × 104



4. 5 × 105 × 2

9 × 104



105

104

 10

So if your calculator gave you 101.2753...then you know you have

slipped up on a decimal point.

(ii)

√

254 × 104 + 28764. 5

2. 01 × 10 − 2. 54 × 10 −^6



√

254 × 104 + 3 × 104

2 × 10

neglecting 2. 54 × 10 −^6 in comparison with 2. 01 × 10

=

√

257 × 104

2 × 10



√

256 × 104

2 × 10

=

√

162 × 104

2 × 10

on replacing 257 by 256 for easy square rooting:

=

16 × 102

2 × 10

= 80

The answer to two decimal places is in fact 79.74.