Understanding Engineering Mathematics

• evaluation and properties of limits


• continuity


• slope of a curve and theory of differentiation


• infinite sequences


• iteration and Newton’s method


• infinite series


• infinite power series


• convergence of sequences and series

Motivation

You may need the material of this chapter for:

• understanding when particular mathematical methods are applicable


• solving equations by iteration


• testing convergence of numerical methods

14.1 Continuity and irrational numbers

What do we mean by a ‘continuous curve’? Plotting the result of an experiment on a graph
will lead to a series of points separated by gaps (which may be small, but will always be
there). We can neverplotacontinuous curve, although we usually draw one through the
points. In fact, the idea of a continuous curve is a convenient mathematical abstraction
which allows us to use geometry to talk about slopes and rates of change. When we draw a
continuous curve we tacitly assume that the curve passes through every point on it (which
it doesn’t – there will at least be gaps between the molecules in the chemicals of the ink!).
Toplota point on a curve we mustmeasureits distance from some origin. This can
only be done to a certain level of accuracy and so can only be done using a terminating
decimal, e.g. 3.412. As noted in Chapter 1 such numbers can always be written asrational
numbers– those which can be expressed in the formm/nwheremandnareintegers.

Problem 14.1

Write the decimal 3.412 as a rational number.

Quite simply in this case we have

3. 412 =

3412

1000

However, there are numbers which cannot be written in this form as fractions – i.e. there
exist quantities which we cannevermeasure exactly and yet which do have a real exist-
ence! These can never be plotted on a graph and so represent ‘holes’ in the apparently
continuous curve. An elementary example of such a number is the diagonal of the unit
square. By Pythagoras’ theorem (154
➤
√ ) we know that the unit square has diagonal
2 units. Clearly, this number ‘exists’, otherwise we couldn’t cross a square diagonally.