The Chemistry Maths Book, Second Edition

20.2 Errors 561

so that, in the present case,

andx

2

has only one significant figure.

However, because the roots of a quadratic have the property x

1

x

2

1 = 1 c 2 a, an

alternativeformula for the second root isx

2

1 = 1 c 2 x

1

a,

with error of 1 in the least significant figure.

This example shows that the correct procedure for the solution of a quadratic

equation, irrespective of the number of significant figures used, is

(20.3b)

0 Exercise 6

EXAMPLE 20.4Table 20.1 shows some results obtained for the function

f(x) 1 = 1 (1 1 − 1 cos 1 x) 2 x

2

with a 10-digit pocket calculator.

Table 20.1 Values off(x) 1 = 1 (1 1 − 1 cos 1 x) 2 x

2

x Computed f(x) True f(x)

0.1 0.4995 8346 3 0.4995 8347 22

0.01 0.4999 95 0.4999 9583 33

0.001 0.4999 0.4999 9995 83

0.0001 0.49 0.4999 9999 96

0.00001 0 0.5000 0000 00

An alternative, and correct, procedure to eliminate the differencing errors is to use the

truncated MacLaurin expansion of the function,

which, by Taylor’s theorem, has error bound2.5 1 × 110

− 11

for|x| 1 ≤ 1 0.1.

0 Exercises 7, 8

11

246

01

2

24

−

≈

!

−

!

• 
  

!

≤.

cosx

x

xx

for | |x

if bxthen

b

a

b

a

c

a

x

c

x

=−−













−













0 ,=

22

1

2

2

1

aa

if bxthen

b

a

b

a

c

a

x

c

x

<=−+











 −











0 ,=

22

1

2

2

1

aa

x

2

2 000

35 94

= 0 05565

.

.

=.

x

2

= −18 322 18 00 17 94 0 06=.−.=.

x

1

= +18 322 18 00 17 94 35 94=.+.=.