Numerical Methods and Optimization

42 Numerical Methods and Optimization: An Introduction

2.1.2 Conversion of fractions

Consider a positive real number x. We can represent x as:

x = x

I

+ x

F

,

where x

I

= x is the integer part x,andx

F

= {x} is the fractional part of

x. For example, for x =5.3wehavex

I

= 5 and x

F

=0.3. We can convert x

I

and x

F

to another base separately and then combine the results. Note that

the x

F

part can be written in the form

x

F

=

∞



k=1

a

−k

10

−k

,

where a

−k

∈{0, 1, 2,...9},fork =1,...,∞.Ifa

−k

=0fork ≥ N,whereN

is some finite integer, then the fraction is said to terminate. For example,

1

8

=0.125 = 1 × 10

−1

+2× 10

−2

+5× 10

−3

terminates, whereas

1

6

=0.16666 ...=1× 10

−1

+6× 10

−2

+6× 10

−3

+ ··· ...