10 Chapter 1Numbers, variables, and units
where 10
−n1 = 11210
nand 10
01 = 11 (see Section 1.6).
0 Exercises 37–42
A number with a finite number of digits after (to the right of ) the decimal point
can always be written in the rational formm 2 n; for example1.234 1 = 1123421000. The
converse is not always true however. The number 123 cannot be expressed exactly as
a finite decimal fraction:
the dots indicating that the fraction is to be extended indefinitely. If quoted to four
decimal places, the number has lower and upper bounds 0.3333 and 0.3334:
where the symbol <means ‘is less than’; other symbols of the same kind are ≤for ‘is
less than or equal to’, >for ‘is greater than’, and ≥for ‘is greater than or equal to’.
Further examples of nonterminating decimal fractions are
In both cases a finite sequence of digits after the decimal point repeats itself
indefinitely, either immediately after the decimal point, as the sequence 142857 in
12 7, or after a finite number of leading digits, as 3 in 1 2 12. This is a characteristic
property of rational numbers.
EXAMPLE 1.9Express 1 2 13 as a decimal fraction. By long division,
The rational number 12131 = 1 0.076923 076923=is therefore a nonterminating decimal
fraction with repeating sequence 076923 after the decimal point.
0 Exercises 43–46
0 07692307
13 1 00
91
90
78
120
117
30
26
40
39
100
. ...
).
1
7
0 142857142857
1
12
=. ......, =.0 083333333333
0 3333
1
3
.<<.0 3334
1
3
=.0 333...