Calculus: Analytic Geometry and Calculus, with Vectors

5.6 Sequences, series, and decimals 339

Proof of this fact can be based on the ordinary process by which "long
division" is used to divide one positive integer, say P, by another, say Q.
At each sufficiently advanced stage of the process, we obtain a representa-
tion of P/Q of the form

(5.681) Q = N+ 0.d1d2. do + 10Q,

where N is an integer, the d's are digits, and P,, is an integer remainder
for which 0 < P. < Q. After the long division has progressed past the
place where no digits other than zero are "brought down," the remainders
and hence also the d's run through cycles that produce the repeating
decimal. A cycle begins when a remainder becomes equal to a previous
remainder, and this must happen because 0, 1, 2, -,Q - 1 are the
only values that remainders can have. Dividing 365 by 7 shows an
application of the ideas. The long division process never produces
decimal expansions which, from some place onward, consist exclusively of
nines, but these expansions are clearly repeating decimals.
The elementary arithmetical consequences of Theorem 5.68 are enor-
mous. We can easily write nonrepeating decimals, examples being

0.1234567891011121314151617

where the positive integers are written in order, and

0.101001000100001000001 -.

These decimals converge to real numbers that are not rational and are

called irrational (not ratio-nal). This proves existence of irrational

numbers. Moreover, we can easily generate the idea that if the digits in

04d24d4 ...

are selected in some random way, then it is highly unlikely (or even

almost impossible) that the decimal would be a repeating decimal.

This leads us to the idea that "almost all" real numbers are irrational, and

there are different ways in which this idea can be made precise.

Problems 5.69

1 Show that if a, b, and c are digits, then

a 10a+b

(a) 0.aaaa (b) 0.ababab 99

(c) b.aaaa 9b + a 99c+10a+b

• 9

(d) c.ababab. 99

9b+a 99c+10a+b

(e) 0.baaa 90 (f) 0.cabab -