The Real And Complex Number Systems

Then S = N.
Proof: (A ⇒ B): If S ≠ N, then N − S ≠ φ. So, by (A), there exists
the smallest integer w such that w ∈ N − S. Note that w > 1 by (1), so we
consider w − 1 as follows.
Since w − 1 /∈ N − S, we know that w − 1 ∈ S. By (2), we know that
w ∈ S which contadicts to w ∈ N − S. Hence, S = N.
(B ⇒ C): It is obvious.
(C ⇒ A): We have proved it by this exercise.
Rational and irrational numbers
1.7 Find the rational number whose decimal expansion is 0.3344444444....
Proof: Let x = 0.3344444444..., then
x = 3 10 + 3
10 + 4
2 10 + ... + 4 + .., where n ≥ 3
3 10n = 33
10 + 4 (
1 + 1 2 10 3 10 + ... + 1 )
10 + .. n
= 33
10 + 4 ( 1
2 10 3 1 − 1
10
= 33
10 + 4
2 900
= 301
900 .
)
1.8 Prove that the decimal expansion of x will end in zeros (or in nines)
if, and only if, x is a rational number whose denominator is of the form 2 n 5 m ,
where m and n are nonnegative integers.
Proof: (⇐)Suppose that x =
k , if n ≥ m, we have
2 n 5 m
k5 n−m
2 n 5 n = 5n−m k
10 n .
So, the decimal expansion of x will end in zeros. Similarly for m ≥ n.
(⇒)Suppose that the decimal expansion of x will end in zeros (or in
nines).
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