Assignment 2, Math 313
Due: Wednesday, February 3rd, 2016
1 Express each of the following numbers as a finite simple continued
fraction :
43/30, 30/43, −43/30, −30/43.
2 Let α ∈ Q with α > 1. Prove that if α = [a0 , a1 , a2 , · · · , an ], then
1
= [0, a0 , a1 , a2 , · · · , an ].
α
√
3 Find
the
infinite
simple
continued
fraction
expansion
of
13 and
√
(3 + 13)/2.
4 Let α be a real irrational number. Prove that of any two consecutive
convergents of the infinite simple continued fraction expansion of α, at
least one, say pi /qi , satisfies
p
i
α − < 1 .
qi 2q 2
i
1