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The method of proof used here is often referred to as Cantor’s diagonalization argument. In particular this shows that R is much bigger that the familiar subset Q ⊆ R, however it can be hard to identify particular elements of the complement R − Q. In fact the subset of all real algebraic numbers is countable, where such a real number is a root of a monic polynomial of positive degree, X n + an−1 X n−1 + · · · + a0 ∈ Q[X]. 60 4. FINITE AND INFINITE SETS, CARDINALITY AND COUNTABILITY Problem Set 4 4-1.
K m=n Hence (α ∗ β) ∗ γ(n) = α ∗ (β ∗ γ)(n) for all n, so (α ∗ β) ∗ γ = α ∗ (β ∗ γ). b) We have δ ∗ θ(n) = δ(d)θ(n/d) = θ(n), d|n and similarly θ ∗ δ(n) = θ(n). c) Take t1 = 1. We will show by Induction that there are numbers tn for which td θ(n/d) = δ(n). d|n Suppose that for some k > 1 we have such numbers tn for n < k. Consider the equation td θ(k/d) = δ(k) = 0. d|k Rewriting this as tk = − td θ(k/d), d|k d=k ˜ we see that tk is uniquely determined from this equation. Now define θ˜ by θ(n) = tn .
An ] with A > 1, show that 1/A = [0; a0 , a1 , . . , an ]. Let x > 1 be a real number. Show that the n th convergent of the continued fraction representation of x agrees with the (n−1) th convergent of the continued fraction representation of 1/x. 1 1 1-18. Find the continued fraction expansions of √ and √ . Determine as many conver5 5−1 gents as you can. √ 1 1-19. Investigate the continued fraction expansions of 6 and √ . Determine as many con6 vergents as you can. 1-20. [Challenge question] Try to determine the first 10 terms in the continued fraction expansion of e using the series expansion 1 1 1 e = 1 + + + + ··· .
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