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Chapter 2: Problem 13
(a) Give the first four digits in the binary representation of \(\frac{1}{3}\). (b) Give the complete binary representation of \(\frac{1}{3}\).
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If \(0 \leq a
Assuming the existence of roots, show that if \(c>1\), then \(c^{1 / m}
Solve the following equations, justifying each step by referring to an appropriate property or theorem. (a) \(2 x+5=8\), (b) \(x^{2}=2 x\), (c) \(x^{2}-1=3\), (d) \((x-1)(x+2)=0\).
If \(S \subseteq \mathbb{R}\) is a nonempty bounded set, and \(I_{S}:=[\inf S\), sup \(S]\), show that \(S \subseteq I_{S}\). Moreover, if \(J\) is any closed bounded interval containing \(S\), show that \(I_{S} \subseteq J\).
Show that sup \(\\{1-1 / n: n \in \mathbb{N}\\}=1\)
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