Exploding Dots
2.3 \(x\)mals
Who said we need to stay with a \(1 \leftarrow 10\) machine?
The following picture shows that in a \(1 \leftarrow 5\) machine, \(1432 \div 13 = 110 R 2\).
If we work with reciprocals of powers of five we can keep unexploding dots and continue the division process:
We get:
\(1432 \div 13 = 110.111\cdots\).
as a statement in base five! (What is this really saying? We have that “\(1432\)” is the number \(1 \times 125 + 4 \times 25 + 3 \times 5 + 2 \times 1 = 242\) and “\(13\)” is \( 1 \times 5 + 3 \times 1 = 8\). What’s \(110.111\cdots\)?)
Question 69:
a) Compute \(8 \div 3\) in a base \(10\) machine and show that it yields the answer \(2.666\cdots\).
b) Compute \(1 \div 11\) in base \(3\) and show that it yields the answer \(0.0202020202\cdots\).(In base three, “\(11\)” is the number four, and so this question establishes that the fraction \(\frac{1}{4}\) written in base three is \(0.\bar{02}\).)
c) Show that the fraction \(\frac{2}{5}\) (here written in base ten) has, in base \(4\), “decimal” representation \(0.1212121212\cdots\).
d) CHALLENGE: What fraction has decimal expansion \(0.3333\cdots\) in base \(7\)? Is it possible to answer this question by calling this number \(x\) and multiplying both sides by \(10\)? (Does “\(10\)” represent ten?)

Question 70: Use an \(1 \leftarrow x\) machine and \(x\)mals to show that \(\dfrac{1}{x1} = \dfrac{1}{x} + \dfrac{1}{x^{2}} + \dfrac{1}{x^{3}} + \dfrac{1}{x^{4}} + \cdots\).

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