## 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 $$\dfrac{1}{x-1} = \dfrac{1}{x} + \dfrac{1}{x^{2}} + \dfrac{1}{x^{3}} + \dfrac{1}{x^{4}} + \cdots$$.

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