## Exploding Dots

### 8.7 Decimals in Other Bases

Who said we need to stay with a $$1 \leftarrow 10$$ machine?

The following picture shows that $$1432 \div 13 = 110 \; R2$$ in a $$1 \leftarrow 5$$ machine.

If we work with reciprocals of powers of five we can keep unexploding dots and continue the division process.

Here goes!

(This reads as $$1432 \div 13=110.1 \; R0.2$$  if you like.)

(This reads as $$1432 \div 13=110.11 \; R0.02$$ if you like.)

And so on. We get, as a statement of base $$5$$ arithmetic,

$$1432 \div 13=110.111\cdots$$.

To translate this to ordinary arithmetic we have that

$$1432$$ in base five is $$1 \times 125 + 4 \times 25 + 3 \times 5 + 1 \times 1 = 242$$,

$$13$$ in base five is $$1 \times 5 + 3 \times 1 = 8$$,

$$110.111\cdots$$ in base five is  $$30 + \dfrac{1}{5}+\dfrac{1}{25}+\dfrac{1}{125}+\cdots$$,

so we are claiming, in ordinary arithmetic, that

$$242 \div 8 = 30 + \dfrac{1}{5}+\dfrac{1}{25}+\dfrac{1}{125}+\cdots$$.

Whoa!

Question: What does the geometric series formula from chapter 7 say about the sum $$\dfrac{1}{5}+\dfrac{1}{25}+\dfrac{1}{125}+\cdots$$? Does it equal a quarter? (To be clear, in chapter 7 we showed that

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

Multiplying through by $$x$$ then gives

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

Perhaps it is this second version of the geometric formula we need to work with now.)

Here are some (challenging) practice questions if you are up for them.

28. Compute $$8 \div 3$$ in a base $$10$$ machine and show that it yields the answer $$2.666\cdots$$.

29. Compute $$1 \div 11$$ as a problem in base $$3$$ and show that it yields the answer $$0.02020202\cdots$$. (In base three, “$$11$$” is the number four, and so this question establishes that the fraction $$\dfrac{1}{4}$$ written in base three is $$0.020202\cdots$$.)

30. Show that the fraction $$\dfrac{2}{5}$$, written here in base ten, has the “decimal” representation $$0.121212\cdots$$ in base four. (That is, compute $$2 \div 5$$ in a $$1 \leftarrow 4$$ machine.)

31. 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?)

32. 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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