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Decimalia
Decimalia
Decimalia

Station  8F
Decimals in Other Bases 

Who said we need to stay with a 1101 \leftarrow 10 machine?

Question 1

The following picture shows that 1432÷13=110R21432 \div 13 = 110 R 2 in a 151 \leftarrow 5 machine.
I8S8F-image305

If we work with reciprocals of powers of five we can keep unexploding dots and continue the division process.
I8S8F-image306

Here goes!
I8S8F-image307

(This reads as 1432÷13=110.1R0.21432 \div 13 = 110.1 R 0.2 if you like.)
I8S8F-image309

(This reads as 1432÷13=110.11R0.021432 \div 13 = 110.11 R 0.02 if you like.)
I8S8F-image311

And so on.
We get, as a statement of base 55 arithmetic,

1432÷13=110.1111...1432 \div 13 = 110.1111....

To translate this to ordinary arithmetic we have that

14321432 in base five is 1×125+4×25+3×5+1×1=2421 \times 125 + 4 \times 25 + 3 \times 5 + 1 \times 1 = 242,

1313 in base five is 1×5+3×1=81 \times 5 + 3 \times 1 = 8,

110.111...110.111... in base five is 30+15+125+1125+...30 + \dfrac{1}{5} + \dfrac{1}{25} + \dfrac{1}{125} + ...,

so we are claiming, in ordinary arithmetic, that

242÷8=30+15+125+1125+...242 \div 8 = 30 + \dfrac{1}{5} + \dfrac{1}{25} + \dfrac{1}{125} + ....

Whoa!

What does the geometric series formula from Infinitia say about the sum 15+125+1125+...\dfrac{1}{5} + \dfrac{1}{25} + \dfrac{1}{125} + .... Does it equal a quarter?

To be clear, on Infinitia we showed that

1+x+x2+x3+...=11x1+x+x2+x3+...=\dfrac{1}{1-x}.

Multiplying through by xx then gives x+x2+x3+...=x1xx+x2+x3+...=\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.

Compute 8÷38 \div 3 in a base 1010 machine and show that it yields the answer 2.666...2.666....

Question 2

Compute 1÷111 \div 11 as a problem in base 33 and show that it yields the answer 0.0202020202...0.0202020202....
(In base three, “1111” is the number four, and so this question establishes that the fraction 14\dfrac{1}{4} written in base three is 0.0202020202...0.0202020202....)

Question 3

Show that the fraction 25\dfrac{2}{5}, written here in base ten, has the “decimal” representation 0.121212...0.121212... in base four. (That is, compute 2÷52 \div 5 in a 141 \leftarrow 4 machine.)

Question 4

CHALLENGE

What fraction has decimal expansion 0.3333...0.3333... in base 77? Is it possible to answer this question by calling this number xx and multiplying both sides by 1010? (Does “1010” represent ten?)

Question 5

Use an 1x1 \leftarrow x machine and xx-mals to show that 1x1=1x+1x2+1x3+1x4+...\dfrac {1}{x-1} = \dfrac{1}{x} + \dfrac{1}{x^2} + \dfrac{1}{x^3} + \dfrac{1}{x^4} + ....

You can either play with some of the optional stations below or go to the next island!

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