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Decimalia
Decimalia
Decimalia

Station  8G
Wild Exploration 

Here are some “big question” investigations you might want to explore, or just think about. Have fun!

Question 1

WHICH FRACTIONS GIVE FINITE DECIMAL EXPANSIONS?

We’ve seen that 12=0.5\dfrac {1}{2} = 0.5, 14=0.25\dfrac {1}{4} = 0.25 and 18=0.125\dfrac {1}{8} = 0.125 each have finite decimal expansions. (We’re ignoring infinite repeating zeros now.)

Of course, all finite decimal expansions give fractions with finite decimal expansions! For example, 0.370.37 is the fraction 37100\dfrac {37}{100}, showing that 37100\dfrac {37}{100} has a finite decimal expansion.

What must be true about the integers aa and bb (or true just about aa or just about bb) for the fraction ab\dfrac {a}{b} to have a finite decimal expansion?

Question 2

BACKWARDS? ARE REPEATING DECIMALS FRACTIONS?

We have seen that 13=0.3\dfrac {1}{3} = 0.\overline{3} and 47=0.571428\dfrac {4}{7} = 0.\overline{571428}, for example, and that every fraction gives a decimal expansion that (eventually) repeats, perhaps with repeating zeros.

Is the converse true? Does every infinitely repeating decimal fraction correspond to a number that is a fraction?

Is 0.170.\overline{17} a fraction? If so, which fraction is it?

Is 0.4500.\overline{450} a fraction? If so, which fraction is it?

Is 0.3222220.322222...=0.32= 0.3\overline{2} a fraction?

Is 0.170230.17\overline{023} a fraction?

Indeed, does every repeating decimal have a value that is a fraction?

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

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