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Fractions and their decimal expansions: An exploration

I've been thinking about expanding fractions into decimals recently because I want to do some videos that demonstrate mental Math techniques other than multiplication. I've been reading about the Vedic approaches to division and will cover some of those in future posts or videos.

Related to Vedic division and to decimal expansion of fractions is an exploration I want to suggest that should be accessible to many readers of this blog.

Given two integers, a and b, consider the ratio of the two integers, a/b. Determine for any a and b which of the following conditions is true:

  1. a/b expands to a non-repeating decimal. 5/8 = 0.625 is a non-repeating decimal.
  2. a/b expands to a repeating decimal with n non-repeating digits followed by r repeating digits.
    1/70 = 0.0142857142857142857... It has 1 non-repeating digit, 0, and 6 repeating digits, 142857.
    So, n=1 and r=6.

Assuming that 0 < a < b for integers a and b can you come up with an algorithm that determines n and r given a and b?

For example, d(5,8) = {3,0} because 5/8 = .625 has 3 repeating digits followed by 0 non-repeating digits.
Another example: d(15,99) = {0,2} because 15/99= .15151515..., which has 0 non-repeating digits followed by 2 repeating digits.

For this exploration we won't consider an infinite string of 0's or 9's to be repeating digits.

Here are some ideas to guide your exploration:

  1. Start with a numerator of 1 in all of your fractions when looking for patterns.
  2. Is there a relationship between the number of repeating and non-repeating digits in fractions when the numerator is 1 and when it isn't?
  3. Does the prime factorization of the denominator of a fraction give you any insights?
  4. What do the denominators of all fractions with non-repeating decimal expansions have in common?

I've done some of the exploration of this problem but not all of it. Hopefully this problem isn't harder than I think. Or, if it is then that could be a good thing.


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  1. Ah, of course it is harder, if you let it be. Each year (usually in March) I do an exploration of repeating digits, looking for patterns, but much more directed than this. It’s the opener for the unit on rational and (drumroll) irrational numbers…

    I think this year I will try yours instead.

    Hmm. Teams. Open-ended. Observations are worth more if the other teams don’t make them. Patterns? No. Relationships. Need to focus them on looking for relationships.

    I might revise the guiding/leading questions a bit.

    Thanks for posting this!

  2. re: “For example, d(5,8) = {3,0} because 5/8 = .625 has 3 repeating digits followed by 0 non-repeating digits.”

    think it’s the other way around. I think this because it is the middle of the night and I got really stuck with the notation but now I get it.

    I am playing with it now. I like this because if it leads me to something to help kids and to emphasize the importance of them learning benchmark fractions, it’ll be important.

  3. Andree,

    Of course you’re right. .625 has 3 NON-repeating digits followed by 0 repeating digits.

    Thanks for the catch.

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