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Math Help - Is this a rational number?

  1. #1
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    Is this a rational number?

    2.202002000200002...

    There is a pattern there so part of me wants to say yes. But there is technically no repeat or termination so a bigger part of me wants to say no.

    I am making answer sheets for my students and initially I was going to say no, this is not a rational number due to the fact there is no repeat or termination of the decimal. But looking at the worksheet, it gives no options for the students to label a number as irrational. Just W, I or Q.

    Typically the directions for these sheets are very specific, so that got me thinking that this must be a rational number and that the obvious pattern must allow it to be translated into a fraction somehow, but I am not sure how I would do that. I know how to turn repeating and terminating decimals into fractions but not this if it is in fact possible.

    I know it is probably a basic question but I'm at a loss here.
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  2. #2
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    Re: Is this a rational number?

    I believe a theorem states that every rational number is either a repeating or terminal decimal. Therefore, if it doesn't repeat, it must be irrational.
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  3. #3
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    Re: Is this a rational number?

    That is what I assumed, but the worksheet did not give an option for irrational numbers in the directions, which they are typically VERY specific. So just making sure I am not losing it. Thank you for the reply.
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  4. #4
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    Re: Is this a rational number?

    And what are W and I, just for information? I assume that Q means rational numbers.
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  5. #5
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    Re: Is this a rational number?

    Whole and Integers
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  6. #6
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    Re: Is this a rational number?

    Well, this will probably be meddling in your affairs, but just in case. If W, I and Q are used as labels and are written in thin (regular) font, I guess it's OK. If, however, they denote sets of numbers and are written in thick font like this: \mathbb{Q}, then the universal notation for integers is \mathbb{Z}. Also, Wikipedia claims that "whole number" is a term with inconsistent definitions. It is better to say "natural numbers" and denote their set by \mathbb{N}.
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  7. #7
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    Re: Is this a rational number?

    In the instructions it says
    W = whole
    I = integers
    and Q = Rational


    In the book they also define Whole numbers as {0, 1, 2, 3...}
    Natural as {1, 2, 3, 4...}

    and Rational as any number that can be written in A over B form.

    So I am going based off of the information this company gave me.
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  8. #8
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    Re: Is this a rational number?

    Hello, Jman115!

    \text{Is this a rational number? }\:2.202002000200002\hdots

    There is a pattern there so part of me wants to say yes.
    But there is technically no repeat or termination
    . . so a bigger part of me wants to say no.

    You are right . . . there is a pattern.
    . . But it does not have a repeating cycle.
    As Jman115 suggested, it must be irrational.


    We have: . X \;=\;\frac{2}{10^0} + \frac{2}{10^1} + \frac{2}{10^3} + \frac{2}{10^6} + \frac{2}{10^{10}} + \frac{2}{10^{15}} + \hdots

    The exponents are Triangular Numbers.

    . . That is: . X \;=\;2\sum^{\infty}_{n=1}\frac{1}{10^{\frac{n(n-1)}{2}}}

    The series is neither arithmetic nor geometric, nor a combination thereof.
    I have found no way to evaluate it.

    It can be written as a recurrence: . a_n \;=\;a_{n-1}\!\cdot\!\frac{1}{10^{n-1}},\;\;a_1\,=\,2
    . . but this doesn't help either . . .

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