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B How to convert a fraction into a repeating decimal

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  1. Jul 11, 2016 #1
    For example, say I want to convert 1/7 to its representation as a repeating decimal? Is the fastest way just to do long division, or is there a faster way?
     
  2. jcsd
  3. Jul 11, 2016 #2

    fresh_42

    Staff: Mentor

    You could learn the first ones, say until 11 inclusive, by heart.
     
  4. Jul 11, 2016 #3

    mfb

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    2016 Award

    Staff: Mentor

    Or use a calculator. Where is the point in getting the 7th digit right if you calculate something by hand?
     
  5. Jul 11, 2016 #4

    symbolipoint

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    Do the division operation. You will see the repeating decimal. You can and should be able to do it the long, traditional way. If you do this on 1/7 a couple of times, you may memorize it.

    0 . 1 4 2 8
    _____________________
    7 ) 1. 0 0 0 0 0 0
    7
    ------------
    3 0
    2 8
    --------
    2 0
    1 4
    -------
    6 0

    Keep going. You will find 0.142857, REPEATING those digits.

    The alignment on the page is not working....

    micromass suggested CODE tags...
    Code (Text):

          0 . 1 4 2 8    
         _____________________
    7   )  1. 0 0 0 0 0 0
              7
    ------------
              3 0
              2 8
              --------
                 2 0
                 1 4
                 -------
                    6 0
     
     
    Last edited: Jul 12, 2016
  6. Jul 12, 2016 #5

    chiro

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    Science Advisor

    Hey Mr Davis 97.

    The fraction is found by expanding the numerator and denominator with respect to some base.

    Usually - decimals are written in base 10 and since each digit is written that way you will typically have an X/10^n where n is some integer (positive number).

    You are essentially solving for X.

    You can change the base as you need to (for example - binary, hexadecimal and so on) and even for changing bases (mixed bases) but the idea is the same.
     
  7. Jul 21, 2016 #6
    I have a couple of tricks for remembering the 7ths.
    • The recurring digits of n/7 are the same for any n, the pattern starts on a different digit depending on n. (So you need only remember 1/7.)
    • No digit in 1/7 is divisible by 3.
    • The first three significant digits in 1/7 add up to 7.
    • Use even multiples of 7 to remember the first four digits of 1/7
      • 2*7 = 14: 0.142857...
      • 4*7 = 28: 0.142857...
      • 6*7 = 42: 0.142857...
    • The difference between each successive digit alternates in sign: (e.g. +,-,+,-,+,-,+,-,...)
    • The absolute values of the differences between successive digits is also a repeating pattern.
      • 1/7 = 0.1428571...
        1 [+3] 4 [-2] 2 [+6] 8 [-3] 5 [+2] 7 [-6] 1 ...

    I'm half making this up on the fly, but you could make the denominator a multiple of 10, then decompose the fraction from there. Take 1/6 for example,
    $$
    \begin{align}
    \frac{1}{6} &= \frac{5}{30} \nonumber \\
    &= \frac{3}{30} + \frac{2}{30} \nonumber \\
    &=\frac{1}{10}\left(1 + \frac{2}{3}\right). \nonumber
    \end{align}
    $$
     
  8. Jul 21, 2016 #7

    mfb

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    Don't make it too complicated.

    0.14 28 57 repeating
    2*7=14
    2*14=28
    2*28=56, add 1 which is the overflow from the next step, 2*56=112

    This is basically two steps of a long division each time: 1/7 = 14/98 = 14*(1/98) and 1/98 = 0.0204081...
     
  9. Jul 21, 2016 #8

    fresh_42

    Staff: Mentor

    Whatever might be the answer to this basic question. IMO it simply disguises the fundamental difference between now and then. A few decades ago there haven't been any calculators at school. So it has been necessary to remember a lot of small multiplications and divisions in order to save time. It also helped a lot in everyday tasks like shopping.

    This has fundamentally changed and younger people normally aren't used to numeric solutions anymore. It isn't needed. However, it comes to a prize. E.g. I had to learn to use a slide rule and an essential part of it has been to estimate the order of magnitude of a calculation. Without those kind of training it happens that less and less people have a feeling about the likelihood of a numeric solution. At least I experienced this while tutoring kids. Comparable effects could be said about the usage of units. I can't even estimate how often I requested to pull units through an entire calculation.

    In any case. If one doesn't want to pay this price there is only one way to do (unless blessed with a gift like Ramanujan was): practice, practice, practice, ...
     
  10. Jul 21, 2016 #9
    I would say it is still necessary, to a much lesser extent perhaps. There are so many times I want to do a calculation but don't have my phone at hand, or it seems like too trivial a calculation to go through all the button pressing and screen swiping it takes to open a damn calculator app, or it is just too inconvenient to waggle a phone/calculator around (like when shopping). Times when I go to pay in a shop and I give the person a few extra coins so they can give me back a note instead of a heap of change, a look of fear always sweeps across their face. For example, the other day I paid for $10.50 of items with a $20 note and a 50c and the guy was very unsure about giving me a $10 note in change.

    The units thing I don't understand at all. I love units, they will alert you to most mistakes you will ever make.
     
  11. Jul 21, 2016 #10

    symbolipoint

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    libervurto,
    That is the perfect type of explanation for why to learn basic numerical skills. As regards to units, including them sometimes helps to think straight about your calculations.
     
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