How to convert a fraction into a repeating decimal

[Mr Davis 97]

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?

 

[fresh_42]

You could learn the first ones, say until 11 inclusive, by heart.

 

[mfb]

Or use a calculator. Where is the point in getting the 7th digit right if you calculate something by hand?

 

[symbolipoint]

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.

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micromass suggested CODE tags...

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

 

[chiro]

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.

 

[libervurto]

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,
$$
\begin{align}
\frac{1}{6} &= \frac{5}{30} \nonumber \\
&= \frac{3}{30} + \frac{2}{30} \nonumber \\
&=\frac{1}{10}\left(1 + \frac{2}{3}\right). \nonumber
\end{align}
$$

 

[mfb]

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...

 

[fresh_42]

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, ...

 

[libervurto]

Whatever might be the answer to this basic question. IMO it simply disguises the fundamental difference between now and then... This has fundamentally changed and younger people normally aren't used to numeric solutions anymore. It isn't needed... 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.

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.

 

[symbolipoint]

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.