Notes on Repeating Decimals

To reduce a recurring decimal to a common fraction. -

Subtract the decimal figures that do not recur from the whole decimal including one set of recurring figures; set down the remainder as the numerator of the fraction, and as many nines as there are recurring figures, followed by as many ciphers as there are non-recurring figures, in the denominator.

- Kent's Mechanical Engineer's Pocket Book, 1896

(In this paragraph, by "cipher" Kent means "zero". It was apparently a common term a century ago. Both "zero" and "cipher" come from the Arabic "safira", meaning "it was empty". Likewise, Kent says "recurring figures" were we would say "repeating digits" today. But I digress...)

Thus, to convert 0.79054054054... to a fraction, the non-repeating digits being 79 and the repeating digits being 054:

79054 - 79 = 78975

Since we have three repeating digits (054), and two non-repeating digits (79), the denominator would be three nines and two zeros: 99900.

78975 / 99900 = 117 / 148 = 0.79054054054...

That is the example Kent gave. Here is another: to convert 0.08333... (useful for converting inches to feet) to a fraction, the non-repeating figures being 8 and the repeating figures being 3:

83 - 8 = 75

One repeating digit (3) and two non-repeating digits (08), so the denominator is one nine and two zeros: 900

75 / 900 = 1 / 12 = 0.08333...

Pretty cool, huh? As long as you remember that the repeating digits get the nines in the denominator, it works. Since ninths produce repeating digits (1/9 = 0.1111..., 2/9 = 0.2222.... etc) you can see how it all fits together.

Algebraically, it works like this:

n = 0.42857142857142857...

Multiply by ten to the power of the number of repeating digits, in this case six...

1000000n = 428571.42857142857...

...subtract n, which eliminates the repeating digits...

999999n = 428571

and solve for n.

n = 428571 / 999999 = 3 / 7

If you look carefully, this is the same as Kent's method, just expressed a bit differently. You'll see that if you try it with Kent's example:

n = 0.79054054054...

1000n = 790.54054054054...

999n = 789.75

n = 789.75 / 999 = 78975 / 99900 = 117 / 148

Look familiar?

References

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© 2005 W. E. Johns