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  Other Topics (2)

  Powers of 2
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> Fractions
  Fractions in Base 2
  Fractions in Base N

Fractions

Here's a table of decimal expansions for the fractions 1/n. It's traditional to draw the line over the repeating digits, but here I'm drawing it underneath, just because that's what I have available.

11
20.5
30.3
40.25
50.2
60.16
70.142857
80.125
90.1
100.1
110.09
120.083
130.076923
140.0714285
150.06
160.0625
170.0588235294117647
180.05
190.052631578947368421
200.05
210.047619
220.045
230.0434782608695652173913
240.0416
250.04
260.0384615
270.037
280.03571428
290.0344827586206896551724137931
300.03

I extended the table as far as I did because 1/27 is nice—a nice number, and also the first expansion with an odd repeat length (greater than one).

I never memorized any of the longer expansions, “longer” meaning six digits or more, except for the lovely expansion of 1/7, which shows up relatively often. The sevenths also have the wonderful property that you can obtain the expansion of m/7 by rotating the six digits of 1/7 around so that the mth smallest digit is first. In other words, 2/7 is 0.285714, 3/7 is 0.428571, and so on.

Actually, any other fraction with full length has the same property, it's just not as useful. I mean, look at 1/17, which is the next fraction with full length. For one thing, there are (necessarily) more than one of some digits, so you can't just order the digits themselves, you also have to look at the digits that follow, so that for example the first “8” counts as larger than the second. Even worse, if you then want to know the expansion of 13/17 (say), you have to know, or figure out, that the 13th smallest digit is the “7” that's followed by the “6”.

I was going to say I regretted knowing only the expansions of 1/n, and not the expansions of m/n for all 0 < m < n, but then I realized I was making things up. I only know the expansions of 1/n up to n = 10, and in that range I do know the expansions of m/n, at least if you count the trick with sevenths as knowing. Fortunately for you, I have almost nothing to say about those other expansions, otherwise I'd probably feel obliged to make a table of them. The one thing I'll point out is that ninths are nice: m/9 has expansion 0.m.

If you work out the expansions by long division, which I did once, but, I admit, not this time, a good way to start is to write out the multiples of n. In spite of the name, you don't need to multiply … you're interested in all the multiples, not just one, so you can just add n repeatedly. And, if you go one further than you need to, up to 10n instead of 9n, you get a nice check on your arithmetic.

That's also a good way to learn the multiples of numbers.

Another thing I like to do is to divide into 0.9 instead of 1.0. If you do that, then when you get to the point where the expansion repeats, you get remainder zero instead of one. So, you don't have to watch for the repeat as carefully. That only works if the number contains no powers of two or five, by the way.

Another thing that's interesting about these expansions is that when the repeating sequence has even length, the digits in the two halves often add to 9 … in other words, the halves are first complements. In 1/13, for example, 076 + 923 = 999.

The halves add up often, but not always; the first counterexample is 1/21. For some theory about when and why they add up, see Complementary Parts. Or, for a simpler view, see Other Denominators, and notice that 1/13 = 77/1001.

Even apart from being the number of Arabian nights, 1001 is a very nice number. Unlike 999, which factors uselessly into 33ื37, 1001 is the product of three successive small primes: 7ื11ื13. So, not only can 1/13 be written in terms of it, so also can 1/7 and 1/11. In fact the famous “142” of 1/7 should really be thought of as a “143”: 11 ื 13 = 143, 1/7 = 143/1001.

 

  See Also

  Expansion Calculator
  Fractions in Base 2
  History and Other Stuff
  Negative Digits
  Repeat Length
  Square Roots

@ April (2004)