Repeating Decimals

The Basis for Cyclic Numbers

Most of you are already familiar with the repeating decimal digits of fractions like one third (1/3) or two thirds (2/3) which have these never ending strings of threes and sixes:

1 / 3 = 0.3333333333...   and   2 / 3 = 0.6666666666...

At some point (preferably before high school), a Math teacher should have explained the convention of placing a bar over repeating decimal digits, or possibly underlining them (when making a web page), for example:   1/6 = 0.1666 .   For electronic texts, its acceptable to merely bracket the repeating digits; such as: 1/6 = 0.166(6) .

Math teachers should never allow you to simply round-off  the digits to 0.167; unless you are specifically studying some method of doing so such as the use of Math in the sciences.

Note: In the sciences, where numbers often depend upon the accuracy of some measuring tool -- from a simple meter stick to the most advanced scientific instruments available, it's quite reasonable to agree that none of your results should ever be expressed with more digits than the number obtained from the least accurate tool used. For example, although 6.485 times 1.07 times 4.392 should be equal to 30.4758684 in a math class. In a science class, this would most likely be expressed as just 30.5; using a 'round-up' method, since the accuracy of the 1.07 measurement was only three digits.
      In a Mathematics class, however, you should never state that 1/6 = 0.167 (they are not equal). If a person feels its necessary not to include every digit in some result (due to limitations of space to record the digits or time to calculate all of them), I'd like to see that expressed with a sign of approximation (or at least a statement to that effect at the beginning of your paper). One way for doing this is to use a tilde mark (~) in place of the bars of the equal sign. Another is by placing a dot above the equal sign. You could also define some other way to show an approximate value in any of your own papers.

As you'll soon see below, it can sometimes be very difficult to determine the digits that repeat in the decimal expansion of a repeating fraction.

The first non-trivial (more than a single digit) repeating decimal fraction is that of one seventh:

1/7 = 0.142857142857 

Even the calculator program that came with Windows® 95 (with its 12 to 13 digit display) would give most people the idea that the same six digits might be repeating here all the time. But what's a person to do if the number of digits that comprise a repeating decimal fraction end up being 20, 30, 60 or even hundreds of digits?! This is why we must have some agreed upon notation for expressing an approximate number when that number must be written as a value with digits to the right of the decimal point. (On a web page, one might use a single tilde mark, stating for example that 1/49 ~ 0.0212766 and read this as "approximately equal to," or state in full that the value is an approximation.)

The Decimal Equivalents
for All Fractions
From 1/2 through 1/66

General Rule of Exact Decimal Equivalents

Note that all the fractions that are powers of 1/2 (1/4, 1/8, 1/16, 1/32, 1/64, etc.) have exact decimal equivalents. ("Hey, I thought we were working with decimal numbers not binary!") Well, we are... Just remember that in fractions, dividing by powers of two gives us digits similar to successive powers of 5 (such as: 0.25, 0.125, 0.0625, 0.03125, etc.) As a matter of fact, every fraction that has an exact decimal equivalent consists only of powers of 1/2, or the product of 1/5 times a power of 1/2 or some multiple of these fractions (such as 3 times 1/4 = 3/4 = 0.75); there is no other way for an exact decimal equivalent to exist. Also note that 2 and 5 are the only coprimes of 10.

Observations on Repeating Decimals

• From 1/7, 1/17, 1/19, 1/23, 1/29, 1/47, 1/59 and 1/61 we observe that if a fraction is composed of repeating decimal digits, the number of digits that repeat will never be greater than one subtracted from the number in the fraction's denominator.
  
  
• We also observe that if the number of digits in a repeating decimal fraction is equal to (denominator - 1) digits, then its denominator must be a prime number.
  
  ( But remember that this is not true for all "prime number fractions." Only about half of all the 'prime fractions' between 1/7 and 1/66 have this property. But the ratio could easily change if more 'prime fractions' were included. Here are a few more 'prime fractions' that I eliminated from the special group:

  1/73 = 0.0136986301369863 1/101 = 0.00990099 1/137 = 0.0072992700729927

  You'll have to try the other primes on your own: 67, 71, 79, 83, 89, 97, 103, 107, 109, 113, 127, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc. )
  
  If you're really interested in carrying out such calculations, there's a web page with a JavaScript program which makes things very easy, because it will actually identify the repeating portion within the fraction for you! Download it from here:
  http://www.math.fau.edu/Richman/Liberal/decimal.htm
  Note: If a fraction contains more than 200 repeating digits, you must change the value 200 in the source code line "var maxblockdisp = 200;" to a value large enough to display all of the repeating digits.
  
  
• The fraction 1 / 7  is quite amazing... almost every product of this fraction with another (1/7 times 1/2, 1/3, 1/4, 1/5, etc.) has only six repeating decimal digits; although it may take one, two or even three digits (as in the case of 1/56 or 1/84) before we reach the beginning of the digit string that repeats, it will always be:  142857 .
  

This number, 142857, is the first and most famous of a group in the Cyclic Number Sequence, since it's comprised of only 6 digits; easily memorized, and follows a simple rotation pattern in its multiples:

Multiplying 142857 by 8, 9, etc., leads to some interesting results as well:

Hmm... how far can we extend this? Well, each time we hit a multiple of 7 (14, 21, 28, etc.), we should get something like '999999', but let's check that multiple + 1 (or  29, 36, 43, etc.) for the same patterns we observed above (in rows x 8, x 15 and x 22):

4142853 [x 29; nothing new here; 3 + 4 = 7; so still like 14285(7).]
5142852 [x 36; nothing new here; 2 + 5 = 7; so still like 14285(7).]
6142851 [x 43; nothing new here; 1 + 6 = 7; so still like 14285(7).]
7142850 [x 50; nothing new here; 0 + 7 = 7; so still like 14285(7).]
8142849 [x 57; now we must change 9 + 8 = 17 into a 1 and a 7 to give us: 1428(4+1=5)(7).]
9142848 [x 64; again, this requires 8 + 9 = 17 to be a 1 and a 7 to give us: 1428(4+1=5)(7).].]

We can also see that a definite pattern has emerged in the multiples themselves: Note the sequentially increasing and decreasing digits at both the beginning and end of each new multiple! This is true for every 7th multiple you compare. When we mutiply 142857 by 71, we get: 10142847 which does resemble our pattern once we steal a 1 from the '7' to arrive at 1428(4+1=5)(6+1+0=7). Is there a multiplier that produces a result so different, that our original cyclic number is completely lost in the digital fuzz? Let's try 7 to the 4th, 5th and 6th powers, plus 1, as multipliers:

343142514 [x (7^4)+1 = 2,402; maybe "142" is part of the original, but it's quite fuzzy. However, I can still use my intuition(?) and imagine that '5' + '1' + 2 (of the '4') = 8, leaving 2 + '3' = 5 and 4 + 3 = 7 for 142(8)(5)(7).]
2401140456 [x (7^5)+1 = 16,808; here we can only guess "14" is what's left of our original pattern! I'm not even going to bother imagining how to get back to it from here!]
16807126050 [x (7^6)+1 = 117,650; I think that's sufficiently fuzzy! And using (7^7)+1, gives us a result of: 117648882351 in which it's impossible to even guess at the starting '1' for our original pattern.].

The next few cyclic numbers, however, not only contain increasingly longer strings of 16, 18, 22, etc. digits, but each of them begin with a leading zero:

0588235294117647 (16 digits; see 1/17 in table above)
052631578947368421 (18 digits; see 1/19 in table above)
0434782608695652173913 (22 digits; see 1/23 in table above)

Likewise, the multiples of the these numbers have increasingly difficult patterns of rotations to determine! For example, some of the rotations of the next two cyclic numbers are:

So few people ever study more than the first cyclic number of 142857.

For Further reading:

Cyclic Nubers

Recurring (or Repeating) Decimals

Since you've made it this far, I'm going to tell you where to find a FREE Windows Calculator that can compute (and copy into the clipboard buffer) as many as 10,000 decimal places!!! It defaults to 500 digits which should be quite sufficient for any repeating decimals you'll ever study! Take this offsite link: http://www.wordsmith.demon.co.uk/downloads/ to get a copy of the MIRACL Calc multi-precision calculator by Geoff Wilkins.

NOTE: Do not trust at least the last 5 to 10 digits of the number Pi (3.14159265...) from this calculator, nor possibly of any other value you calculate with it. It's always a good idea to carry out a calculation to many digits beyond what you'll actually use, since the digits at the end may be incorrect! And because it produces 500 to 10,000 digits (depending on how you set it up), you can easily afford to forget about the last 10 or so digits!

Last Update: October 6, 2007 (2007.10.06)

The Starman's Math Index