As many of you may already know, dividing by 9 gives you a repeating decimal. For example, 1 divided by 9 equals .1111111… When you divide by 99, 999, etc. you find a similiar pattern. For example, 123 divided by 999 equals .123123123… But while playing around with a calculator, I found another astounding pattern involving the number 9. Look at the following example: when you divide 11 by 90, you get .12222… The first number is one, and the second repeating number is one plus one, from 11 of course. Now look what happens when the two numbers added together equal more than a single digit: 55 divided by 90 equals .61111… The repeating digit, one, of course comes from five plus five, which equals 10. Then you add for 10, one plus zero to get the repeating one. But notice since 5 plus 5 equals two digits, 10, the first number of the fraction is 6, not 5. The one is then added to the 5.

I sometimes use this unique pattern to simplify repeating decimals. I don’t know about the rest of you, but I sometimes have a hard time simplifying repeating decimals. Now I have something that helps a little, at least. For example, take the familiar repeating decimal .16666… Since I know it is something divided by 90, I simply multiply it by 90 to get 15. 15 divided by 90 then simplifies easily to 1/6. Neat, no?

The pattern continues with 900, 9000, etc. For example 111 divided by 900 equals .12333… But I’ll just leave it there and let you figure out the rest for yourselves .

Dividing by 90 is equivalent to dividing by 9, then by 10 (or vice versa).

If you divide a two-digit number by 9, you’re going to get a quotient and a remainder.
For example, 38 / 9 = 4 and 2/9, so in decimal form it would be 4.222222222…
Then divide this by 10 and it becomes .422222…

Everything you need to know about simplifying repeating decimals comes from very simple algebra: Suppose you have the number K = 0.(pattern)(pattern)(pattern)… Multiply it by the appropriate power of 10 to get 10^c * K = (pattern).(pattern)(pattern)(pattern)… = (pattern) + K. Now subtract K from both sides and divide out the power of 10 to get K = (pattern)/(10^c - 1). Thus, for example, if K was 0.123123123…, then c would be 3 and (pattern) would 123, so we’d get 10^3 * K = K+123, so K = 123/(10^3 - 1) = 123/999 = 41/333. [Obviously, the essentially same thing works for any base, not just decimal.]

Furthermore, if your number isn’t strictly of the form K = 0.(pattern)(pattern)(pattern)…, but, rather, has an initial nonrepeating part followed by a repeating pattern, you can still easily transform it to one of the right form to figure out its value. For example, take 0.166666… This is 0.66666… / 10 + 0.1. So once you know what 0.66666 is, you can tell what 0.1666… is. And, by the above method, we know that, letting L = 0.66666…, we have that L * 10 = L+6, so L = 6/9 = 2/3. Therefore, 0.16666… = (2/3)/10 + 0.1 = 2/30 + 1/10 = 5/30 = 1/6. Ta-da!

Or, to put my thing from above another way, no matter what the number is, if it eventually hits a cyclic pattern, then you can figure out its value with simple algebra as follows: multiply it by two different powers of 10 (or whatever your base is), the first bringing it so that the first iteration of its cyclic pattern starts right after the decimal point and the second so that the second iteration starts right after the decimal point. These two values will differ by an integer, and algebra will handle the rest.

Thus, for a random example, consider K = 0.53[1234][1234][1234]…, where I’ve bracketed the cyclic pattern. We see that 10^2 * K = 53.[1234][1234]… while 10^6 * K = 531234.[1234][1234]…; therefore, 10^6 * K - 10^2 * K = 531234 - 53 = 531181. Therefore, K = 531181/(10^6 - 10^2) = 531181/999900. In this case, that fraction happens to already be in lowest terms, but in general, you may get something not in lowest terms; you can, of course, reduce it to lowest terms using whatever your favorite method is.

You don’t normally think of them this way, but every decimal representation repeats. It’s just that if the repeating part is a string of 0s, we tend not to write it.

Well, not every decimal representation, of course… [Since irrational numbers still have decimal representations that don’t lapse into cyclic patterns.]