In the first musing post, I mentioned that *any* decimal that ends or repeats — even if it doesn’t repeat immediately, can be expressed as a fraction. This post will describe how to find these fractions.

## Case 1 – Terminating Decimals

The first case is pretty easy. Any decimal that terminates — that is to say *ends* — can be written as a fraction with a power of 10 in the denominator. First, consider the sequence of fractions 1/10, 1/100, 1/1000, etc. The corresponding decimals are 0.1, 0.01, 0.001, and so on. The next key thing is to note that any terminating decimal could be expressed as a product between one of these decimals and a whole number. For example, 0.2457 can be rewritten as 2457 x 0.0001. The 0.0001 can be written as the fraction 1/10,000. Therefore, 0.2457 = 2457 x 1/10,000 = 2457/10000.

## Case 2 – Repeating Decimals

You probably remember that 0.3333… = 1/3 and 0.6666… = 2/3. You might also know that 0.1111… = 1/9. This last fact is the important one. If the repeating decimal is all one number, like 0.2222…, we can rewrite it as 2 x 0.1111… = 2 x 1/9 = 2/9. Therefore, 0.3333… would equal 3/9, which reduces to 1/3.

If the repeating portion is longer, we need to use 1/99, 1/999, 1/9999, or something of those forms. For each additional 9 in the denominator, we get an additional 0 in the repeating decimal: 1/99 = 0.010101…, 1/999 = 0.001001001… and so on. To find the fractional form of a number like 0.123123123…, we can remove the common 123 and rewrite the decimal as 123 x 0.001001001… = 123 x 1/999 = 123/999. Another example: 0.145214521452… = 1452 x 0.000100010001… = 1452 x 1/9999 = 1452/9999 = 44/303.

## Case 3 – Eventually Repeating Decimals

Not all repeating fractions repeat immediately. Consider 1/6 = 0.16666… To find the fractional forms of decimals like these, we break it up a bit more. First, we can observe that the decimal 0.16666… = 0.1 + 0.06666… To get the extra 0 at the beginning of the 0.06666… we need to tweak the 6/9 form that 0.6666… yields. To do this, we just add a zero to the denominator: 0.06666… = 6/90. Therefore, 0.16666… = 0.1 + 0.06666… = 1/10 + 6/90 = 15/90 = 1/6.

Three more examples:

- 0.12454545… = 0.12 + 0.00454545… = 12/100 + 45/9900 = 1233/9900 = 157/1100.
- 0.3339339339339… = 0.3 + 0.0339339339… = 3/10 + 339/9990 = 3336/9990 = 556/1665
- 0.4523678678678… = 0.4523 + 0.0000678678678… = 4523/10000 + 678/9990000 = 4519155/9990000 = 301277/666000

These are easy enough to check on your calculator. Make up a decimal, figure out the fraction, and type in the fraction. Keep in mind that your calculator doesn’t go on forever, and so at some point the calculator will round off your decimal.