Numbers with repeating decimal digits are whole number ratios

The so-called rational numbers; aka the number set ℚ, are numbers that can be written as the ratio (fraction, quotient) of two whole numbers. For instance 2.5 = 5 / 2, 3.1 = 31 / 10, etc.

Sometimes, however, those numbers have endlessly repeating groups of decimals; for example:

• 10 / 3 = 3.3  ^*
• 5118 / 999 = 5.123
• 22 / 7 = 3.142857

We can wonder if the reverse is also true; i.e., can all numbers which end in endlessly repeating groups of decimals be written as a ratio of whole numbers? The answer is 'yes' and the proof goes as follows:

We can rewrite a number that has any repeating digits as that same number with just a single occurrence of those repeating digits; thus

• 100.111 = 100.1
• 100.123123123123 = 100.123
• 100.5161616 = 100.516
• etc.

Identically, we can rewrite a number that has only one repeating digit group as that same number with an extra occurrence of that repeating digit group; thus
• 100.1 = 100.11
• 100.123 = 100.123123
• 100.516 = 100.51616
• etc.

If we follow these steps in order —rewrite with just one group and then add a second group— we always end up with two numbers that have the identical value. A few examples:

Number	Repeating group	One-group version	Two-group version
99.99999999	9	99.9	99.99
99.1616	16	99.16	99.1616
99.123123123	123	99.123	99.123123
99.56123123123	123	99.56123	99.56123123

Next, we multiply the two-group version by 10ⁿ where n is the number of digits in the repeating group. For example: if 99.123123123 is our number, 123 is our repeating group and 99.123123 is our two-group version of the number. Since the repeated group has three digits, we multiply by 10³:
1000 × 99.123123 = 99123.123

Subtract the one-group version of the number:
99123.123 - 99.123 = 99024

Therefore:
999 × 99.123 = 99024, and thus

99.123 = 99024 / 999 = 33008 / 333

And hence 99.123 can be written as a ratio (fraction) of two whole numbers.

We can of course apply the same technique with any other number with a repeating group of decimals.

You can play with this below. Enter a number with one or more groups of repeating decimals and hit the Do it! button.

number (must contain a decimal part):     Do it!

^*The 'overbarred' digits indicate endless repetition. So 13.16 means that the 16 pattern endlessly repeats.