How is it that you can turn a recurring decimal into a fixed fraction if the decimal is infinite?

In: 2

Infinite decimals are an artifact of the base of your numbering system. 1/3 is still 1/3, but you can’t express 1/3 as a clean decimal in a base 10 system (what we use) – in a base 12 system, it is perfectly expressable.

Conversely, you can’t express 1/5 as a clean decimal in base 12, but you can do it just fine in base 10 (0.2).

The way that I think makes the most sense is to approach it like this:

Let’s say we want to find a fraction that’s equivalent to 0.123123123…. Let’s call the mystery value x. So we have x = 0.123123….

We then multiply both sides by 1000. This gives us 1000x = 123.123123….

Note that the right-hand side is equivalent to 123 + x: we’ve just shifted everything over one cycle. So now we have that 1000x = 123 + x. We can subtract x from both sides to get 999x = 123, and then we can divide both sides by 999 to get x = 123/999.

Here’s a little experiment for you to try on your calculator:

As you probably already know, if you punch `72 / 100` you get a decimal of `0.72`. But nothing is repeating…. Now what happens if you put in `72 / 99` instead? The answer: you get `0.7272727272…` recurring. Similarly `7 / 9` is `0.77777`, and `123 / 9999` would be `0.012301230123…` so now we have a way to build a repeating decimal pattern from a fraction, and a very obvious way to reverse the process.

Basically the number of 9’s tells you how many digits are in the recurrence and the first number is the pattern.

Now using that and other basic fraction math, you can turn anything from a decimal number into a fraction. Let’s actually try one:

Example: `0.0123456456456…` has a 3-digit combo of “456” repeating. So that’s `456 / 999`. However we need to shift it right by 4 digits as well so we divide by 10000, so `456 / 9990000`. Finally we need to add the “0123” at the beginning which is just `123 / 10000`. So add those two fractions together and you’ll get what you want.

The rest of the math is up to you, but I hope this convinces you that it can be done and is actually easy.

Fractions and decimals are just notation systems, they are just ways of writing things.

Fractions represent the number exactly, 1/3, and for decimal notation you can represent the number exactly with [a notation](https://en.wikipedia.org/wiki/Repeating_decimal#Notation) such as 0.(3) (where the parentheses indicate that the 3 is repeating), or you can indicate the “degree of precision” that you’re using, for example 0.333 meters has 3 digits of precision and refers to 333 millimeters.

It depends on what you mean by “infinite”. For example, 1/9 = 0.11111111111… for an infinite, unending series of 1s. Conversely, for an infinite pattern, 1/11 = 0.0909090909090909…

That’s just how the math works out.