Are All Repeating Decimals Fractions? Or What Is One? (Baby, Don't Round Me)

In math, we often categorize numbers. All numbers fall under the term complex numbers, numbers which have both real and imaginary components (though at least one of which may be zero). The real numbers can be further divided into the irrational numbers and the rational numbers. The terms rational and irrational do not refer to how well the numbers can reason; rather, they come from the word ratio. Irrational numbers cannot be written as a ratio, while rational number can. In other words, rational numbers are numbers that can be rewritten as fractions.

What makes a number rational? To determine whether or not a number is rational, all we have to consider is the number after the decimal. If the decimal terminates (or repeats), then the number is rational. Other wise, it is irrational.

Which decimals terminate? One example is 4. Since there are no numbers after the decimal (other than zero), we can say it terminates. The number 1.2345 also terminates, but the number 1.243531723... does not terminate. Neither does 3.14159... or 2.71828... But what about a number like 3.333333.... or 0.123123123... or 2.45321321... or 7.9999....? They don't terminate, but they do repeat.

I could easily say 4/1 is the fraction of 4, or that 1 and 2345/10000 is fraction of 1.2345. Those repeating decimals are not so easy. We've already said that repeating decimals indicate that the number is rational, so there must be a way to convert a repeating decimal into a fraction. But how?

When I was in 7th grade, I had a two-page "cheat sheet" of mathematical formulas to memorize for my scholars' bowl team. Two of the formulas on there related to turning a repeating decimal into a fraction. When I was in 10th grade, though, my Algebra II/Trig and Math Team teacher, Suzanne Ingram, taught me a method for finding these fractions. The best part is that the method requires only Algebra I level skills.

Since most of you reading know that 0.333... is 1/3, we'll start with 3.33... so there's some familiarity to build upon. We start by setting our decimal, 3.33..., equal to x.

Now we want to multiply x by 10 enough times so that the number after the decimal doesn't change (but at least once). Here, we can multiply by 10 one time and the number after the decimal is the same.

Now we have two equations. We want to subtract the smaller from the larger.

This leaves us with 9x on the left and 30 on the right.

Since we have an equation, we just divide both sides by 9. This gives us x on the left and 30/9 on the right, which reduces to 10/3. If we work that out, we find that 10/3 does equal 3.333...

Let's try 0.123123... for x. How many 10s will it take? Since we have three numbers before the repeat, we will have to multiply by 10 three times, or 1,000.

Again, we subtract the smaller from the larger to get 999x on the left and 123 on the right.

Dividing by 999 gives us 123/999, which reduces to 41/333. If we were to work that out, we would find that it does equal 0.123123123...

In the next case, the decimal doesn't repeat right away. We have 2.45321321... Since this problem is different, we should treat it slightly different. The first step is to multiply by 10 enough times so that all of the decimal repeats. In other words, we need to get rid of that 45. To do so, we will multiply by 100 (10 two times). Now, we have to multiply by 10 enough times so that the number after the decimal doesn't change like we did before. Since there are three numbers, 3, 2 and 1, we will have to multiply by 1000 (10 three times). But we're not multiplying the original x; we're multiplying the new 100x to get 100000x.

We have to take the 100000x and subtract 100x. This will give us 99900x on the left and 245076 on the right.

Dividing both sides by 99900 gives us 245076/99900, which reduces to 20423/8325. If we work it out, we find that does equal 2.45321321...

What about 7.999...? This one looks simple like the first case, and it is. However, it is much more interesting. Again, we only have to multiply by 10 one time to maintain our repeating decimal.

Subtracting the two yields 9x on the left and 72 on the right.

Dividing both sides gives us 72/9, which reduces to 8.

Hold on, now. We just showed that 7.999.... = 8. If we subtract 7 from both sides, we are effectively saying: 0.9999.... = 1. Could that be true?

Let's walk through the divisions by nine to convince ourselves.

1/9 = 0.111....

2/9 = 0.222...

3/9 = 0.333... = 1/3

4/9 = 0.444...

5/9 = 0.555...

6/9 = 0.666... = 2/3

7/9 = 0.777...

8/9 = 0.888...

Even though we know that 9/9 is equal to 1, the pattern implies that 9/9 = 0.999... The existence of the pattern does not prove that 0.999... = 1. It does, however, provide us some insight as to whether or not the solution we came up with makes sense.

Some people prefer formulas; I'm okay with them, but I'd much rather see a process than memorize a formula.

Anyway, I hope you got something out of this. If not, well, there's always next time.

If you have any questions, or if anything isn't clear, just let me know in the comments.

I first saw this method from by Algebra II/Trig, Precalculus and Math Team teacher Suzanne Ingram. I first saw the 0.999... = 1 problem online my senior year of high school and asked my Calculus teacher Jan Haynes about it. 

The images here were produced by me using Noteshelf for iPad.