Episode 50: February 11, 2011
by Jason Marshall
Over the last several articles we’ve learned that many of the numbers we deal with in our daily lives are what are known as rational numbers. The fact that these numbers are rational means that we can write them either as terminating decimals that stop after some number of digits or as repeating decimals with a pattern of digits that repeats forever. In the last episode we learned how to turn rational numbers that can be written as terminating decimals into fractions. Today, we’re going to continue where we left off and talk about how to turn repeating decimals into fractions.
Recap: How to Convert Terminating Decimals to Fractions
Before we get too far into today’s topic, let’s take a minute to recap what we learned last time. The quick and dirty summary is that terminating decimals are numbers that have decimal representations that eventually stop. For example, the fractions 1/2 and 5/16 have decimal representations of 0.5 and 0.3125—both of which stop after some number of digits. On the other hand, repeating decimals are numbers whose decimal representations don’t stop, but instead repeat some pattern forever. For example, 1/3=0.3333… and 2/7=0.285714285714…. The first repeats after one digit, and the second requires six digits before it starts repeating.
To convert a terminating decimal into a fraction, you just need to remember what decimal notation means. Namely, the first digit to the right of a decimal point is the number of tenths, the next digit to the right is the number of hundredths, the next is the number of thousandths, and so on. With this in mind, you can see that 0.5 just means 5/10 (which is equal to 1/2 after reducing it) and 0.3125 is equal to the fraction 3,125/10,000 (which can be reduced to 5/16).
How to Turn Repeating Decimals Into Fractions
Okay, it’s now time to figure out how to do the same type of conversion with repeating decimals. For example, how do you convert a decimal number like 0.1111… into an equivalent fraction? I’ll start by giving you the quick and dirty tip, and then we’ll talk about why it works. Here’s the tip: Any decimal with a single repeating number that begins right after the decimal point is equal to the fraction that has the repeating digit in its numerator and nine in its denominator.
For example, since the numeral 1 is doing all the repeating in the decimal 0.1111…, this tip tells us that the equivalent fraction must have a numerator of 1 and a denominator of 9. In other words, 0.1111… = 1/9. Go ahead and try dividing 1 by 9 with a calculator and make sure it’s true. How about a number like 0.6666…? Well, since the number 6 repeats over and over, we can immediately conclude that 0.6666… = 6/9—which, after dividing both the numerator and denominator by 3, you’ll see is equivalent to 2/3.
Why Does this Repeating Decimal Tip Work?
Any decimal with a single repeating number is equal to the fraction that has the repeating digit in its numerator and nine in its denominator.
But why does this work? Well, let’s think about the repeating decimal 0.1111…. First, let’s multiply this number by 10 to get the new repeating decimal 1.1111….
Now, let’s subtract the original repeating decimal, 0.1111…, from this new number, like this: 1.1111… – 0.1111….
That just leaves the number 1 since the decimal parts subtract away. But now let’s look at the problem this way: What do you get when you subtract 1 of “something” from 10 of “something”? Well, 10 of “something” minus 1 of “something” is just equal to 9 of “something”.
And that means that so far we’ve figured out that 9 of “something” in this problem has to be equal to 1. But if 9 of “something” is equal to 1, then that “something” must just be equal to 1/9. Which means that the repeating decimal 0.1111… is equal to 1/9—precisely the answer given to us by our efficient and convenient quick and dirty tip.
You can go through the same series of steps with any other decimal that has a single repeating digit which begins right after the decimal point. For example, let’s look at 0.4444…. First multiply it by 10 to get 4.4444…, and then subtract 0.4444… from this result. The answer is the number 4. Now, as before, we can look at this in another way too: Subtracting 1 of “something” from 10 of “something” leaves you with 9 of “something”. So 9 of “something” is equal to 4 in this problem, which means that “something” must equal 4/9…exactly as we find for the repeating decimal 0.4444… using our quick and dirty tip.
But does this tip only work for decimals with a single repeating number? What about a decimal number like 0.8181… that has two numbers which repeat over and over again? How do you turn that into a fraction? Well, unfortunately, we’re out of time for today. Which means that we’ll tackle these more complex repeating decimal conversions next time.
But before we finish, here are some practice problems for you to try to help you make sure you’re up to speed with converting simpler repeating decimals like the ones we talked about today:
0.2222… = ______
0.3333… = ______
0.8888… = ______
You can find the answers to these questions at the very end of the article. After checking your answers, feel free to leave a comment at the bottom of the page and let me know how you did.
If you have questions about how to solve these practice problems, or any other math questions you might have, please email them to me at firstname.lastname@example.org, send them via Twitter, or become a fan of the Math Dude on Facebook and get help from me and the other math fans there.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading math fans!
Practice Problem Answers
0.2222… is equal to the fraction with 2 in its numerator (since that’s the single number after the decimal point that’s repeating over and over again) and 9 in its denominator. In other words, 0.2222… = 2/9.
Using the logic from the last problem, 0.3333… = 3/9. We can reduce this fraction (a process that we’ll talk more about in a future article) by noticing that we can divide both the numerator and denominator by 3 to get 0.3333… = 3/9 = 1/3.
Similar to the first problem, 0.8888… = 8/9.