Learn what terminating and repeating decimals are and how to convert these rational numbers from decimal to fractional form.
As we’ve discovered in the last two articles, the numbers we use in our daily lives can be broken up into two main groups: rational and irrational numbers. We’ve found that irrational numbers cannot be written out exactly in decimal form since you’d need an infinite number of decimal digits to do so. But we’ve found that rational numbers can be written as decimal numbers that either stop after some number of digits or keep repeating some pattern of digits forever. In today’s article we’re going to learn how to take a decimal representation of a rational number and turn it into an equivalent fraction.
What are Terminating and Repeating Decimals?
Before we get into the details of how to actually convert terminating and repeating decimals into fractions, we’d better make sure we understand what it means for a rational number to be a “terminating” or “repeating” decimal in the first place. To see what the difference is, let’s take a look at a few examples of decimal representations of rational numbers:
1/4 = 0.25 is a terminating decimal since it has a finite number of decimal digits
1/3 = 0.3333… is a repeating decimal since the number 3 goes on forever
3/5 = 0.6 is another terminating decimal number
7/9 = 0.7777… is a repeating decimal since 7 goes on forever
9/11 = 0.818181… is another repeating decimal since the pattern of digits “81” repeat forever.
So a repeating decimal is a rational number whose decimal representation has some repeating pattern, and a terminating decimal is a rational number whose decimal representation eventually stops. (Remember, a decimal that just goes on and on with no repeating pattern is irrational.)
Can a Terminating Decimal Be Written as a Repeating Decimal?
If you think about it though, you’ll see that any terminating decimal number can actually be written as a repeating decimal too. How? Well, since you can always attach an infinite number of zeros to the very end of a number without changing its value, you can put an infinitely long string of zeros on the end of an otherwise terminating decimal…and you’ll have turned it into a repeating decimal!
For example, you can think of the terminating decimal 0.25 as 0.25000… instead. But in this case none of this really matters since the value of the number is exactly the same no matter how it’s written. And that’s why usually when we say “repeating decimal,” we mean a decimal number where something other than only zeros are doing the repeating!
How to Convert Decimals to Fractions
Now that we know the lingo and can tell the difference between a terminating and repeating decimal, let’s figure out how to convert them into fractions. In other words, in the examples we gave earlier, we said things like “the fraction 1/4 is equal to the terminating decimal 0.25” and “the fraction 7/9 is equal to the repeating decimal 0.7777…,” and so on. But now let’s figure out how to do this problem backwards so that we can take a decimal number, like 0.818181…, and convert it into a fraction with an equivalent value.
How to Convert Single Digit Decimals to Fractions
Let’s start by converting a simple terminating decimal number like 0.5 into a fraction. As we learned back in the article called “What are Decimals?,” a decimal number like 0.5 means “five of the fraction one-tenth.” Which, of course, is just equal to the fraction 5/10. And believe it or not, that’s the answer to the problem! So, the decimal 0.5 is equivalent to the fraction 5/10. Easy, right?
Well, it turns out that we can actually do a bit more with this fraction. We’ll talk about this in a future article, but this fraction can be rewritten so that it’s what’s called “reduced to lowest terms.” Without going into too much detail, the basic idea is that we can divide both the numerator and denominator of the fraction 5/10 by 5 to find that it has an equivalent and simpler representation of 1/2. But don’t worry if that all sounds like a bunch of crazy talk right now—we’ll look at it in more detail soon enough.
How to Convert Terminating Decimals to Fractions
Okay, we’re now ready to move on to a more complex problem. Let’s convert a terminating decimal like 0.875 into a fraction. As we talked about in the earlier article on decimals, the 8 in 0.875 represents 8/10, the 7 represents 7/100, and the 5 represents 5/1000. So the decimal number 0.875 is equal to
8/10 + 7/100 + 5/1000
But instead of worrying about how to actually add up all of these fractions (which is another topic that we’ll talk about in a future article), we can simplify things by first writing 0.875 as
0.875 = 0.800 + 0.070 + 0.005
When we do this, you can see that 0.875 is equal to the sum of 800/1000 + 70/1000 + 5/1000 for a grand total of (800 + 70 + 5)/1000 = 875/1000. And that’s the answer!
As with our earlier problem, it turns out that we can reduce this fraction to lowest terms by dividing its numerator and denominator by 125. Doing so, we find that 0.875 = 875/1000 is equivalent to 7/8. But, again, don’t worry if you don’t understand how reducing to lowest terms works for now…we’ll come back and talk about that in more detail soon.
So that’s how you convert a terminating decimal into an equivalent fraction. How about a repeating decimal number such as 0.333… or 0.818181…? Well, unfortunately, we’re out of time for today. Which means that we’ll tackle repeating decimals next time. But to make sure you’re up to speed with converting terminating decimals, here are a few practice problems for you to try.
1.4 = ______
0.125 = ______
0.800 = ______
You can find the answers at the very end of the article.
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
1.4 = 1 and 4/10. We can reduce this to lowest terms by dividing the numerator and denominator or 4/10 by 2 to get the equivalent fraction 1 and 2/5.
0.125 = 125/1000. We can reduce this to lowest terms by dividing the numerator and denominator by 125 to get the equivalent fraction 1/8.
0.800 = 8/10. Once again, we can reduce this to lowest terms by dividing the top and bottom by 2 to get the equivalent fraction 4/5. The zeros on the end of 0.800 don’t change anything about the problem since they simply tell us that there are zero hundredths and zero thousandths!