Learn what repeating decimal number are and how to convert basic repeating decimals from decimal come fractional form.

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Over the last several short articles we’ve learned that plenty of of the number we resolve in our daily lives room what are recognized as reasonable numbers. The fact that this numbers are rational way that we deserve to write them either as terminating decimals that avoid after some variety of digits or together repeating decimals with a pattern of digits the repeats forever. In the last illustration we learned how to revolve rational numbers that can be written as end decimals into fractions. Today, we’re walking to continue where us left off and also talk about how to turn repeating decimals into fractions.

Recap: just how to convert Terminating decimal to Fractions

Before we acquire too much into today’s topic, let’s take a minute come recap what us learned last time. The quick and dirty an overview is the terminating decimals room numbers that have decimal representations that at some point stop. For example, the fountain 1/2 and also 5/16 have decimal representations of 0.5 and 0.3125—both the which stop after some number of digits. Top top the other hand, repeating decimals room numbers who decimal representations don’t stop, yet instead repeat some pattern forever. Because that example, 1/3=0.3333… and also 2/7=0.285714285714…. The first repeats ~ one digit, and the 2nd requires six digits prior to it beginning repeating.
To convert a end decimal right into a fraction, you just need to remember what decimal notation means. Namely, the an initial digit come the right of a decimal suggest is the variety of tenths, the following digit come the best is the variety of hundredths, the following is the number of thousandths, and also so on. V this in mind, you can see the 0.5 just means 5/10 (which is equal to 1/2 after ~ reducing it) and 0.3125 is same to the fraction 3,125/10,000 (which deserve to be diminished to 5/16).

How to revolve Repeating Decimals into Fractions

Okay, it’s currently time to figure out just how to execute the same kind of conversion with repeating decimals. Because that example, how do you transform a decimal number favor 0.1111… right into an indistinguishable fraction? I’ll begin by offering you the quick and also dirty tip, and then we’ll talk about why that works. Here’s the tip: any type of decimal v a solitary repeating number that starts right after ~ the decimal point is same to the portion that has the repeating digit in the numerator and nine in its denominator.For example, due to the fact that the character 1 is doing every the repeating in the decimal 0.1111…, this reminder tells united state that the equivalent fraction must have a numerator of 1 and also a denominator the 9. In other words, 0.1111… = 1/9. Walk ahead and try dividing 1 through 9 v a calculator and also make certain it’s true. How around a number like 0.6666…? Well, since the number 6 repeats over and also over, us can instantly conclude that 0.6666… = 6/9—which, after dividing both the numerator and denominator by 3, you’ll view is tantamount to 2/3.Why go this Repeating Decimal tip Work?
Any decimal through a single repeating number is equal to the fraction that has the repeating digit in its numerator and also nine in the denominator.
But why go this work? Well, let’s think around the repeating decimal 0.1111…. First, let’s main point this number by 10 to get the brand-new repeating decimal 1.1111….Now, let’s subtract the initial repeating decimal, 0.1111…, indigenous this new number, favor this: 1.1111… – 0.1111….That just leaves the number 1 due to the fact that the decimal components subtract away. But now let’s look at the difficulty this way: What execute you obtain when you subtract 1 that “something” from 10 that “something”? Well, 10 that “something” minus 1 of “something” is simply equal to 9 that “something”.And that method that so much we’ve determined that 9 of “something” in this trouble has to be same to 1. However if 9 of “something” is same to 1, then that “something” must simply be same to 1/9. Which method that the repeating decimal 0.1111… is same to 1/9—precisely the answer provided to us by our efficient and also convenient quick and dirty tip.You have the right to go v the same collection of actions with any type of other decimal that has a single repeating digit which starts right ~ the decimal point. Because that example, let’s look in ~ 0.4444…. An initial multiply it by 10 to acquire 4.4444…, and then subtract 0.4444… from this result. The price is the number 4. Now, together before, we can look in ~ this in another way too: individually 1 of “something” indigenous 10 of “something” pipeline you through 9 that “something”. Therefore 9 that “something” is same to 4 in this problem, which method that “something” should equal 4/9…exactly together we discover for the repeating decimal 0.4444… making use of our quick and dirty tip.

Practice Problems

<But walk this guideline only work for decimals with a single repeating number? What about a decimal number prefer 0.8181… that has actually two numbers which repeat over and also over again? exactly how do you rotate that right into a fraction? Well, unfortunately, we’re the end of time for today. Which method that we’ll tackle these more complex repeating decimal conversions next time.But before we finish, below are some practice problems for girlfriend to shot to aid you make certain you’re up to rate with converting simpler repeating decimals like the ones we talked about today:0.2222… = ______0.3333… = ______0.8888… = ______You can discover the answers come these concerns at the really end the the article. After ~ checking her answers, feel complimentary to leave a comment at the bottom of the page and let me know how you did.

Wrap Up

If you have questions about how to resolve these exercise problems, or any other math questions you could have, please email them to me at mathdude
cg-tower.com, send castle via Twitter, or end up being a pan of the mathematics Dude on Facebook and also get aid from me and also the various other math pan there.Until next time, this is Jason Marshall with The math Dude’s Quick and also Dirty tips to do Math Easier. Thanks for reading math fans!

Practice trouble Answers

0.2222… is same to the fraction with 2 in its molecule (since that’s the single number after ~ the decimal allude that’s repeating over and also over again) and 9 in the denominator. In other words, 0.2222… = 2/9.Using the logic from the last problem, 0.3333… = 3/9. We deserve to reduce this portion (a process that we’ll talk an ext about in a future article) by noticing that we have the right to divide both the numerator and also denominator through 3 to obtain 0.3333… = 3/9 = 1/3.Similar come the an initial problem, 0.8888… = 8/9.

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Jason Marshall is the writer of The math Dude's Quick and Dirty overview to Algebra. He provides clear explanations of math terms and principles, and also his basic tricks because that solving an easy algebra troubles will have even the many math-phobic human looking front to functioning out every little thing math difficulty comes their way.