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Lesson 91: Repeating and Non-Repeating Decimals
This lesson introduces you to what are called repeating decimals--decimals that continue forever, following a pattern. You'll also learn about non-repeating ones.
Important: Study this lesson simultaneously with Lesson 89.
Recall from pervious lessons that to convert a fraction to a decimal, just compute numerator divided by denominator. The fraction line is really a dividing line. Let's convert 1/3 (one-third) to a decimal. We'll do 1 ÷ 3. We can do the division manually like we learned, or on a calculator.
What you'll see is that the answer is 0.3333, and the threes go on forever. We call this a repeating decimal, because the pattern of threes continues forever. Even if your calculator stops displaying the threes because it runs out of room, they still continue forever. What we do is write 0.3 with a line over the 3, to show that the threes continue. This is not the same as 0.3 without a line. 0.3 without a line is actually 3/10, which is not equivalent to 1/3.
Let's try 2/3. If you compute this on a calculator, it might show something like 0.66666667. It's rounding up the last digit, but the truth is, the sixes continue forever. This is a repeating decimal, and we would write it as 0.6 with a line over the 6 to show that the sixes continue.
Sometimes you'll see repeating decimals with a different pattern. For example, if you convert 1/7 to a decimal, you will see that the answer is 0.142857142857, with that pattern continuing forever. We would just write 0.142857 with a line over the entire decimal portion, to show that the pattern continues on and on. This is very different than if the decimal just stopped after the 7.
Look back at Lesson 89. Repeating decimals are all rational numbers. That means they were calculated from a fraction, which is a ratio. Terminating decimals like 0.5 are also rational numbers, and their denominator will always be a power of 10.
You will also encounter non-repeating decimals. These are decimal numbers that go on forever, but do not follow a pattern at all. An example might be 0.48376922026985321.... If there is no pattern, and no end in sight, it means that the number is a non-repeating decimal.
Be careful, a decimal may follow a pattern such as 0.14114111411114... Even though this is a pattern of sorts, it never actually repeats, so it's considered a non-repeating decimal.
Non-repeating decimals are all irrational numbers. That means that they are not equivalent to any fraction (i.e., ratio). If they were, they would have a repeating pattern, and they would be considered rational.
Sometimes we are asked to convert a repeating decimal into a fraction. In most cases, there is no obvious way to do this, since a repeating decimal never ends. For example, how can we convert 0.77777.... into a fraction? We know that it will be somewhere between 7/10 and 8/10, and since it's bigger than 0.75, we know that it will be bigger than 3/4. Still, these are not precise answers.
What we must do is a type of "trick" which will stop the decimal from repeating. Then it will be easy to convert. What we do is call the repeating decimal N. N=0.777777..... We then multiply N by 10, and see that 10N = 7.77777. The reason why we do this is because we can then subtract 1N from 10N. That gives us 9N = 7. Notice how by doing this, the repeating decimal portion cancelled out, leaving us with just an integer. Now it is easy to see that N = 7/9.
Let's try converting 0.123123123..... into a fraction. We can call that N, but now we'll have to work with 1000N, which is 123.123123123.... The reason for this is that we have to move the decimal point to the right of the digits that repeat. Now we can do 1000N minus N to give us 999N = 123. N = 123/999, which reduces to 41/333.
Let's try one more: Convert 0.833333.... into a fraction. Here we'll call the number N, but we'll use 10N to help us eliminate the repeating decimal portion. We want the decimal point to move past the 8, so that everything to the right of it repeats. We have 10N=8.33333.... What we now have to do find 100N, which is 83.33333.... Now we have two numbers with identical repeating decimals. We can do 100N - 10N, and the repeating decimal drops off. We have 90N = 75. N = 75/90, which reduces to 5/6.
You'll learn more about this topic in later lessons. For now, memorize the terms presented, and make sure that you understand the concepts.