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In Part A of this session, you learned that the decimal representation for every rational number was either a terminating or a repeating decimal. You also learned how to find the decimal representation for any rational number. Is the converse of that statement true? That is, is every terminating or repeating decimal a rational number? The answer is yes. And any non-terminating, non-repeating decimal cannot be a rational number. So, for instance,  is an irrational number, as is  .
So how do we find the fractional representation of a decimal? Note 2 If the decimal is terminating, it's already a fraction; you just can't see the denominator. For example, 0.25 means 25/100, which reduces to 1/4. However, if the fraction is repeating, the process isn't quite so simple. To find the fractional representation for 0.232323..., for example, here's what you need to do.
First, choose a letter to represent the fraction you are looking for; let's say, F. This fraction, F, represents your repeating decimal; that is, F = 0.232323.... Now we need to think of a way to get rid of those repeating parts. To do this, multiply F by 10n, where n equals the size of the period. In this case, the period is two, so multiply F by 102, or 100. Finally, subtract F. The problem looks like this:
Since 99F = 23, F = 23/99.
This worked out nicely, didn't it? But it does raise some questions:
• | Why can we do this? We can do this because we subtracted equal quantities from both sides of an equation. |
• | How did we know to multiply by 100? The period of this decimal is two, so if we multiply by 102, the repeating part will "move over" two places and the repeating parts then "line up" under each other. In other words, if the period is p, we can multiply by 10p. |
• | What if the decimal doesn't repeat right away? Then we need to modify the process. Let's look at another decimal number, 0.45545454.... We know that F represents the repeating decimal number; that is, F = 0.45545454.... Once again, we need to think of a way to get rid of those repeating parts. To do this, we again find 100 times F (because the repeating part has a period of two) and then subtract F:
So, since 99F = 45.09, F = 45.09/99.
Notice that, unlike in the previous example, the first couple of digits didn't "line up," which resulted in having a terminating decimal number in the numerator. To simplify this fraction that contains a decimal point, multiply both top and bottom by 100, which gives us F = 4,509/9,900 = 501/1,100.
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