Converting Repeating Decimals into FractionsDate: 10/22/2003 at 00:31:18 From: mike Subject: rational numbers I am looking for a rational number with a repeating/non-terminating decimal expansion where the repeating pattern is 12 digits long. How can I find such a number? Date: 10/22/2003 at 02:01:08 From: Doctor Luis Subject: Re: rational numbers Hi Mike, I'll teach you a simple method to convert repeating decimals into a rational number form. Let's say I want the five digit pattern 12345 to repeat itself over and over. _____ x = 0.12345 or 0.12345123451234512345.... I can exploit the fact that the decimal repeats itself by multiplying this decimal by 10^n, where n is the length of the pattern. In our case, we multiply by 10^5 = 100,000. That will move the decimal point 5 places to the right: _____ _____ x = 0.12345 = 0.1234512345 _____ 100000 * x = 12345.12345 Note that the decimal part is the same as in the original number. Hmm..so what would happen if we subtracted these two equations? _____ 100000 * x = 12345.12345 _____ - 1 * x = 0.12345 ========================== 99999 * x = 12345 All the repeating decimals cancel when we subtract, and we get an integer on the right side of the equation! On the left side, 100,000x - 1x = 99,999x. Now we divide both sides by 99999 and we find that x, our original number, is equal to 12345/99999 or 4115/33333 (expressed as a reduced fraction). Thus, we know that: _____ 12345 4115 0.1234512345 = ----- = ----- 99999 33333 You can check this result by dividing either fraction on your calculator to see if you get the repeating decimal. For numbers like y = 6.0121212..., where there's a finite pattern before the repeating one, you have to be a bit more careful. You need to bring out the repeating decimal pattern like this y = 6 + 0.01212... = 6 + (1/10)*(0.121212...) Now you can work with x = 0.1212... and apply the method you just learned to express it as a fraction (x = 12/99 = 4/33). Then substitute it back into the expression for y: y = 6 + (1/10)*(4/33) = 6 + 4/330 = 6 + 2/165 = 992/165 You should now be able to apply this method help you find a rational number with a 12 digit long repeating pattern. I hope this helped! Let us know if you have any more questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/ |
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