Repeating Decimals - Rational or Irrational?Date: 09/11/2001 at 19:26:09 From: Mirko l. Subject: Rational or irrational numbers? In my 7th grade class we are working on rational and irrational numbers. My teacher wants an opinion about whether these numbers are rational or irrational: 0.252252225 0.125126127 Date: 09/11/2001 at 21:57:22 From: Doctor Peterson Subject: Re: Rational or irrational numbers? Hi, Mirko. I assume you mean decimals that continue these patterns forever, not the terminating decimals you show. In each case, althought there is a pattern, it is not the special kind of pattern that rational numbers have, namely an exact repetition of a given group of digits forever. It is only in that case that the decimal can be converted to a fraction, which is the true definition of a rational number. So these numbers are irrational; the first has a "repetition" of a changing number of digits, and the second has changing numbers within the "repeating" pattern. There is a way you could interpret the second number that would make it rational. Since both numbers are defined only by example, and without clearly stating the rule, we can't be sure. Here's how to make it rational: 0.125 126 127 ... 998 999 000 001 002 ... 124 125 ... \_________________________________________/ If we think of the three-digit numbers as wrapping around within themselves, so that 999 is followed by 000, rather than somehow carrying over into the previous section, then the indicated section is in fact a repetend, and this is rational. The irrational version I had in mind is something like this: 0.125 + 0.000 126 + 0.000 000 127 ... + 0.000 000 000 ... 997 998 999 + 0.000 000 000 ... 000 000 001 000 + 0.000 000 000 ... 000 000 000 001 001 ... --------------------------------------- 0.125 126 127 ... 997 999 000 001 002 This way there is no true repetition. But even that is hard to be sure of. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 09/12/2001 at 10:03:27 From: Doctor Peterson Subject: Re: Rational or irrational numbers? Hi again, Mirko! I thought some more about your problem, and realized it is even more subtle than I realized. The "irrational" interpretation I suggested for the second number turns out also to be rational, using some special tricks. I suggested taking the number as 0.125 + 0.000 126 + 0.000 000 127 ... + 0.000 000 000 ... 997 998 999 + 0.000 000 000 ... 000 000 001 000 + 0.000 000 000 ... 000 000 000 001 001 ... --------------------------------------- 0.125 126 127 ... 997 999 000 001 002 But this is the same as 0.125 125 125 ... 125 125 ... + 0.000 001 002 ... 872 873 ... which in turn can be broken up as 0.125 125 125 ... 125 125 ... = 125/999 + 0.000 001 001 ... 001 001 ... + 1/999 * 1/1000 + 0.000 000 001 ... 001 001 ... + 1/999 * 1/1000^2 ... + 0.000 000 000 ... 001 001 ... + 1/999 * 1/1000^872 + 0.000 000 000 ... 000 001 ... + 1/999 * 1/1000^873 ... + ... This gives 125/999 + 1/999 * 1/999, or 124876/998001. That's rational. The lesson here is that, though 0.125126127... can not be shown to be rational BASED ON THE OBVIOUS THREE-DIGIT PERIOD, that doesn't mean it might not be a rational number with a longer period. It's very hard to show that a given number is irrational! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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