Proving Irrational NumbersDate: 05/10/2000 at 22:48:43 From: Robert Dawson Subject: Irrational numbers I know that there are only three types of decimals: 1) Terminating 2) Repeating 3) Non-terminating and non-repeating I know and can prove that terminating and repeating decimals are always rational. I know that non-terminating and non-repeating decimals are irrational. But I don't know how to prove that this is the case. How do you prove it? Sincerely, Rob Dawson Date: 05/11/2000 at 02:27:25 From: Doctor Floor Subject: Re: Irrational numbers Hi, Rob, Thanks for writing. To prove that case 3 always yields irrational numbers we use indirect reasoning: We can show that rational numbers - fractions - are always in category (1) or (2). This can be done by converting the fraction into a decimal by long division. After each step you get a remainder. After a couple of initial steps, you have to attach a zero to the remainder to continue on. Since the remainder is always smaller than the denominator of the fraction, there is only a limited number of numbers you can have to work with. The remainder might be 0 at some point; then you have case (1). If 0 does not appear, then at some point you will get a remainder that you have had before, and repetition is born, leading us to case (2). If you need more help, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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