First 100 Irrational NumbersDate: 10/26/2002 at 10:49:01 From: Eric Subject: First 100 irrational numbers Could you please tell me the first 100 irrational numbers? If you can't, can you tell me where I can find it? Date: 10/26/2002 at 11:26:00 From: Doctor Ian Subject: Re: First 100 irrational numbers Hi Eric, Pick some starting point, e.g., zero. Now imagine that you've found the 'first' irrational number, i.e., the one closest to the starting point. No matter which number you choose, you'll always be able to find one closer to the starting point - for example, by taking the square root of the the number you chose. Here's another way to think about it: Which of the numbers in the following sequence is the smallest? 2^(1/2), 2^(1/2^2), 2^(1/2^3), 2^(1/2^4), ..., 2^(1/2^n), ... Since there's no largest value of n, there can't be a smallest item in the sequence. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 02/09/2007 at 11:52:57 From: Paul Subject: Irrational Numbers close to Zero Is there an irrational number that is closer to zero than any other irrational? If there isn't, is there a sequence that can be identified to show the irrational numbers approaching zero? I cannot understand how to formulate a sequence of irrational numbers that approaches zero, when it is so easy to find one that does. For example, think of the square root of 7 over 10000. This is pretty close to zero, right? But to get closer, can't we just throw some more zeros onto the end of the denominator? I get lost after this point... Date: 02/09/2007 at 12:21:18 From: Doctor Douglas Subject: Re: Irrational Numbers close to Zero Hi Paul. Your thinking is on the right track. The number sqrt(7)/10000 is certainly irrational [can you prove it, perhaps starting by assuming "sqrt(7) is irrational" as a given?]. And you can get closer to zero with sqrt(7)/100000 [it's easy to prove that this is indeed closer to zero], a number which is also irrational [again, a straightforward proof]. And you can get closer still with sqrt(7)/1000000, sqrt(7)/10000000,..., and so on. So you've formulated a sequence of irrational numbers that clearly approaches zero: sqrt(7)/10^k k = 1,2,3,... The step needed to make your argument complete is to convince yourself that each of these numbers is indeed irrational. Instead of the above sequence, you could have also chosen pi/10^k k = 1,2,3,... or indeed ANY irrational number q for the numerator: q/10^k k = 1,2,3,... [why are these all irrational?] or a similar expression whose denominator increases with k: q/k k = 1,2,3,... q/(k^2 + 1) k = 1,2,3,... q/k! k = 1,2,3,... Do you see how each of these sequences approaches zero as k approaches infinity? And do you see how each number in each sequence is irrational? I hope that my questions have helped to solidify your understanding about this subject. However, if you have more questions about it, please write back. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ |
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