First 100 Irrational Numbers

Date: 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/