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 Post subject: Exponential function
PostPosted: Sun, 27 May 2012 02:48:38 UTC 
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Let s be a positive irrational number and r be a positive rational number. Does {e^s} = r hold only if s = \ln (r)? Are there any other values of s that make {e^s} rational?


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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 04:19:17 UTC 
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glebovg wrote:
Let s be a positive irrational number and r be a positive rational number. Does {e^s} = r hold only if s = \ln (r)? Are there any other values of s that make {e^s} rational?


That would be the definition of an inverse function.

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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 04:25:58 UTC 
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Are there any cases when {e^s} is rational? The Lindemann-Weierstrass theorem states that if \alpha is algebraic, then {e^\alpha } is transcendental, but when is {e^s} rational?


Last edited by glebovg on Sun, 27 May 2012 04:27:32 UTC, edited 2 times in total.

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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 04:26:54 UTC 
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glebovg wrote:
Are there any cases when {e^s} is rational? The Lindemann-Weierstrass theorem states that if \alpha is algebraic, then {e^\alpha } is transcendental, but when is {e^s} rational?


\alpha=0?

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 04:28:38 UTC 
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But 0 is rational. Look at my original post. I defined s to be a positive irrational number.


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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 04:31:38 UTC 
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glebovg wrote:
But 0 is rational. Look at my original post. I defined s to be a positive irrational number.


No he meant that you are wrong, 0 is algebraic and e^0 is NOT transcendental, if you want to keep your original context you should say so.

In any case again, all you need is the definition of an inverse function.

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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 04:32:29 UTC 
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Hence the use of different letters. \alpha is supposed to be a nonzero algebraic number. Sorry.


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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 04:35:36 UTC 
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So, there are no other rational numbers of the form {e^s}? Is there a theorem?


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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 04:51:04 UTC 
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glebovg wrote:
So, there are no other rational numbers of the form {e^s}? Is there a theorem?


I would not call that result "theorem", it's just immediate from the definition of an inverse function, or of a 1-1 function, or of a lot of other things.

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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 05:30:16 UTC 
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So, is {e^e} irrational?


Last edited by glebovg on Sun, 27 May 2012 05:52:10 UTC, edited 1 time in total.

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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 05:31:08 UTC 
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glebovg wrote:
So, is {e^e} is irrational?


unless e is the log of a rational number.

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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 06:00:46 UTC 
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What do you mean? Does this also work for all {n^{{{\log }_n}r}}?


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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 06:03:45 UTC 
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glebovg wrote:
What do you mean? Does this also work for all {n^{{{\log }_n}r}}?


Assuming you're dealing with positive, real n not equal to 1.

I mean, this is really basic stuff from algebra II in high school, look at the definition of the log function. If you still cannot convince yourself use the derivative to show the exponential (respectively log) are 1-1 on their domains, there are tons of ways to see this.

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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 07:00:05 UTC 
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But why is it unknown whether {2^e} is irrational?


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 Post subject: Re: Exponential function
PostPosted: Sun, 27 May 2012 07:01:12 UTC 
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glebovg wrote:
But why is it unknown whether {2^e} is irrational?


Because that's equivalent to knowing if \log_2(e) is rational...a very difficult thing to decide in general. Understanding which numbers we can easily express are or are not algebraic is a VERY difficult prospect in general.

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