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MathGroup Archive 2005

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Re: Re: Types in Mathematica thread


On 8 Dec 2005, at 17:24, Andrzej Kozlowski wrote:

>>>
>>> Exp[I Pi]==-1.
>>
>> Yes.
>>
>>> In that case would you say that you can also "derive" I form Pi and
>>> E? What do you mean by deriving a number from another number?
>>
>> I meant to say that Pi can be defined in terms of E.  I am  
>> assuming the
>> definition of complex numbers as a prerequisite.  It's something  
>> that's
>> been in the back of my mind for quite some time.
>
>
> But there is no reason at all to think that this would  help in  
> deducing that Pi is transcendental form the fact that E is. The  
> fact that neither Lindemann's nor Hilbert could do this using this  
> ancient formula of Euler, which they certainly new, would make most  
> people hesitate in claiming that it should be done once you have  
> got complex numbers "as a prerequisite".
>
> But if you really have an idea how to deduce that Pi is  
> transcendental from the fact that E is then perhaps you might wish  
> to prove that E+Pi is transcendental because somehow nobody has so  
> far been able to do it.
>
> Andrzej Kozlowski


Although, I have to admit, that if you make use of the Lindemann- 
Weierstrass theorem (which was proved by Weierstrass much later than  
Lindemann's proof of the transcendentality of Pi) than indeed  
transcendentality of Pi does follow from the transcendentality of E  
and the relationship Exp[I Pi]= -1. But the Liendemann-Weierstrass  
theorem is far from easy to prove; and certainly requires more  
prerequisites than "complex numbers", see for example Alan Baker,  
"Transcendental Number Theory" (which I just have been looking at).

Andrzej Kozlowski


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