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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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