- #1
n_kelthuzad
- 26
- 0
Gelfond–Schneider theorem can be seen here(http://en.wikipedia.org/wiki/Gelfond's_theorem) wiki.
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So what happens when b is a transcedental number?
since b is transcedental;
it no longer can be described as a fraction.
so the above 3 rules doesn't apply.
So I pick one of the most common transcedental numbers for b : b = e
and (-1) for a.
so the equation is: (-1)^e
it is known that e=1+1/1!+1/2!=1/3!+...
so it becomes: (-1)^1*(-1)^1*(-1)^(1/2)*(-1)^(1/6)...
and for all exponents starting from the 3rd one: the above (III) rule applys.
so the answer to the equation is : -1*-1*i*i*i...
=i^∞
(so this is what I was working up to now)
so there can be 2 conjectures:
1. (-1)^e = 1 or -1 or i or -i
2. the equation is undefined.
but either way, we will just leave it alone for a moment.
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If the Gelfond–Schneider theorem is true:
then (-e)^e must be transcendental.
and (-e)^e is equal to [(-1)^e]*[e^e]
still if the theorem is true; e^e must be transcendental.
then assume 1 from the above conjecture is true:
there are 2 complex answers of (-e)^e
or assume 2 is true:
(-e)^e is undefined
so clearly either way, it is a paradox.
By Victor Lu, 16
Burnside High School, Christchurch, NZ
496200691@qq.com
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Given a simple calculation:
a^b
where a<0;
and let b be a fraction : u/v
so there are 3 possible ways of u and v s' arrangement:
I. u even, v odd
II. u odd, v odd
III. u odd, v even.
in I, a^b>0
in II, a^b<0
in III, a^b is a complex number.
then the part of my study kicks in:a^b
where a<0;
and let b be a fraction : u/v
so there are 3 possible ways of u and v s' arrangement:
I. u even, v odd
II. u odd, v odd
III. u odd, v even.
in I, a^b>0
in II, a^b<0
in III, a^b is a complex number.
So what happens when b is a transcedental number?
since b is transcedental;
it no longer can be described as a fraction.
so the above 3 rules doesn't apply.
So I pick one of the most common transcedental numbers for b : b = e
and (-1) for a.
so the equation is: (-1)^e
it is known that e=1+1/1!+1/2!=1/3!+...
so it becomes: (-1)^1*(-1)^1*(-1)^(1/2)*(-1)^(1/6)...
and for all exponents starting from the 3rd one: the above (III) rule applys.
so the answer to the equation is : -1*-1*i*i*i...
=i^∞
(so this is what I was working up to now)
so there can be 2 conjectures:
1. (-1)^e = 1 or -1 or i or -i
2. the equation is undefined.
but either way, we will just leave it alone for a moment.
------------------------------------------------------------------------
If the Gelfond–Schneider theorem is true:
then (-e)^e must be transcendental.
and (-e)^e is equal to [(-1)^e]*[e^e]
still if the theorem is true; e^e must be transcendental.
then assume 1 from the above conjecture is true:
there are 2 complex answers of (-e)^e
or assume 2 is true:
(-e)^e is undefined
so clearly either way, it is a paradox.
By Victor Lu, 16
Burnside High School, Christchurch, NZ
496200691@qq.com