I believe that is an unsolved mystery of mathematics... it might even be listed among the great problems of out time. I remember reading about that when I was looking up he millennium prize problems.
It has nothing to do with them being constants.
π and e are both transcendental numbers... meaning that they each are not algebraic numbers.
Transcendental numbers are tautologically defined as any number that isnt an algebraic number... and algebraic numbers are tautologically defined as any number that isnt a transcendental number. They are mutually exclusive sets, but together (unioned sets) are all-encompassing for all numbers.
Algebraic numbers are numbers, rational or irrational, imaginary, complex, or real, that can be written as (or found as) a root of polynomial. Transcendental numbers cannot be found as the root of a polynomial... if its a real transcendental number, it is guaranteed to be irrational.
Anyway... pi and e are both transcendental. But that doesnt mean they are algebraically independent from one another.
Given a single algebraic equation or function, pi may relate as the input while e is the output, or vice versa.
Believe it or not,
e^(i·π) + 1 = 0
But this doesnt conform to the definition of the type of function required to prove algebraic co-dependence.
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I believe that is an unsolved mystery of mathematics... it might even be listed among the great problems of out time. I remember reading about that when I was looking up he millennium prize problems.
It has nothing to do with them being constants.
π and e are both transcendental numbers... meaning that they each are not algebraic numbers.
Transcendental numbers are tautologically defined as any number that isnt an algebraic number... and algebraic numbers are tautologically defined as any number that isnt a transcendental number. They are mutually exclusive sets, but together (unioned sets) are all-encompassing for all numbers.
Algebraic numbers are numbers, rational or irrational, imaginary, complex, or real, that can be written as (or found as) a root of polynomial. Transcendental numbers cannot be found as the root of a polynomial... if its a real transcendental number, it is guaranteed to be irrational.
Anyway... pi and e are both transcendental. But that doesnt mean they are algebraically independent from one another.
Given a single algebraic equation or function, pi may relate as the input while e is the output, or vice versa.
Believe it or not,
e^(i·π) + 1 = 0
But this doesnt conform to the definition of the type of function required to prove algebraic co-dependence.
Yes, they are constants.