It is known that constants π and e are irrational numbers but also transcedental.Where consist difference between irrationality and transcedentality. How we know that given irrational number is not tanscedental?

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It is known that constants   π and   e are irrational numbers but also transcedental.Where consist difference between irrationality and transcedentality. How we know that given irrational number is not tanscedental?

budd99055uruey · Accepted Answer

A number   x is irrational if there are no integers       a    0    ,      a    1   such that       a    1    x  +      a    0    =  0. That is, if there is no integer polynomial   P of degree 1 with   P  (  x  )  =  0.A number   x is transcendental if there is no positive integer   n and no integers       a    0    ,  …      a    n   such that       a    n        x    n    +      a          n      −      1            x          n      −      1        +  …  +      a    0    =  0. That is, if there is no integer polynomial   P of any degree   n with   P  (  x  )  =  0.All transcendental numbers are irrational, because we can take   n  =  1. Not all irrational numbers are transcendental. Non-transcendental numbers are called algebraic.       2   is irrational, but not transcendental, because   (      2        )    2    −  2  =  0. (That is,   n  =  2  ,      a    2    =  1  ,      a    1    =  0  ,      a    0    =  −  2.)Nobody knows methods that work in general to show that a particular number is rational, irrational, or transcendental.

It is known that constants π

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