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-   -   4 times pi squared (http://mymathforum.com/number-theory/332361-4-times-pi-squared.html)

 Adam Ledger June 2nd, 2016 08:40 AM

4 times pi squared

is this number transcendental?

 Country Boy June 2nd, 2016 08:51 AM

Yes, of course. If it were not it would satisfy a polynomial equation, say nth degree, with integer coefficients. But since any integer power of 4 is an integer that would then give a polynomial equation, of degree 2n, with integer coefficients, satisfied by $\displaystyle \pi$,which is impossible because $\displaystyle \pi$ is transcendental.

 Adam Ledger June 2nd, 2016 09:00 AM

1 Attachment(s)
ok sweet so if I then divide it by three, still transcendental?

 Country Boy June 2nd, 2016 04:12 PM

Seriously? Since you simply asked whether a specific number was "transcendental", I assumed you knew what "transcendental number" meant! Are you now saying that you do not?

A number, a, is "transcendental" if and only there exist a polynomial equation with integer coefficients (equivalently "rational coefficients") having a as a root. If a3 were NOT transcendental, then there would exist a Polynomial, of degree n, having a/3 as a root. Then that same polynomial, multiplied by 3 to the nth power, would have x as a root showing that a is not transcendental.

 Country Boy June 4th, 2016 10:56 AM

Quote:
 Originally Posted by Country Boy (Post 538014) Seriously? Since you simply asked whether a specific number was "transcendental", I assumed you knew what "transcendental number" meant! Are you now saying that you do not? A number, a, is "transcendental" if and only there exist a polynomial equation with integer coefficients (equivalently "rational coefficients") having a as a root.
This should have been "if and only if there does not exist...", of course.

Quote:
 If /a3 were NOT transcendental, then there would exist a Polynomial, of degree n, having a/3 as a root. Then that same polynomial, multiplied by 3 to the nth power, would have x as a root showing that a is not transcendental.

 Adam Ledger July 31st, 2016 04:13 AM

yea its ok don't stress I was taking the piss over the gamma(2/3) thing.

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