- #1

- 223

- 9

- Thread starter caters
- Start date

- #1

- 223

- 9

- #2

jgens

Gold Member

- 1,581

- 50

Yes. For your first question note that √2^{2} = 2. For your second question all number of this form should be algebraic.

Edit: As Mark44 pointed out below I need to add the caveat that the exponent be non-zero for my second claim to hold.

Edit: As Mark44 pointed out below I need to add the caveat that the exponent be non-zero for my second claim to hold.

Last edited:

- #3

pwsnafu

Science Advisor

- 1,080

- 85

The Gelfond-Schneider theorem says: if ##a## and ##b## are algebraic, with ##a \neq 1,0## and ##b## irrational, then ##a^b## is transcendental.

This means that ##\sqrt{2}^{\sqrt{2}}## is transcendental (hence irrational). Raise this to the power of ##\sqrt{2}## and you get 2.

- #4

Mark44

Mentor

- 34,151

- 5,766

A very simple example is ##\pi^0 = 1##.

##\pi## is irrational, and 0 is rational.

- #5

- 9

- 0

example:√2^4=4

- #6

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 956

- #7

DrClaude

Mentor

- 7,428

- 3,701

algebraic. A transcendental number raised to a rational power cannot be rational. In fact, it must still be transcendental.

Isn't there a contradiction here?A very simple example is ##\pi^0 = 1##.

##\pi## is irrational, and 0 is rational.

- #8

arildno

Science Advisor

Homework Helper

Gold Member

Dearly Missed

- 9,970

- 132

I think you just became a candidate for the Fields medal!Isn't there a contradiction here?

- #9

DrClaude

Mentor

- 7,428

- 3,701

No , I just became the candidate for another coffee!I think you just became a candidate for the Fields medal!

Sorry Mark44, completely missed your humor there...

- #10

Mark44

Mentor

- 34,151

- 5,766

That was arildno...No , I just became the candidate for another coffee!

Sorry Mark44, completely missed your humor there...

- #11

DrClaude

Mentor

- 7,428

- 3,701

I thought you were trying to be funny by proposing the trivial case "to the power of 0".That was arildno...