- #1
caters
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Is it true that in some cases an irrational number raised to a rational power is rational? If so what kinds of irrational numbers?
Irrational^rational = rational
- Context: Undergrad
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Discussion Overview
The discussion explores whether an irrational number raised to a rational power can result in a rational number, examining specific cases and types of irrational numbers involved. It includes theoretical considerations and examples.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants propose that certain irrational numbers raised to rational powers can yield rational results, with examples such as √22 = 2 and π^0 = 1.
- It is noted that the irrational number must be algebraic for the claim to hold, and a caveat is added that the exponent should be non-zero.
- Another participant mentions the Gelfond-Schneider theorem, stating that if a and b are algebraic with b irrational, then a^b is transcendental, implying that √2 raised to √2 remains irrational.
- One participant suggests that raising an irrational number to a power that is a multiple of its root results in a rational number, providing the example of √2^4 = 4.
- There is a repeated emphasis that a transcendental number raised to a rational power cannot be rational and must remain transcendental.
- Some participants express confusion or humor regarding the trivial case of raising a number to the power of 0, questioning whether it contradicts earlier claims.
Areas of Agreement / Disagreement
Participants express multiple competing views regarding the conditions under which an irrational number raised to a rational power can be rational. The discussion remains unresolved with respect to the implications of transcendental versus algebraic numbers.
Contextual Notes
Limitations include the dependence on definitions of algebraic and transcendental numbers, as well as unresolved nuances regarding the conditions under which the discussed properties hold.
- #2
jgens
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Yes. For your first question note that √22 = 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
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It's also worth pointing out that irrational to the irrational can be rational.
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.
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
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A very simple example is ##\pi^0 = 1##.caters said:
##\pi## is irrational, and 0 is rational.
- #5
rama
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if the order of the root is say x, and the power you raise are multiples of x , then it becomes a rational number(or if the power is 0)
example:√2^4=4
example:√2^4=4
- #6
HallsofIvy
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Note that the irrational number has to be algebraic. A transcendental number raised to a rational power cannot be rational. In fact, it must still be transcendental.
- #7
DrClaude
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HallsofIvy said:
Mark44 said:
Isn't there a contradiction here?
- #8
arildno
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DrClaude said:
I think you just became a candidate for the Fields medal!
- #9
DrClaude
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No , I just became the candidate for another coffee!arildno said:I think you just became a candidate for the Fields medal!
Sorry Mark44, completely missed your humor there...
- #10
Mark44
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That was arildno...DrClaude said:
- #11
DrClaude
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I thought you were trying to be funny by proposing the trivial case "to the power of 0".Mark44 said:
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