- #1
Hertz
- 180
- 8
I haven't had much practice with proving things, so I'm not sure of exactly what would go into this, but can you prove that x is irrational in the following equation?
x^x = 2
x^x = 2
jedishrfu said:start with the assumption that sqrt(2) is rational hence sqrt(2) = a / b where a and b are integers (definition of rational number) and that there are no common factors between a and b.
from there we get a^2 / b^2 = 2 or a^2 = 2 * b^2 from which we conclude that a is even.
since a is even then we can say a = 2 *c and we plug it into conclude what about b?
jedishrfu said:its the same thing x^2 = 2 --> x=sqrt(2) the question is: is x or the sqrt(2) rational?
to continue with the proof if you plug 2 *c in for a you can conclude that b is even too.
Isn't that a contradiction?
Diffy said:Again he is not asking about the square root of two.
He is asking about x^x = 2
which x = 1.5596104...
Hertz said:I haven't had much practice with proving things, so I'm not sure of exactly what would go into this, but can you prove that x is irrational in the following equation?
x^x = 2
jedishrfu said:got it. I misread x ^ x to be x * x = 2. okay.
So going back the sqrt(2) proof... I think a proof by contradiction would still apply.
assume x = a/b with a and b having no common factors
a^a = 2^b * b^a from which we conclude a is even
using similar logic and some added caveats about a, b and c we should be able to conclude that b is even and that x is then irrational.
Diffy said:Again he is not asking about the square root of two.
He is asking about x^x = 2
which x = 1.5596104...
jedishrfu said:got it. I misread x ^ x to be x * x = 2. okay.
So going back the sqrt(2) proof... I think a proof by contradiction would still apply.
assume x = a/b with a and b having no common factors
a^a = 2^b * b^a from which we conclude a is even
using similar logic and some added caveats about a, b and c we should be able to conclude that b is even and that x is then irrational.
Curious3141 said:Not strictly relevant to the OP's question, but once you prove that x is irrational, you can immediately prove it's transcendental as well.
Assume to the contrary that x is algebraic. By Gelfond-Schneider, x^x has to be transcendental. But x^x = 2, by definition. This is a contradiction, hence x is not algebraic, and therefore transcendental.
Neat.Citan Uzuki said:2^(1/x) is a rational power of 2 and therefore an algebraic integer.
Citan Uzuki said:It's pretty easy, actually. Suppose that x is rational and x^x = 2. Then x = 2^(1/x) is a rational power of 2 and therefore an algebraic integer. But x is rational, so x must actually be an ordinary integer, and it's easy to see that no integer can satisfy that equation.
jedishrfu said:Just to be clear my post was designed to hint at the solution not provide a complete answer due to forum rules.
Diffy said:I cheated. I also use Wolfram Alpha.
Hertz said:I see how x is a rational power of 2, but I don't see why that means that it is an algebraic integer (Considering polynomials are defined as only having integer exponents).
I also don't see why if it is an algebraic integer, that means that it is an ordinary integer.
I can think of many rational powers of two that are rational, but not integers :S
Citan Uzuki said:It is a well-known theorem that every algebraic integer in the rational numbers is an integer. The proof is as follows:
Hertz said:I haven't had much practice with proving things, so I'm not sure of exactly what would go into this, but can you prove that x is irrational in the following equation?
x^x = 2
No, scientific theories cannot be proven definitively. They can only be supported by evidence and repeatedly tested and verified, but they are always subject to change as new evidence is discovered.
Scientists use the scientific method, which involves making observations, forming a hypothesis, designing and conducting experiments, and analyzing the results. If the results consistently support the hypothesis, it is considered to be true.
No, not all scientific experiments are 100% accurate. There is always a margin of error and room for improvement in experimental design. However, scientists strive to minimize error and improve accuracy through rigorous testing and peer review.
Yes, there is a difference between proof and evidence in science. Proof suggests absolute certainty, which is not possible in science. Evidence, on the other hand, is data that supports a hypothesis and can be used to make predictions and draw conclusions.
Yes, scientific findings can be trusted as they are based on rigorous testing and peer review by other scientists. However, they are always subject to change as new evidence is discovered.