How can you prove that the square root of two is irrational?
Mar 14, 2016 · The irrationality of the square root of 2 follows from our knowledge of how Pythagorean triples behave, specifically, that for positive integers x, y, and z, if x^2 + y^2 = z^2, then x is not equal …
radicals - What rational numbers have rational square roots ...
Trivially, a rational number has a rational square root if and only if it's the square of some rational number. As the other answers note, various other characterizations can be given, e.g. that $\frac ab$ …
rationality testing - Is $\sqrt {2}^ {\sqrt {2}}$ rational ...
Jun 28, 2016 · Closed 9 years ago. In §2.2 of her essay on mathematical morality, Eugenia Cheng includes the following example: Why is it possible for an irrational to the power of an irrational to be …
Square root of $2$ is irrational - Mathematics Stack Exchange
Aug 4, 2017 · I am studying the proof that $\\sqrt 2$ is an irrational number. Now I understand most of the proof, but I lack an understanding of the main idea which is: We assume $\\frac{m^2}{n^2} = 2$. …
Constructive proof of the irrationality of $\sqrt {2}^ {\sqrt {2}}$
Theorem: If $\alpha\neq 0,1$ is algebraic, $\beta$ is positive and rational, not a perfect square, then $\alpha^ {\sqrt {\beta}}$ is transcendental. Unfortunately the paper is in Russian and I failed to find …
Proof that $\\sqrt{2}$ is irrational - Mathematics Stack Exchange
If the square root of $2$ is rational, then by definition it can be written as a ratio of two whole numbers. But then, as long as both numbers are even, you can divide both sides by $2$, and get two new …
Proving that $\sqrt {2}$ is not rational. - Mathematics Stack Exchange
Mar 31, 2021 · The new proof is in the texts below. I am trying to prove that $\sqrt {2}$ is not a rational number on my own using proof by contradiction. However, I am not sure whether my proof is correct …
elementary number theory - Understanding the proof of "$\sqrt {2}$ is ...
Then they say that because of this contradiction $\sqrt {2}$ cannot be a rational number. What I do not understand is that how the contradiction proves that $\sqrt {2}$ cannot be a rational number. The …
If the square root of two is irrational, why can it be created by ...
Oct 25, 2016 · The square root of two cannot exactly be written out on a computer screen in decimal notation – the digits extend forever. You're looking at an approximation accurate to a lot of digits. The …
. Prove that the square root of 2 is an irrational number with
Jun 18, 2025 · Find an answer to your question . Prove that the square root of 2 is an irrational number with proper steps