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I know there are plenty of posts on proving that it exists. I know you can, for e.g. by defining $x$ to be the supremum of all rationals $r$ such that $r^2<2$ and proving that $x^2=2$, or by applying the intermediate value theorem.

Timothy Gowers asks us to imagine that if you did not know any advanced mathematics and were confronted by somebody who denied the existence of the square root of two. What would you say?

Xam
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TripleA
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    It is usually not possible to convince these people. See this – Henricus V. Dec 27 '16 at 20:58
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    The ratio of the length of the diagonal of a square to the length of one of its sides is the square root of $2$ by the pythagorean theorem. I agree with Henry, but this might at least make the person want to believe. – Callus - Reinstate Monica Dec 27 '16 at 21:01
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    Because the range of the square function is all positive numbers, there exists a number that you can square, such that you get 2. Therefore, there exists a square root of 2. It's not terribly complicated, I don't see how someone can miss this. – Kaynex Dec 27 '16 at 21:06
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    If there were no square root of 2 then we would make it up because it's useful to have it, just like the imaginary unit. – Kamil Maciorowski Dec 27 '16 at 21:18
  • It does not work just to define $\sqrt{2}$ that way. A bare definition has no impact on the question of existence. To prove existence you no doubt have to invoke properties of $\mathbb{R}$ that these people do not "believe in" in the first place. Better to just admit that if they insist on their own private assumptions, then they are probably right. And point out why mathematics does not agree with them. – Tommy R. Jensen Aug 24 '19 at 14:06

4 Answers4

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Take two squares of area 1. Cut them diagonally and assemble the 4 pieces into a square of area 2.

What is the side length of the square?

Doug M
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  • Ahhh, this is beautiful – TripleA Dec 27 '16 at 21:25
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    "But in reality, any square you take won't have area exactly 1! It won't even be a perfect square! You can't use something that doesn't exist!" – Wojowu Dec 27 '16 at 21:46
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    @Wojowu That is something my brother would say. He says that lines do not exist because light moves in a wave. To which I say math is not physics and the wave form of light is irrelevant. But, if you cannot make basic definitions, then you cannot define length, area, or square root. If they cannot be defined, do they exist? And, you can take the attitude that not only does $\sqrt 2$ not exist, no numbers exist, as they do not have a physical being. But if that is what you are up against, then there really is no point. – Doug M Dec 27 '16 at 22:22
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I would ask him to prove that a wudget isn't purple. You can't prove anything about something that isn't defined. What is the definition of a number? If you don't have a def'n of "number" you can't prove that one of them is not $\sqrt 2,$ or not purple either.

DanielWainfleet
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I would first use a minor variant of the Archimedian Property, in which you can create at least one number between any two numbers by taking their average to prove the continuity of the real numbers, then I would use a binary search to find an approximation for the square root of two, then I would bring up the fact that these results look like they are converging to a specific number ($1.414...$), and then I would argue that this converges to a specific value.

AlgorithmsX
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I might similarly ask, "Does 1/3 exist"

The unspoken argument here appears to be that a number that cannot be written in decimal form by a finite number of digits does not exist.

William Barton
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