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Formalising a new proof that the square root of two is irrational

lawrencecpaulson.github.io

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Re: Formalising a new proof that the square root of two is irrational

#81

Post by thaumasiotes »

simiones wrote: post #65
thehappypm wrote: post #64

Earlier quoted context omitted.

Um, I don’t really think that’s right. using the square root to convert area to perimeter, for example, is an application of square roots, one where the negative root is kinda useless. That’s not the definition of the operation. The definition of the operation is the solution to y = x x. And sure for many applications the negative root is useless, but you can’t argue against that both -2 -2 and 2*2 = 4.

You may not think I'm right, but that is the actual history. This operation was invented at a time where geometry was the main way mathematics was done. The square root is a much older concept than negative numbers (edit: at least in the Hellenistic world; Chinese and Indian mathematics may have had different histories). So, by definition, the square root of 4 is +2. x * x = 4 has two solutions, which we dub +sqrt(2)…

> Diophantus, in Arithmetica, c. 280 AD, was claiming that the equation 4x + 20 = 4 doesn't have any solutions/is absurd. So, the square root is more than 600 years older than negative numbers in the Hellenistic tradition.

The gap is much wider than that; double-entry bookkeeping speaks of "credit" and "debit" (defined to be exactly the same concept as positive and negative numbers) because negative numbers were still nearly unknown to Europeans in the 15th century.

Re: Formalising a new proof that the square root of two is irrational

#82

Post by thaumasiotes »

nanis wrote: post #5
adunk wrote: post #4

Earlier quoted context omitted.

The first sentence in the blog post contains a link to the proof itself, which seems to have been posted to tumblr: https://theapproximatepresent.tumblr.com/post/51484587425/a-... I'm not sure it is necessary for the author also post the link to the tweet that referred to the tumblr post, since the tumblr post is where the actual proof is.

The tweet is the "secondary source" whereas the Tumbler post is the "primary source" in this context[1]: > For example, suppose you are reading an article by Brown (2014) that cites information from an article by Snow (1982) that you would like to include in your essay. For the reference list, you will only make a citation for the secondary source (Brown). You do not put in a citation for the primary source (Snow) in…

> For the reference list, you will only make a citation for the secondary source (Brown). You do not put in a citation for the primary source (Snow) in the reference list. For the in-text citation, you identify the primary source (Snow) and then write "as cited in" the secondary source (Brown).

That's not so you can give credit to Brown for helpfully pointing you toward Snow. It's a requirement that you admit, when you cite Snow, that you never actually read Snow. If you read Brown, find a pointer to Snow, and then read Snow, you don't cite Brown at all.

Re: Formalising a new proof that the square root of two is irrational

#83

Post by thaumasiotes »

BeetleB wrote: post #72
zeroonetwothree wrote: post #54

Earlier quoted context omitted.

I can’t believe you are downvoted. In math sqrt is defined as the positive one. It seems HN readers need a refresher.

Most people are taught that there are two values for the square root. It was only in later math courses (e.g. precalculus) that they reserved the square root symbol exclusively for the positive square root.

That is not what people are taught. If that were true, the numerator in the quadratic equation would be "-b + √(b² - 4ac)", not "-b ± √(b² - 4ac)".

Re: Formalising a new proof that the square root of two is irrational

#84

Post by BeetleB »

thaumasiotes wrote: post #83
BeetleB wrote: post #72

Earlier quoted context omitted.

Most people are taught that there are two values for the square root. It was only in later math courses (e.g. precalculus) that they reserved the square root symbol exclusively for the positive square root.

That is not what people are taught. If that were true, the numerator in the quadratic equation would be "-b + √(b² - 4ac)", not "-b ± √(b² - 4ac)".

Your statement assumes the educational system values consistency, which is a clearly false assumption.

Re: Formalising a new proof that the square root of two is irrational

#85

Post by graycat »

raldi wrote: post #74
graycat wrote: post #18

Let's have a simple proof so that we don't have to argue about more complicate issues: For integers p, q with q not 0, suppose 2 = (p/q)^2 Then 2 q^2 = p^2 and the left size has an odd number of factors of 2 and the right side has an even number of factors of 2, a contradiction. Therefore such p, q do not exist, and 2 does not have a rational square root. Done.

> and the left size has an odd number of factors of 2 and the right side has an even number of factors of 2, a contradiction. Can you explain this step further?

If I had explained it further, then the proof would have been longer!

Right, we need the Fundamental Theorem of Arithmetic, that each integer factors uniquely into a product of prime numbers; right, that statement leaves out a little about the order of the factors.

Well, 2 is prime number. So however many factors of 2 q has, q^2 has twice that many and, thus, an even number of factors of 2. Same argument for p^2; it also has an even number of factors of 2. But then

2 q^2

has an odd number of factors of 2.

Uh, more generally, we can keep arithmetic simple or make it complicated. E.g., multiplication of integers is associative so that

(2 x 3) x 4 = 2 x (3 x 4)

The intermediate results are different, but the final results are the same. So there is a little something there (matrix multiplication is also associative, with more there -- right, easily function composition is associative so regard a matrix as a function and matrix multiplication as function composition).

We can keep the Fundamental Theorem of Arithmetic simple, e.g., ask a 4rd grader, or make it complicated: To make arithmetic complicated, we can take a college course in abstract algebra that proves all this. Then for still more complicated we can go through axiomatic set theory before the abstract algebra course. Then in the set theory course we can get to undecidable issues.

I went through all that. Instead, what I wanted was differential geometry as background for general relativity, partial differential equations for the various applications, but I'd put up with exterior algebra, of course the inverse and implicit function theorems (proved in Fleming's book and an exercise in Rudin's), fluid flow, Maxwell's equations, Schroedinger's equation, Fourier theory, Lebesgue integration, Leibniz's rule (differentiation under the integral sign; there's a simple proof if assume enough continuity and a more general proof from Lebesgue theory), how come, with Lagrange, action is stationary, on and on. I was willing to take arithmetic as okay!

Re: Formalising a new proof that the square root of two is irrational

#86

Post by raldi »

graycat wrote: post #85
raldi wrote: post #74

Earlier quoted context omitted.

> and the left size has an odd number of factors of 2 and the right side has an even number of factors of 2, a contradiction. Can you explain this step further?

If I had explained it further, then the proof would have been longer! Right, we need the Fundamental Theorem of Arithmetic, that each integer factors uniquely into a product of prime numbers; right, that statement leaves out a little about the order of the factors. Well, 2 is prime number. So however many factors of 2 q has, q^2 has twice that many and, thus, an even number of factors of 2. Same argument for p^2; it…

Sounds like the crux is that 2(x^2) = y^2 can never be true for any integers x and y because any square of an integer must have an even number of 2's in its prime factorialization, and any doubled square of an integer must have an odd number.

Re: Formalising a new proof that the square root of two is irrational

#87

Post by thaumasiotes »

BeetleB wrote: post #84
thaumasiotes wrote: post #83

Earlier quoted context omitted.

That is not what people are taught. If that were true, the numerator in the quadratic equation would be "-b + √(b² - 4ac)", not "-b ± √(b² - 4ac)".

Your statement assumes the educational system values consistency, which is a clearly false assumption.

Textbooks aren't that bad. Here's what Pearson's Algebra 1 says when introducing the novel concept of the radical sign:

> The radical symbol √ indicates a nonnegative square root, also called a principal square root.

I quoted that one because it's available on libgen ( http://library.lol/main/BE522034F8D3680B248162CD2C35D455 , page 16), but obviously every basic algebra textbook covers this. The symbol is never discussed -- or used -- in any other terms than that it uniquely refers to a positive (or zero) square root.

Re: Formalising a new proof that the square root of two is irrational

#88

Post by graycat »

raldi wrote: post #86
graycat wrote: post #85

Earlier quoted context omitted.

If I had explained it further, then the proof would have been longer! Right, we need the Fundamental Theorem of Arithmetic, that each integer factors uniquely into a product of prime numbers; right, that statement leaves out a little about the order of the factors. Well, 2 is prime number. So however many factors of 2 q has, q^2 has twice that many and, thus, an even number of factors of 2. Same argument for p^2; it…

Sounds like the crux is that 2(x^2) = y^2 can never be true for any integers x and y because any square of an integer must have an even number of 2's in its prime factorialization, and any doubled square of an integer must have an odd number.

Right. But, with this approach to a proof, to be strict, for "prime factorization" need that it is unique, and that is the Fundamental Theorem of Arithmetic -- need that.

Re: Formalising a new proof that the square root of two is irrational

#89

Post by BeetleB »

thaumasiotes wrote: post #87
BeetleB wrote: post #84

Earlier quoted context omitted.

Your statement assumes the educational system values consistency, which is a clearly false assumption.

Textbooks aren't that bad. Here's what Pearson's Algebra 1 says when introducing the novel concept of the radical sign: > The radical symbol √ indicates a nonnegative square root, also called a principal square root . I quoted that one because it's available on libgen ( http://library.lol/main/BE522034F8D3680B248162CD2C35D455 , page 16), but obviously every basic algebra textbook covers this. The symbol is never disc…

Your evidence is consistent with my claims. Or rather: I never claimed there aren't textbooks that define it as the positive square root.

Re: Formalising a new proof that the square root of two is irrational

#90

Post by nanis »

orra wrote: post #8
nanis wrote: post #5

Earlier quoted context omitted.

The tweet is the "secondary source" whereas the Tumbler post is the "primary source" in this context[1]: > For example, suppose you are reading an article by Brown (2014) that cites information from an article by Snow (1982) that you would like to include in your essay. For the reference list, you will only make a citation for the secondary source (Brown). You do not put in a citation for the primary source (Snow) in…

You keep saying “author”, but putting a screenshot of a Tumblr post on Twitter is stretching the word quite considerably. Besides, the Tumblr post is already a secondary source.

We do not know what was in the tweet simply because it is not referenced/cited.
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