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Euclid's Proof that √2 is Irrational
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Re: Euclid's Proof that √2 is Irrational
#62[deleted]
Re: Euclid's Proof that √2 is Irrational
#63Earlier quoted context omitted.
By “usual integer”, I mean what people usually refer to as an integer: …, -2, -1, 0, 1, 2, … As opposed to “algebraic integer”, which is a more general notion.
If I wasn't familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread. The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).
> The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this
So, what term do propose they use for this? “Hutyfreklop” or “gensym_167336871904” would probably be unique, but wouldn’t tell anything about the subject itself, “roots of polynomials with integer coefficients and a leading coefficient of one” would get cumbersome soon.
Re: Euclid's Proof that √2 is Irrational
#64For those who are interested in connections to more advanced mathematics, there is a sense in which √2 is still an integer, even though it is irrational. Specifically there is the notion of “algebraic integers”, which are the set of all complex numbers expressible as the root of a monic polynomial: x^n + a_{n-1}x^(n-1) + … + a_1x + a_0. Here each a_i is a usual integer in ℤ, and monic refers to the leading coefficien…
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Re: Euclid's Proof that √2 is Irrational
#65Earlier quoted context omitted.
By “usual integer”, I mean what people usually refer to as an integer: …, -2, -1, 0, 1, 2, … As opposed to “algebraic integer”, which is a more general notion.
If I wasn't familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread. The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).
Yeah, you might as well call any countable set "integers" because you can find a 1:1 mapping between them. This is silly.
Re: Euclid's Proof that √2 is Irrational
#66And then in 1737 Euler (another name starting with Eu, definitely a good name) showed that the constant e is irrational.
His proof exploited the fact that the continued fraction representation of any rational number terminates. The CF representation for e does not.
Re: Euclid's Proof that √2 is Irrational
#67Earlier quoted context omitted.
https://en.wikipedia.org/wiki/Proof_by_contradiction , https://ncatlab.org/nlab/show/proof+by+contradiction , https://web.stanford.edu/class/cs103/guide_to_proofs#proof-b... all agree (the first three things that came up when I googled for "proof by contradiction"): a proof by contradiction is specifically a proof which shows that P is not false, and concludes that it is true. There is already a perfectly cromulent t…
> a proof by contradiction is specifically a proof which shows that P is not false, and concludes that it is true. And this proof matches that description exactly, with P = "√2 ∉ ℚ". The only case where these would be different ideas is the case where ¬¬P ≠ P. And of course, that can never happen.
Your statement is, I think, being wilfully sloppy.
To quote the outline of the proof: "First Euclid assumed √2 was a rational number.". To quote the proof itself: "Euclid's proof starts with the assumption that √2 is equal to a rational number p/q.". In two different places, the proof explicitly states that it is showing that "√2 in ℚ" is false. It is not showing that "√2 ∉ ℚ" is not false; such a proof would begin "Suppose that it were not the case that √2 ∉ ℚ", which is obviously not how the proof starts (and for good reason, because that would be much more confusing).
By all means argue that "nobody cares about excluded middle"! You're probably right, and when I insist that "proof by contradiction" has a meaning that is correctly stated by Wikipedia and the nlab, I'm just like one of the old fogeys complaining about things like "could care less" or "irregardless"! But don't misquote arguments and say that they support your case when they don't.
Re: Euclid's Proof that √2 is Irrational
#68Earlier quoted context omitted.
By “usual integer”, I mean what people usually refer to as an integer: …, -2, -1, 0, 1, 2, … As opposed to “algebraic integer”, which is a more general notion.
If I wasn't familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread. The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).
I don't think that's fair.
Mathematicians pretty reliably say "algebraic integer" or "integer in [some specific class of numbers that has non-rational integers in it]" when they are talking about the broader notion, and if they're doing something where that broader notion is often relevant they will generally say something like "rational integer" when they mean the narrower notion. So in practice there is seldom any confusion.
And algebraic integers really _are_ like ordinary integers in important ways. Inventing a completely new term would not obviously be an improvement.
It's not like this sort of thing is unique to mathematics. Once upon a time a "language" was a thing human beings used to communicate with one another. Then along came "programming languages" which are not languages in that sense. And then things like "hypertext markup language" which isn't a language in the programming sense either.
(Arguably this is partly mathematicians' fault since I think they were the first to use "language" to refer to purely formal constructs. But I think the use of "language" in computing arose mostly by analogy to human languages.)
And it happens plenty outside "the exact disciplines". A republican is someone who favours a mode of government that doesn't have monarchs, but if you call someone a "Republican" in the US you mean something rather more specific and a few "Republicans" would actually quite like a system hard to distinguish from monarchy. A window is a transparent thing placed in a wall to let light in, but a window of opportunity is something quite different. A czar is the absolute ruler of Russia, but when someone says (rightly or wrongly) that Kamala Harris was "border czar" they don't mean that. A star is a gigantic ball of stuff undergoing nuclear fusion and producing unimaginable amounts of energy, but even the most impressive rock stars don't do that, and some people called "rock stars" have never played or sung a note of rock music in their lives.
Re: Euclid's Proof that √2 is Irrational
#69Earlier quoted context omitted.
Perhaps a translation issue? I've always referred to that set as "algebraic numbers" ( https://en.wikipedia.org/wiki/Algebraic_number ). Since they are equipotent with integers, you _can_ call them that, but it's misleading.
As already mentioned by another poster, algebraic numbers are more general than algebraic integers, because the leading coefficient of the polynomial does not have to be one, similarly to the difference between rational numbers and integer numbers, where for the former the denominator does not have to be one, like for the latter.
Ahh, that would explain why the intersection of algebraic integers and Q is Z. I wasn’t convinced of that when I had the notion of algebraic numbers in place of algebraic integers.
I like teaching this kind of stuff to my grade 9 and 10 advanced math classes. It’s not that hard to understand and yet it gives students a sense of wonder about how math works. I might try to show the grade 10s algebraic integers now.
Re: Euclid's Proof that √2 is Irrational
#70Earlier quoted context omitted.
By “usual integer”, I mean what people usually refer to as an integer: …, -2, -1, 0, 1, 2, … As opposed to “algebraic integer”, which is a more general notion.
If I wasn't familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread. The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).
Terminology is often times used to encapsulate a lot of information in a single word or phrase. It’s sort of a compression of information to facilitate communication. Things like “roots of f” is a shorter way to say that: “the set of all x such that f(x) = 0”. As you get deeper into a subject the more terminology you encounter. This is why research papers are generally unintelligible to those with no training in the areas that the research is about. To not use terminology would make papers insanely long and far too tedious to read.