These days I much prefer a different proof of this fact, attributed to Conway: https://www.youtube.com/watch?v=wNOtOPjaLZs -- I'm actually about half-way through teaching it to my kids right now!
I hadn't seen this before, it's fantastic! Thanks!
Euclid's Proof that √2 is Irrational
81–90 of 100 posts
Re: Euclid's Proof that √2 is Irrational
#82We can also assume that the p/q=√2 is already the simplest form of the fraction, since every fraction must have one, as in the first section of the article. Then if we figure out that both p and q are even, it means that p/q can be simplified (by dividing p and q by 2), which contradicts the assumption about the simplest form - and we don't need to use the infinite descent.
I was going to say this. The proof, unfortunately, sidelines into some very deep mathematics that it didn’t need to. I imagine but don’t know for sure that Euclid stopped where you do.
I’m not sure when infinite descent would be considered to have been formally proven to a modern mathematician but I bet it wasn’t in Euclid’s time!
Re: Euclid's Proof that √2 is Irrational
#83Earlier quoted context omitted.
> the set of such roots is actually closed under multiplication, addition, and subtraction, and there is even an analogue of prime factorization if you squint I did a maths undergrad, but I don’t think I ever studied algebraic integers. That’s something I shall have to remedy now, thanks!
If you took abstract algebra (which presumably you did as a math major), you certainly encountered these at least in the exercises as groups of the form ax + b where x is some irrational number (or imaginary) and a and b are integers are a staple of chapter 1–2 proofs. Gaussian integers ( ai + b ) are a special case that are loads of fun to play with it. They are not unique factorization domains like the integers (e.…
Nit: while it is not generally the case that rings of algebraic integers must be unique factorization domains, it is the case for Gaussian integers! In your example, 5 is uniquely factorizable up to units as (1-2i)(1+2i).
Re: Euclid's Proof that √2 is Irrational
#84Earlier quoted context omitted.
Yes, but did God make the algebraic integers? Because this looks suspiciously like the work of man.
This depends on your philosophy of mathematics. If you believe god gave us the nonnegative integers then in a very natural way one is led to the notion of algebraic integers. Whether this would be god’s creation or man’s creation depends on your view of who gets the credit in such a situation.
I think next time I’d better make the apologies to Kronecker explicit.
Re: Euclid's Proof that √2 is Irrational
#85We can also assume that the p/q=√2 is already the simplest form of the fraction, since every fraction must have one, as in the first section of the article. Then if we figure out that both p and q are even, it means that p/q can be simplified (by dividing p and q by 2), which contradicts the assumption about the simplest form - and we don't need to use the infinite descent.
I was going to say this. The proof, unfortunately, sidelines into some very deep mathematics that it didn’t need to. I imagine but don’t know for sure that Euclid stopped where you do. I’m not sure when infinite descent would be considered to have been formally proven to a modern mathematician but I bet it wasn’t in Euclid’s time!
Isn't the proof just claiming that you can't divide an integet infinitely many times by 2, and still expect it to be an integer?
Re: Euclid's Proof that √2 is Irrational
#86We can also assume that the p/q=√2 is already the simplest form of the fraction, since every fraction must have one, as in the first section of the article. Then if we figure out that both p and q are even, it means that p/q can be simplified (by dividing p and q by 2), which contradicts the assumption about the simplest form - and we don't need to use the infinite descent.
That assumes that every fraction has a unique simplest form. The first section of the article makes no such claims about the existence of a simplest form of fraction. The proof uses just algebraic manipulation, the fact that a sequence of strictly decreasing positive integers is finite in length, and the definition of a rational number (there exist integers p, q (q != 0) such that the number can be expressed as p/q).
Re: Euclid's Proof that √2 is Irrational
#87We can also assume that the p/q=√2 is already the simplest form of the fraction, since every fraction must have one, as in the first section of the article. Then if we figure out that both p and q are even, it means that p/q can be simplified (by dividing p and q by 2), which contradicts the assumption about the simplest form - and we don't need to use the infinite descent.
I was going to say this. The proof, unfortunately, sidelines into some very deep mathematics that it didn’t need to. I imagine but don’t know for sure that Euclid stopped where you do. I’m not sure when infinite descent would be considered to have been formally proven to a modern mathematician but I bet it wasn’t in Euclid’s time!
I think the idea is that any strictly decreasing sequence of positive integers starting at N is a subsequence of (N, N-1, N-2, ..., 1) which has N elements, so any such sequence must have a finite number of elements.
So if you manage to produce an infinite sequence of strictly decreasing positive integers starting at a particular positive integer, then you've reached a contradiction.
Re: Euclid's Proof that √2 is Irrational
#88We can also assume that the p/q=√2 is already the simplest form of the fraction, since every fraction must have one, as in the first section of the article. Then if we figure out that both p and q are even, it means that p/q can be simplified (by dividing p and q by 2), which contradicts the assumption about the simplest form - and we don't need to use the infinite descent.
That assumes that every fraction has a unique simplest form. The first section of the article makes no such claims about the existence of a simplest form of fraction. The proof uses just algebraic manipulation, the fact that a sequence of strictly decreasing positive integers is finite in length, and the definition of a rational number (there exist integers p, q (q != 0) such that the number can be expressed as p/q).
The article explicitly claims
> Rational numbers or fractions must have a simplest form.
They make no claim about uniqueness, but that is not needed in the argument.
Re: Euclid's Proof that √2 is Irrational
#89Earlier 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).
This seems a bit harsh. Mathematicians tend to be mostly unambiguous when they write. This isn't even on the same planet as the empirical disciplines.
Re: Euclid's Proof that √2 is Irrational
#90These days I much prefer a different proof of this fact, attributed to Conway: https://www.youtube.com/watch?v=wNOtOPjaLZs -- I'm actually about half-way through teaching it to my kids right now!
The cool thing is this argument immediately generalizes to zeroes of monic polynomials with integer coefficients, i.e. x^n + a_0 x^{n-1} + ... + a_n.
In commutative algebra one says "the ring Z is integrally closed in its field of fractions Q", or short "Z is normal".[1]
[1] https://en.wikipedia.org/wiki/Integrally_closed_domain#Norma...