[HN Gopher] Euclid's Proof that [?]2 is Irrational
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Euclid's Proof that [?]2 is Irrational
Author : thunderbong
Score : 36 points
Date : 2024-08-21 20:39 UTC (2 hours ago)
(HTM) web link (www.mathsisfun.com)
(TXT) w3m dump (www.mathsisfun.com)
| taeric wrote:
| I thought this was basically the same proof that got someone
| killed in the Pythagorean cult.
| https://www.scientificamerican.com/article/how-a-secret-soci...
| shows it was a different proof in a similar vein, though. Fun
| times.
| cornstalks wrote:
| > _Likewise if a number is even and is a square of an integer,
| then its square root must be even._
|
| The proof would be more compelling if this was proven instead of
| being taken as an obvious fact.
| colechristensen wrote:
| It is really trivial though.
| DiggyJohnson wrote:
| Even x Even results Even
|
| Even x Odd irrelevant if squaring
|
| Odd x Odd results Odd
| tines wrote:
| Let n = 2r, and n = xx for some integers r and x, because n is
| even and n is a square. So xx = 2r.
|
| Because of the fundamental theorem of arithmetic, we know that
| x must be representable as the product of a unique string of
| prime numbers.
|
| Because 2 is prime, then since xx = 2r, there must be a 2 in
| the string of primes for xx.
|
| But since 2 is prime, it must be in x as well, because a prime
| cannot come out of nowhere. In other words, if there is a given
| prime P in xx, there must be at least two P in xx, because
| there was at least one in x, and the number of each one got
| doubled in xx.
|
| Therefore xx = 2r = 2*2*y = 4y for some integer y.
|
| Therefore n = 4y and sqrt(n) = sqrt(4y) = sqrt(4)sqrt(y) =
| 2sqrt(y) which is an even number.
|
| Therefore sqrt(n) is even.
| Smaug123 wrote:
| FTA is massive overkill. For every number n, either n can be
| expressed as 2k for some k, or 2k+1 for some k, but not both
| (proof: by induction); in particular the square root can too.
| If the square root is (2k+1), then the square is 4k^2 + 4k +
| 1 = 2(2k^2+2k) + 1, which is by definition odd, not even as
| we supposed.
| tines wrote:
| True, but the FTA proof is just really intuitive for me and
| I like it.
| Hackbraten wrote:
| Thanks for the proof, it was fun to follow, and I agree
| that it's quite intuitive.
|
| I think that it would be helpful to mention why sqrt(y)
| must be an integer.
|
| (I know that it is, but it also feels a bit glossed over,
| given that all the other steps of the proof were
| explained so thoroughly.)
| thaumasiotes wrote:
| The FTA proof is the one that's obvious, though. If it's
| really easy to do something using a basic tool, why worry
| that the basic tool is complex to describe?
| abstractbill wrote:
| 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!
| Smaug123 wrote:
| 1) it's not a proof by contradiction, it's a proof of a negation
| :grump:
|
| 2) I am not a fan of this phrasing "we can't simplify forever".
| Why can't we? It's obvious if you phrase it in the usual way as
| "the denominator is strictly smaller than it was before", but the
| "simplify" operation is kind of complex! They don't even mention
| "decreasing" until the very final Note box where they say offhand
| that actually it's an infinite descent (which is a critical part
| of the proof they've otherwise handwaved).
| thaumasiotes wrote:
| > 1) it's not a proof by contradiction, it's a proof of a
| negation :grump:
|
| I don't get your complaint. It is a proof of a negation, yes,
| the conclusion is that [?]2 [?] Q.
|
| But the proof is done by contradiction; "it's not a proof by
| contradiction" is flat-out false. "Proof by contradiction"
| describes the method of the proof, and "proof of a negation"
| describes its conclusion, which is why one of those phrases
| uses _by_ and the other one uses _of_.
| Smaug123 wrote:
| 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 term for proofs of negations: "refutation by
| contradiction"
| (https://ncatlab.org/nlab/show/refutation+by+contradiction),
| which admittedly nlab says it prefers to its other name
| "proof of negation".
| renewiltord wrote:
| The two seem isomorphic. Just sub R for \not P. Doesn't
| seem to change anything interesting about the proof
| structure.
| Smaug123 wrote:
| As I said, according to the three sources above, which
| are the first sources I clicked on which didn't seem like
| blogspam, the phrase "proof by contradiction" is a term
| of art which means "uses the law of excluded middle to
| conclude the truth of a statement given a proof that its
| negation is false". It may be _unfortunate_ that the
| mathematical world has standardised on the phrase "proof
| by contradiction" for this, but it _has_ standardised on
| that phrase!
| Jun8 wrote:
| If you know a child in middle school, this is a great way to get
| them started on "cool mathematical thinking", which I called
| "Mathematia" when I discussed with my son when he was young (to
| distinguish from the horrible Math being taught in school):
| 1. Introduce Z & Q - this is easy. Perhaps, fingers and slices of
| pizza. Now s/he's ready to be as surprised as the members of the
| Phythagorean cult 2. Go over the classical proof for
| [?]2 given here. We now have a number that's not in Z or Q!
| 3. It's one thing to show a result, a very different thing to
| *grasp* it. Why is (2) a big deal? It smashes the simple notion
| Greeks had that *any* two lengths (rational numbers) are
| commensurable, which is a perfectly simple and obvious (and
| wrong) thing to believe: "Have one stick for one side of a square
| and another for the diagonal. You cannot cut both sticks into
| pieces of the same length, no matter what length you choose."
| *This is amazing* 4. We only discovered one such weird
| number. Are there others? Motivated by the above, how about
| checking [?]3. Show that it's weird, too. 5. [?]4 is
| just 2. How about [?]5? OMG, that's weird, too. 6. So
| the square root of an integer is either an integer or one of
| these weird numbers. It cannot be of the general Q form p/q. This
| is an interesting proof. (While thinking about that with the
| youngster you can think about another generalization: roots
| higher than second. Turns out it's true for those, too:
| https://math.stackexchange.com/questions/4467/how-to-prove-if-a-
| b-in-mathbb-n-then-a1-b-is-an-integer-or-an-irratio)
| 7. How do we work these weird numbers? For example, can we add
| them up, e.g. [?]2 + [?]3? How do we do that? Is that another
| weird number or could it ever be an integer? Some facts about
| these sums are trivial to prove:
| https://math.stackexchange.com/questions/157245/is-the-sum-and-
| difference-of-two-irrationals-always-irrational 8.
| Using the wacky notion of adding two numbers as "mating" you can
| generally outline some higher algebra concepts, e.g. if a lion
| mates with a lion the result is always a lion. What if it mates
| with a tiger? (depends, liger or tigon). Can we think of adding a
| Q to one of these weird numbers the same way? Such intuitions may
| be misleading (remember the Greeks?) but are fun.
| glial wrote:
| Where in Euclid's Elements does this proof appear? I can't seem
| to find it.
| Smaug123 wrote:
| Euclid X prop 9.
| http://aleph0.clarku.edu/~djoyce/java/elements/bookX/propX9....
| nickt wrote:
| (300 BCE)
| AnotherGoodName wrote:
| Couldn't you stop the proof at the statement q^2 must equal 2m^2
| since it's obvious there's no solutions to q^2 = 2m^2.
|
| To explain why it's obvious, squares always have an even number
| if factors of two (an even multiple of any prime factor since
| it's a square but just focus in on 2 here for now).
|
| A square times two always has an odd number of factors of 2 since
| it's the above (an even number of factors of two) plus one more
| factor.
|
| An odd number of prime factors on one side can't be equal to an
| even number of factors on the other side. q^2 can never equal
| 2m^2 for any integer value of a or m. Therefore it's irrational.
|
| This ends the proof much earlier right?
| Smaug123 wrote:
| Yep, that does work, although it needs a bunch of extra
| machinery (the fundamental theorem of arithmetic, which
| guarantees existence and uniqueness of prime factorisation). If
| you're happy to take that machinery as having already been
| proved - it's not entirely trivial, and it's definitely not
| obvious! - then you can indeed stop there.
|
| Why do I claim that it's not obvious? Consider the ring of
| integers with sqrt(-5): that is, all complex numbers of the
| form `a + b sqrt(-5)` with a, b integers. This is a ring - it
| has all the nice additive and multiplicative properties that
| the integers do - but it _doesn 't_ have unique factorisation,
| because 6 has two distinct factorisations.
| AnotherGoodName wrote:
| That's reasonable. I've been taught unique prime
| factorization is a thing since primary school but never
| considered the history behind that knowledge. I feel anyone
| with that basis could reasonably stop at the third line here.
| In fact they could quickly create a generalization since a^2
| could clearly never equal b(c^2) unless b was also a square
| for integer a and c but that obviousness is based on a lot of
| other knowledge.
| Y_Y wrote:
| > To explain why it's obvious
|
| Have you considered a career in mathematics?
| wging wrote:
| It's an interesting exercise to find the right generalization of
| this proof to sqrt(n) for arbitrary numbers n that are not
| perfect squares, and for kth roots for m >= 2. I.e. prove that if
| kth_rt(n) is rational, then n is a perfect kth power (or
| equivalently, that if n is not a perfect kth power, then
| kth_rt(n) is irrational).
|
| (I'm talking about adapting the ideas of this divisibility-based
| proof. abstractbill's post
| https://news.ycombinator.com/item?id=41314547 about Conway's
| method, https://www.youtube.com/watch?v=wNOtOPjaLZs, is a
| completely different (and very cool) way to do this that I hadn't
| seen before today.)
| bbno4 wrote:
| slightly related but mathsisfun is a goated website. one of the
| all time greats. myself and many other people only got through
| maths because of this site.
| CaptainNegative wrote:
| This proof has almost nothing to do with Euclid. The Pythagoreans
| knew about it more than a century before his birth (Hippasus was
| apocryphally killed for divulging this proof), and the proof is
| widely believed to have only been inserted into Elements by
| others after Euclid's death.
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