More on proofs

[ciphergoth]

Some really interesting answers on what proofs you know. The most common proofs people mention are the ones I name, plus Cantor's diagonalisation argument that |R| > |N| - cool. More specific comments follow.

wildbadger -- product of two compact spaces is compact

A strong opening! I looked it up and found Tychonoff's Theorem - looks interesting.

cryptodragon -- A number of them from crypto stuff

Interesting, name us a favourite?

simple_epiphany -- As a third-year maths student, I'm required to know quite a lot of them, but the one for the Bolzano-Weierstrass theorem is quite nice.

Bolzano–Weierstrass theorem on Wikipedia. I think I could remember that proof. Cool, thanks!

aegidian -- Cantor's Diagonalisation, proving there are as many rational numbers as there are integers.

Ah, the proof that |Q| = |N| rather than the proof that |R| > |N|?

olethros -- Maybe I lied. I can prove (by recursion) that all marbles in the world are the same colour.

I know that proof :-) Oh go on, there must be a *valid* proof you like!

keirf -- Bolzano-Weierstrass theorem - every bounded sequence in R{n} has a convergent subsequence

A second showing for this theorem!

ajva -- that 0.999...=1

Don't you need to get into the construction of the real numbers to explain this one?

ergotia -- The infinite number of primes/hotel at the end of the universe one

Two proofs for the price of one :-)

nikolasco -- irrationality of sqrt(2) (fundamental theorem of arithmetic, even/odd, well-ordered)

What proofs are you referring to with "even/odd, well-ordered"?

Thanks all, please keep commenting :-)

Comments

[marnanel.livejournal.com]

"But it MUST be a different number! It's written differently!"

"Okay, subtract 0.999 from one. What's the answer?"

"Z..."

"Aha!"

"Er. The next real number after zero. See, got you."

"But there is no such number."

"Who says?"

et cetera.

The most comments I ever got on an LJ post, something like seventy of them, was by people attempting to disprove this when I asserted it in the course of talking about something else.

[emarkienna.livejournal.com]

"Okay, subtract 0.999 from one. What's the answer?"

"Z..."

I love it when they reply "0.0000 ... 1"

[valkyriekaren.livejournal.com]

And that's not the answer because...?

[ciphergoth.livejournal.com]

Because there's nowhere to put the "1". Every decimal place has to be at some numbered position; there are no extra places "at infinity", even though there are infinitely many places.

[marnanel.livejournal.com]

I try to tell people "you're asking for an infinite number of zeros and then a one. Have you considered what 'infinite' means and the problems in following it with 'and then'?"

[ciphergoth.livejournal.com]

You could do that just fine if you index the decimal places with something like 2ω instead of the natural numbers - I think this gives you the dual numbers (actually not quite but I think you can get there with some extra tweaking). We do have number spaces that have some of the behaviours people expect of the reals here, but the ordinary reals are not they.

[emarkienna.livejournal.com]

What ciphergoth said. Or to put it another way, "0.9..." or "0.9 recurring" [*] has a specific mathematic definition (an infinite sum), but "0.0...1" does not - and whenever I ask such a person for its definition, they are unable to provide one.

(Just to clarify - I don't mean to mock people who don't see that 0.9recurring = 1 - after all, the rigorous proof is rather high level, and the "simple proofs" aren't really rigorous. I mean more that the issue tends to attract some people who insist they are right, but are unable to either justify it, or find a fault in the proof that shows they are equal. I think it's a bit like the evolution of the mathematical world, except not with all the political and religious dispute.)

[*] I think one of the points that causes misunderstanding is the use of "...", which probably gets misunderstood as simply meaning "a lot of 9s". I'd prefer to use the notation of putting a dot or bar over the 9, but of course that's not so easy when writing online...

[ciphergoth.livejournal.com]

Nice analogy - there's a huge difference between "I don't understand how this feature evolved and I'd like to, it seems counter to my intuition because..." and "This seems counter to my intuition so EVILUTION IS A LIE!"

[elsmi.livejournal.com]

I tend to feel like arguing about 0.999... is a bit of a cheat, because really it isn't so much that the rigorous proof is obscure, it's that the *rigorous definition* is obscure. And then I feel like I'm telling people "ah-hah, you are wrong, because I am using a different definition than you! And normally switching around definitions is not a valid rhetorical technique, but it is valid in this case because this is Maths, which you were taught in school is the incomprehensible domain of people who know the secret code, which I do! So you must trust me!"

*cough* Right, err, I'll go lay down for a bit.