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Author Topic: Random Stuff  (Read 1162520 times)

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blueblimp

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Re: Random Stuff
« Reply #400 on: May 17, 2013, 01:33:10 am »
+2

Exercise: show that there exist irrationals a and b such that ab is rational.
Not as elegant as your solution:

Observe that sqrt(2) ^ (2 * log_2(3)) = 3.

Let's assume we know sqrt(2) is irrational already. Claim: log_2(3) is irrational. Pf: Otherwise, there would exist positive integers p,q such that 2^(p/q) = 3, so 2^p = 3^q, which is impossible by unique prime factorization.
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DStu

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Re: Random Stuff
« Reply #401 on: May 17, 2013, 02:11:44 am »
+1

eln(2) = 2
The trouble then is to show that log 2 is irrational.  You'd be amazed if it wasn't, but it's not obvious.

If you know e is transcendent it's clear, but of course that proof isn't easy.
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Re: Random Stuff
« Reply #402 on: May 17, 2013, 02:52:46 am »
+1

eln(2) = 2
The trouble then is to show that log 2 is irrational.  You'd be amazed if it wasn't, but it's not obvious.

If you know e is transcendent it's clear, but of course that proof isn't easy.

If you want to start with proof that e is transcendent, you need to invent the universe first.
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DStu

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Re: Random Stuff
« Reply #403 on: May 17, 2013, 03:26:48 am »
+1

eln(2) = 2
The trouble then is to show that log 2 is irrational.  You'd be amazed if it wasn't, but it's not obvious.

If you know e is transcendent it's clear, but of course that proof isn't easy.

If you want to start with proof that e is transcendent, you need to invent the universe first.
Don't understand.  But another one:
Take a^f(a)=2, where f(a) is defined implicitly by that equation.  Obviously, f(a) is strictly decreasing in a, and therefore injective.  In particular, the set {f(a)| a irrational} is uncountable. Therefore, they can't be all rational, and there exists a pair (a,f(a)) s.t. a,f(a) are irrational and a^f(a)=2.

Edit: ... which is just the unconstructive way of saying "Replace e with an arbitrary transcendent number in the argumenr e^log_e(2)=2"
« Last Edit: May 17, 2013, 03:29:01 am by DStu »
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Thisisnotasmile

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Re: Random Stuff
« Reply #404 on: May 17, 2013, 07:10:45 am »
+2

This proof is the most elegant I have ever seen. Guy's a genius.
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DStu

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Re: Random Stuff
« Reply #405 on: May 17, 2013, 07:15:31 am »
0

This proof is the most elegant I have ever seen. Guy's a genius.

Yup, gotta make me understand...
« Last Edit: May 17, 2013, 07:16:54 am by DStu »
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Axxle

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Re: Random Stuff
« Reply #406 on: May 17, 2013, 02:38:53 pm »
+2

Didn't let me down.
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WanderingWinder

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Re: Random Stuff
« Reply #407 on: May 17, 2013, 02:55:09 pm »
0

eln(2) = 2

The trouble then is to show that log 2 is irrational.  You'd be amazed if it wasn't, but it's not obvious.

There's a fun argument that gives existence easily, but isn't constructive.

First you try √2√2.  If it's rational, you're done.  If it's irrational, then (√2√2)√2 = (√2)2 = 2.

Actually, (√2√2)√2 is NOT the same thing as (√2)2. You want √2(√2√2)

Or maybe that's the thing you intended - but the parentheses placement is important here.

heron

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Re: Random Stuff
« Reply #408 on: May 17, 2013, 03:04:17 pm »
0

eln(2) = 2

The trouble then is to show that log 2 is irrational.  You'd be amazed if it wasn't, but it's not obvious.

There's a fun argument that gives existence easily, but isn't constructive.

First you try √2√2.  If it's rational, you're done.  If it's irrational, then (√2√2)√2 = (√2)2 = 2.

Actually, (√2√2)√2 is NOT the same thing as (√2)2. You want √2(√2√2)

Or maybe that's the thing you intended - but the parentheses placement is important here.

No, qmech is right.

(√2√2)√2=√2√2*√2=2.

But you are correct that parenthesis placement is important.
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WanderingWinder

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Re: Random Stuff
« Reply #409 on: May 17, 2013, 03:44:22 pm »
0

eln(2) = 2

The trouble then is to show that log 2 is irrational.  You'd be amazed if it wasn't, but it's not obvious.

There's a fun argument that gives existence easily, but isn't constructive.

First you try √2√2.  If it's rational, you're done.  If it's irrational, then (√2√2)√2 = (√2)2 = 2.

Actually, (√2√2)√2 is NOT the same thing as (√2)2. You want √2(√2√2)

Or maybe that's the thing you intended - but the parentheses placement is important here.

No, qmech is right.

(√2√2)√2=√2√2*√2=2.

But you are correct that parenthesis placement is important.

Hrmm, 'course he's right. How did I read that as the other thing. Probably just a mis-wired brain.
Apologies.

Tables

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Re: Random Stuff
« Reply #410 on: May 17, 2013, 05:34:38 pm »
0

Exercise: show that there exist irrationals a and b such that ab is rational.

I was going to produce an elegant, non-consntructive solution, but then noticed you'd already posted the same one. Bah.
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

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Re: Random Stuff
« Reply #411 on: May 20, 2013, 08:19:46 pm »
0

I played a game of Dominion with Celestial Chameleon this weekend. I wanted to come back and brag about how I beat him, but he won (although both of us agreed, it wasn't the most interesting kingdom - still enough decisions to give the better player an advantage, but not enough that it's really decisive). Still it was nice to see him again, and the game was great fun, mainly because of the comments he was making - e.g. something along the lines of "My [opening cards] fell to turn 5, which was a big disadvantage, and stopped me emptying the supply as well."

As another totally unrelated topic, I've finally gotten some recording software. For now I'm recording You Have to Win the Game, but I may do some Dominion videos at some point. For now though, I'm only 6 seconds off the world record... and lost maybe 2-3 seconds because I was trying to mess around with WSplit. I wanted an at least decent run, I wasn't expecting something THAT good.
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

Thisisnotasmile

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Re: Random Stuff
« Reply #412 on: May 21, 2013, 07:50:00 am »
0

Just broke the 20,000 points mark on Kongregate. Thanks Snail Bob!
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Kuildeous

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Re: Random Stuff
« Reply #413 on: May 21, 2013, 12:14:35 pm »
+5

You're so vain; you probably think this post is about you.
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mail-mi

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Re: Random Stuff
« Reply #414 on: May 21, 2013, 04:06:54 pm »
0

You are now breathing manually.

Have fun for the next minute or so.
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sudgy

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Re: Random Stuff
« Reply #415 on: May 21, 2013, 04:16:13 pm »
+1

You are now breathing manually.

Have fun for the next minute or so.

You are now aware that when you swallow, there is a popping sound in your ears.

You now know how heavy your head is.

You now know that you can see your nose.
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

Thisisnotasmile

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Re: Random Stuff
« Reply #416 on: May 21, 2013, 04:19:03 pm »
+1

You are now aware that when you swallow, there is a popping sound in your ears.

I had never noticed this before in my life. I am now complete.
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Thisisnotasmile

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Re: Random Stuff
« Reply #417 on: May 21, 2013, 04:20:22 pm »
0

But on the bright side, it's not tongue awareness month until November.
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sudgy

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Re: Random Stuff
« Reply #418 on: May 21, 2013, 04:21:07 pm »
0

My tongue is comfortable.  That's why I didn't say it.
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

mail-mi

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Re: Random Stuff
« Reply #419 on: May 21, 2013, 04:44:29 pm »
0

You are now breathing manually.

Have fun for the next minute or so.

You are now aware that when you swallow, there is a popping sound in your ears.

You now know how heavy your head is.

You now know that you can see your nose.

You are now blinking manually.

You are now aware that your clothes are touching your body.
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I currently imagine mail-mi wearing a dark trenchcoat and a bowler hat, hunched over a bit, toothpick in his mouth, holding a gun in his pocket.  One bead of sweat trickling down his nose.

'And what is it that ye shall hope for? Behold I say unto you that ye shall have hope through the atonement of Christ and the power of his resurrection, to be raised unto life eternal, and this because of your faith in him according to the promise." - Moroni 7:41, the Book of Mormon

theory

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Re: Random Stuff
« Reply #420 on: May 21, 2013, 05:07:02 pm »
+8

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Drab Emordnilap

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Re: Random Stuff
« Reply #421 on: May 21, 2013, 07:53:20 pm »
0


I tried to report this post to a moderator, but the button just laughed at me.  :-[
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Kirian

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Re: Random Stuff
« Reply #422 on: May 21, 2013, 08:16:49 pm »
0

Posit:  the best part of the 11th Doctor's action theme is the swingy oboe riff that sounds completely out of place in a minor key but fits anyway.  Discuss.
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Re: Random Stuff
« Reply #423 on: May 22, 2013, 09:29:54 am »
+4

This proof is the most elegant I have ever seen. Guy's a genius.

I thought that nice man was going to get to the proof eventually, but after a whole hour he didn't!  I don't think your link is correct, tinas.  There wasn't much in the way of mathematical proof there in that presentation, elegant or otherwise.
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Thisisnotasmile

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Re: Random Stuff
« Reply #424 on: May 22, 2013, 09:39:08 am »
+3

This proof is the most elegant I have ever seen. Guy's a genius.

I thought that nice man was going to get to the proof eventually, but after a whole hour he didn't!  I don't think your link is correct, tinas.  There wasn't much in the way of mathematical proof there in that presentation, elegant or otherwise.

Are you saying he let you down? I don't think he did.
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