00:00:08 I decided that |b@| was simply -|b|, since |ab|=|a||b|, unless someone has a counter-example?
00:00:09 well |a@| is obviously |a|*|@|
00:00:11 |@| = -1
00:00:14 so it's a * -1
00:00:32 but yeah it's a case of reducing |a+b|...
00:00:52 Has reducing |a+b| been done, or is it impossible?
00:01:19 I asked wolfram alpha for some inspiration but it wasn't very helpful
00:01:26 I think you'd just have to make shit up until all the properties work...
00:01:27 apart from |x| > 0 ofc :P
00:01:34 >= that is
00:01:46 what is??
00:02:02 ?
00:03:14 Why is |a+bi| defined the way it is?
00:04:27 -!- oerjan has joined.
00:07:48 -!- alise has joined.
00:07:49 -!- AnMaster has quit (Ping timeout: 256 seconds).
00:08:01 wb alise
00:08:05 Sgeo, because we want |-| to be a "metric"
00:08:15 metric?
00:08:25 which means |a-b|+|b-c|>=|a-c| and |a-b|=0 <=> a=b
00:08:37 -!- AnMaster has joined.
00:08:46 oh and |a-b|=|b-a|
00:09:05 * Sgeo wonders how he can use that to define at least the properties of |a+b@|
00:09:13 why do we care about metrics? They make a topology
00:09:19 why do we care about topologies... I don't know yet
00:09:26 what's @
00:09:53 @ is defined such that |@| = -1. Pseudomathematical curiousity that I seem to have interested alise in just now
00:10:15 [Or, probably less peusomathematically, |x| gains an exception in its definition]
00:10:47 -!- alise_ has quit (Ping timeout: 252 seconds).
00:14:15 |a-@| + |@-c| >= |a-c|. If a=0 and c=5, for instance, and this rule is upholdable, |@-5| >= 5
00:14:41 I'm not convinced that it's upholdable given the redefintion, however.
00:15:29 so |@| = -1
00:15:38 |@-1| >= 1
00:15:41 |@-2| >= 2
00:16:07 That seems... broken
00:16:18 |@| = -1, |@-1| = 1 (presumably)
00:16:24 |@-2| = 2
00:16:31 so there's a 0 missing there...
00:16:36 -!- nooga has joined.
00:16:38 -!- augur has joined.
00:17:43 If |@-1| = 1, |@-2| = 3, somehow makes more sense, but there's no evidence for that really
00:18:33 Not "sense", but it's nicely pattern-filling
00:19:20 This number is no number, it's more like a function :P
00:19:26 |a-b|+|b-c|>=|a-c|
00:19:30 lol
00:19:41 fax: /msg???
00:21:32 a=@, b=0, c=1 |@-0| + |0-1| >= |@-1| <=> -1 + 1 >= |@-1|
00:21:53 I think we have a problem
00:23:36 well clearly |@-1| results in a number
00:23:45 ~@-1~
00:23:58 which has these contradictory properties!
00:24:13 Maybe |a-b|+|b-c|>=|a-c| is simply broken
00:25:44 Maybe you are broken :|
00:26:18 Well, when |x|>=0 is suddenly violated, I'd expect anything that relies on that to be broken
00:26:30 fax: /msg :(((((
00:26:37 im not here right now
00:26:41 fax: yes you are
00:28:17 1 = 1 + 0 = 1 + 0*(-1) = 1 + (1+(-1))*(-1) = 1 + 1*(-1) + (-1)*(-1) = 1 + (-1) + (-1)*(-1) = 0 + (-1)*(-1) = (-1)*(-1)
00:29:15 oerjan 1 + 0*(-1) = 1 + (1+(-1))*(-1) ?
00:29:20 ho sorry I see it now
00:29:36 oerjan, that's so.. obvious, looking at it
00:29:43 heh
00:30:06 Hm, that means something major would have to break in order to make a system in which -1*-1=-1
00:30:13 yeah
00:30:17 Maybe 1-1 != 0
00:30:40 well 1-1 == 0 is pretty much the _definition_ of negation
00:31:24 in fact it's the only property of negation used above
00:31:40 (1+(-1) = 0)
00:31:55 1 - 1 = 1 + (-1)
00:32:05 axiom: x + (-x) = i, where i is additive identity
00:32:08 i = 0
00:32:12 x + (-x) = 0
00:32:15 1 + (-1) = 0
00:32:16 1 - 1 = 0
00:32:18 Q.E.D.
00:32:53 You also used a(b+c) = ab+ac
00:32:55 BRB
00:33:12 a-b is just an abbreviation for a+(-b) afaiac
00:33:46 Sgeo: yeah obviously, and several other properties, but none that involve negation
00:33:52 oerjan: yeah it is
00:34:00 and since a b = a*b, a-b = a*-b
00:34:04 therefore a+(-b) = a*-b
00:34:05 :D
00:34:07 back
00:34:08 Q.E.D!
00:34:18 * oerjan swats alise -----###
00:34:20 Which property is the easiest to break?
00:34:25 none of them
00:34:32 otherwise someone will have already done so and formalised everything with it
00:34:40 you have to sacrifice something major, that's why they are fundamental properties
00:36:20 What causes |a-b| + |b-c| >= |a-c| to break, exactly?
00:36:27 (Given |@|=-1
00:36:29 )
00:36:48 nothing you just have to break other things to make it work :P
00:37:12 -!- FireFly has quit (Quit: http://www.qdb.us/60621).
00:37:30 How is that rule derived? That would be easiest to work with
00:38:09 a+0=a, 0*a = 0, a+(-a) = 0, (a+b)c = ac+bc, a+(b+c) = (a+b)+c, 1*a = a, 0+a = a
00:39:04 of those 0*a = 0 _might_ be derivable from something simpler, but the rest are really fundamental (you could drop one of the first and last by adding commutativity)
00:39:11 (a+b)c = ac+bc seems somewhat unintuitive. Maybe in the goal of -1*-1=-1, we can break it
00:39:27 heh in fact that means i'm _not_ using commutativity, not even of addition
00:40:36 Sgeo: the distributive law is pretty powerful yeah. but without it there isn't really any connection _between_ addition and multiplication to speak of.
00:40:42 You are educated stupid
00:41:02 Anyone want to email oerjan's proof to Gene Ray?
00:41:34 he denies the definitions duh
00:43:16 |@@| = |@||@| = 1; |a@+b@@| = |[a+b@]@| = -|a+b@| and this probably gets us nowhere
00:43:54 * Sgeo wonders what |1/@| is
00:44:24 |1/x| = 1/|x|
00:44:27 = 1/-1
00:44:39 = -1
00:44:48 |1/@| = -1
00:44:49 Can we be certain that that rule works given that |x|=-1 exists?
00:45:01 Unless you find a contradiction.
00:45:27 Sgeo: |a-b| + |b-c| >= |a-c| is equivalent to |x| + |y| >= |x+y|
00:45:43 btw
00:45:54 |@| + |n| >= |@+n|
00:45:57 earlier on I was using |z-w| as notation for d(z,w)
00:46:00 |@| + |1| >= |@+1|
00:46:08 |@+1| <= -1 + 1
00:46:11 |@+1| <= 0
00:46:12 d is just a function onto R(eal)
00:46:21 |@+1| = -2, perhaps?
00:46:26 you can prove that d(x,y) >= 0
00:46:31 I remember seeing |x|+|y| >= |x+y| before, and deciding that it was broken, I don't remember why
00:46:47 Sgeo, wormholes I guess
00:46:53 fax: plz respond in /msg :(
00:47:34 x=@, y=-@, |x| + |y| = |@| + |-@| = -2 >= |@+-@| = |2@| = -2
00:47:37 hm
00:47:56 erm, |@+-@| = |0| = 0
00:48:00 So yeah, contradiction
00:49:22 -!- BeholdMyGlory has quit (Remote host closed the connection).
00:52:02 CAN HAS SQL?
00:52:03 DBASE IZ GETDB('db/lcsn.db')
00:52:03 FUNNAHS IZ DBUCKET(&DBASE&,"CAN I PLZ GET * ALL UP IN lollit")
00:52:03 IM IN UR FUNNAHS ITZA TITLE
00:52:03 VOTEZ IZ &TITLE#ups& - &TITLE#downs&
00:52:04 http://imgur.com/noL0U
00:52:07 by what metric is lolcode esoteric again?
00:53:09 sgeo's metric above
00:53:14 It's deliberately strange.
00:53:25 I made a metric for esotericism?
00:54:00 you made a metric for *WHOOSH*
00:54:57 Hmm, @ such that |@| = -1.
00:55:30 You can define the absolute value of a number as the square root of the number times its complex conjugate.
00:55:37 |x| = sqrt(x*x')
00:56:46 Hm, was thinking of sqrt(x*x), but that doesn't work with non-reals. I think the above def. fixes that
00:57:31 well it _is_ the usual one for complex numbers
00:57:47 aka modulus
00:57:54 What's the complex conjugate for @?
00:58:01 And what's @*@?
00:58:32 If you define a complex number as a 2x2 matrix of real numbers, I'm pretty sure the absolute value is the square root of the determinant.
00:58:48 a+bi has complex conjugate a-bi
00:58:59 sqrt(a^2+b^2), of course.
00:59:13 |@| = sqrt(@*@) = @ is not what we want
00:59:22 So @' != @
00:59:31 If sqrt(a^2+b^2) = -1, then a^2+b^2 is an imaginary number.
00:59:35 Assuming we're following that definition, which might not even be a useful one
01:00:00 @' is a unique number such that @*@' = -1^2
01:00:03 = -1
01:00:04 For simplicity, let's assume b = 0. Then a^2 is an imaginary number.
01:00:25 er wait
01:00:25 = 1
01:00:36 hmm now that is a bit of an issue
01:00:47 Why is it a bit of an issue?
01:01:00 @' = 1/@.
01:01:45 Lemme think, the numbers we're familiar with are of the form a + bi. These numbers could simply be a + bi + c@ + di@.
01:02:02 I wonder if this is a quaternion.
01:02:14 Rather, if this is the quaternions.
01:02:24 yes but
01:02:34 If there are usual rules for abs value for quaternions, then those are broken by this
01:02:38 @*@' = -1^2 (because sqrt(@*@')=-1) = 1
01:02:45 but sqrt(1) is actually 1
01:03:17 Heh, right.
01:03:31 We'll have to find a way to decide when sqrt(x) is sqrt(x) and when it's -sqrt(x) instead.
01:04:59 Can't you get a similar paradox without @?
01:05:11 Hey, I have heard something that might accomplish this.
01:06:00 Heard of, rather.
01:07:48 No, I haven't.
01:08:16 It just seems like absolute values are too fundamentally nonnegative.
01:08:57 don't be so nonnegative
01:09:22 oh!
01:09:32 minkowsky inner product!
01:09:33 If we don't find a way to solve this impossible problem, surely humanity will be destroyed!
01:09:40 That's precisely the thing I mentioned.
01:09:59 Or, well, something with the word Minkowski in it.
01:10:13 it has _imaginary_ absolute values, not negative ones, though
01:10:26 (from special relativity theory)
01:10:40 It has imaginary absolute values?
01:10:49 but their _square_ is negative though
01:10:53 This is not quite related, but for the usual 2x2 matrix representation of (a+bi), [a -b; b a], you do have |a+bi|^2 = det([a -b; b a]) = a^2+b^2... if you do a+b% as [a b; b a] you'd have a fancy number % for which |%| = i. (Since |%|^2 = det([0 1; 1 0]) = -1.) That sort of representation would seem to be sort-of closed under addition and multiplication.
01:11:15 uorygl: yes
01:11:52 basically a distance in time has a length that is an imaginary number times a distance in space
01:12:16 (for combinations of time/space, it depends on whether it's faster than light or not)
01:12:26 Hmm. Find a shape that's like a cone, except that one vertical cross section is a V and a perpendicular one is an upside-down V.
01:12:40 * uorygl ponders hyperconic sections.
01:12:56 Or whatever the heck those things are called.
01:13:48 Quadric sections.
01:14:17 hyperboloids?
01:14:28 That is a type of quadric section.
01:15:00 two opposed cones would be a degenerate instance of that
01:15:25 * Sgeo lost the conversation
01:15:26 oh wait that's not what you said
01:15:27 -!- nooga has quit (Ping timeout: 245 seconds).
01:15:34 There is no quadric section, not even a degenerate one, that's that I said.
01:15:52 uorygl: yours is like a degenerate saddle, more like?
01:16:01 Yes.
01:16:27 You know, I'm pretty sure there's only one sane shape that's like that.
01:17:12 Is there a way to define @ in terms of imaginary abs. values?
01:17:14 -!- nooga has joined.
01:17:16 you are sure there's at least one?
01:17:42 What makes a shape sane
01:17:45 ?
01:18:00 Take the absolute value, and then multiply it by the square of the sine of the argument.
01:18:09 s/sine/cosine/
01:18:18 hm actually for timelike distances in minkovski space there _is_ a difference of direction, between forwards and backwards in time
01:18:22 Sgeo: sanity.
01:18:39 The absolute value is actually an insane function, by one formal definition.
01:18:43 * Sgeo is utterly lost
01:19:20 Sgeo: I'm guessing you understand all but one word of "take the absolute value, and then multiply it by the square of the sine of the argument".
01:19:49 BRB
01:21:28 hm a saddle surface is defined to be smooth on wikipedia, so wouldn't include your case, or would it?
01:22:03 no it cannot be smooth, since absolute values are cross sections
01:22:07 Any smooth function would make a silly absolute value.
01:22:25 Since it has a non-differentiable spot at zero.
01:23:03 And |(|x|)| should be |x|, not |x|^2.
01:23:28 What is the absolute value of (, anyway
01:23:47 Gregor: you forgot to close a parenthesis.
01:24:10 i'll close it here )
01:24:39 I don't know what the absolute value of (, anyway i'll close it here) is.
01:24:48 JOKE MURDERED
01:24:55 I understand the words, but not the point
01:25:15 Sgeo: well, visualize that shape.
01:25:38 Can I just plot it on Wolfram?
01:25:54 Probably.
01:25:57 -!- ubunt63 has joined.
01:26:39 but does he _want_ such a tattoo?
01:27:07 I don't know whether Wolfram wants such a tattoo.
01:27:16 I guess even if he had it, we wouldn't be able to see it.
01:28:22 Sgeo: I'm guessing you understand all but one word of "take the absolute value, and then multiply it by the square of the sine of the argument".
01:28:27 which word doesn't he understand?
01:28:37 Sgeo: http://www.wolframalpha.com/input/?i=plot+abs%28x%2By*i%29*cos%5E2%28arg%28x%2By*i%29%29
01:28:38 hm why would you square it?
01:28:49 you'd want it negative sometimes
01:29:13 Trigonometry fail on my part.
01:29:23 Take the absolute value, and then multiply it by twice the cosine of the argument.
01:29:26 Er.
01:29:29 Idle speculation: do you get some sort of sensible structure from matrices of the form [a b; b a], with the usual matrix addition and multiplication? It won't be a field, since there's no multiplicative inverse if a=b there, but is it isomorphic to some other well-known thing?
01:29:34 Take the absolute value, and then multiply it by the cosine of twice the argument.
01:29:55 uorygl: neither works since they don't fulfil f(-x) = f(x)
01:30:17 This doesn't fulfill that?
01:30:26 didn't see your last one
01:30:34 Sgeo: http://www.wolframalpha.com/input/?i=plot+abs%28x%2By*i%29*cos%282*arg%28x%2By*i%29%29
01:30:39 my workbench is almost complete. all it needs is a work surface. :D
01:30:55 fizzie: I think there's a name for those.
01:31:07 win 12
01:31:14 arg?
01:31:23 fizzie: the split-complex numbers.
01:31:33 Sgeo: the argument function. It returns the angle of the Argand diagram.
01:31:43 It does the same sort of thing as the signum function.
01:31:44 * Sgeo mindboggles
01:31:57 So, in a sense, it's the angle of the complex number.
01:32:01 * Sgeo is completely clueless as to what Warrig.. oh
01:32:18 hm wait maybe i'm the one failing trigonometry here
01:32:29 The argument of 1 is 0, the argument of i is pi/2, the argument of -1 is pi, and so on.
01:32:31 * Sgeo will just stay lost
01:32:37 oerjan: but I also failed trigonometry.
01:33:30 fizzie: it's isomorphic to the ring of 2*2 matrices ;D
01:34:14 oh wait now i'm failing reading too
01:35:13 Yes, it seems they've called split-complex numbers indeed.
01:36:28 (And a huge pile of other names.)
01:37:19 It's fun to have both the complex numbers and the split-complex numbers.
01:37:31 You have both i and j such that i^2 = -1 and j^2 = 1.
01:37:49 Then ij = -ji.
01:37:57 Darn, we've lost commutativity.
01:40:08 also
01:40:09 oh man
01:40:21