## Crazy numbers

For the discussion of math. Duh.

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### Crazy numbers

Hey guys. The other day in math class I invented "crazy numbers". They are numbers whose absulote values are less than zero. I was inspired by imaginary numbers. Any thoughts?
mcmesher

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### Re: Crazy numbers

Well for one thing it's not actually an "absolute value" if it doesn't start at zero.

Beyond that, we need more information on these numbers before we can offer any thoughts on them.
In the future, there will be a global network of billions of adding machines.... One of the primary uses of this network will be to transport moving pictures of lesbian sex by pretending they are made out of numbers.
Spoiler:
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gmalivuk
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### Re: Crazy numbers

Well, by definition, the absolute value of a number is the distance that number is from 0.
I doubt you can have a negative absolute value anymore than you can have negative speed.

EDIT: Wait, speed is defined as the absolute value of velocity, so I guess that's kind of redundant.
Arariel

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### Re: Crazy numbers

There is an algebra called the split-complex numbers that you might be interested in.
++$_ Mo' Money Posts: 2370 Joined: Thu Nov 01, 2007 4:06 am UTC ### Re: Crazy numbers Ooooh, those are synonymous with the perplex numbers. Very fun stuff. Many of the properties of the surreal numbers also hold when you disregard the restriction that the elements of the left set must be less than all the elements of the right set. Conway calls these objects 'games', although Knuth just treated them as even stranger numbers, of a sort, which were still in line with the other axioms. I came here to read a cool post, a witty dialogue, a fresh joke, but stumbled upon a "bump"... Way to go, jerk... ~CordlessPen GyRo567 Posts: 107 Joined: Sun Jun 01, 2008 4:59 am UTC Location: Austin, TX ### Re: Crazy numbers Negative "distances" show up in relativity: https://secure.wikimedia.org/wikipedia/ ... wski_space One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total. Yakk Posts: 10064 Joined: Sat Jan 27, 2007 7:27 pm UTC Location: E pur si muove ### Re: Crazy numbers It could be fun to just experiment with these. Let us begin with the crazy number c, which is defined by |c|=-1. In the same way, since for any real number n |n|=|-n|, we may also say that |-c|=-1, because if |-c|=1 than -c would equal -1. Likewise, |2c|=-2; |3c|=-3, etc. Now, in the real numbers, |n|2=n2. This cannot be true of the crazy numbers, where given a coefficient k |kc|2= k2. There are therefore two possibilities. 1. Those two statements are not mutually exclusive, and (kc)2=k2. 2. They are mutually exclusive; the easiest (but by no means the correct) interpretation of (kc)2=k2c. For extra credit, for a given number ℝ, find logecℝ. Come join Dadapedia- the open-source Dadaist novel that anyone can edit. dhokarena56 Posts: 179 Joined: Fri Mar 27, 2009 11:52 pm UTC ### Re: Crazy numbers Yakk wrote:Negative "distances" show up in relativity: https://secure.wikimedia.org/wikipedia/en/wiki/Minkowski_space This. A couple years ago I had an obsession with figuring out how time travel would work(which obviously it didn't...), and these "crazy numbers" came up a lot, especially when I tried to calculate the math of going forwards in time.(The math of going into the past is a lot simpler...) nicomon Posts: 2 Joined: Sat Nov 13, 2010 2:53 am UTC ### Re: Crazy numbers nicomon wrote: Yakk wrote:Negative "distances" show up in relativity: https://secure.wikimedia.org/wikipedia/ ... wski_space This. A couple years ago I had an obsession with figuring out how time travel would work(which obviously it didn't...), and these "crazy numbers" came up a lot, especially when I tried to calculate the math of going forwards in time.(The math of going into the past is a lot simpler...) I would think the math of going forwards in time would be simpler, since it is possible, while going backwards in time is not. I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side. "With math, all things are possible." —Rebecca Watson skeptical scientist closed-minded spiritualist Posts: 5920 Joined: Tue Nov 28, 2006 6:09 am UTC Location: Madison, Wisconsin ### Re: Crazy numbers Between my writing this and your reading it, we all have gone forward in time without having to do any math at all! In the future, there will be a global network of billions of adding machines.... One of the primary uses of this network will be to transport moving pictures of lesbian sex by pretending they are made out of numbers. Spoiler: gmss1 gmss2 gmalivuk Archduke Vendredi of Skellington the Third, Esquire Posts: 19438 Joined: Wed Feb 28, 2007 6:02 pm UTC Location: Here, There, Everywhere (near Boston, anyway) ### Re: Crazy numbers My mistake, I should have been more specific. I meant "accelerating" forwards in time, past the natural "speed".(This feels worded incorrectly, but I can't think of better wording.) nicomon Posts: 2 Joined: Sat Nov 13, 2010 2:53 am UTC ### Re: Crazy numbers I'm keeping to the notation of having c be the crazy unit. I'll assume we're starting with the real numbers and formally adjoining c to get a new ring. in the crazy numbers we'll have to accept one of three things: either * the inequality ||x| -|y|| <= |x + y| <= |x| + |y| no longer holds in the crazy extension, or * there are nonzero numbers x + xc for which |x + xc| = 0, or * in fact x + xc = 0 for any real number x, meaning subtraction is not well defined. I can't decide which of these is the best to accept, but I'm pretty sure it's not the last. http://polytopal.wordpress.com --- a continuous injection of pure mathematics SunAvatar Posts: 200 Joined: Sat Dec 15, 2007 3:36 pm UTC Location: Dover, NH ### Re: Crazy numbers dhokarena56 wrote:if |-c|=1 than -c would equal -1. This does not follow. Small Government Liberal Qaanol Posts: 2258 Joined: Sat May 09, 2009 11:55 pm UTC ### Re: Crazy numbers What is |1 + c|? Mike_Bson Posts: 252 Joined: Mon Jul 12, 2010 12:00 pm UTC ### Re: Crazy numbers I wonder if you can create such an algebra using matrices. The determinant of matrices can positive or negative. Let 1 = ((1,0),(0,1)) [the 2x2 unit matrix] and c = ((0,1),(1,0)). Define a crazy number as z = a * 1 + b * c, where a is the real part, and b is the crazy part. Now identify an ordinary number x with Sqrt[x] * 1. And define the absolute value of a crazy number z as Det[z]. Now for any real number x we have |x| as a crazxy number is |x| as a real number. And |c| = -1. That's what we wanted isn't it? This does mean that |1 + c| = 0. That's rather strange edit: I forgot to mention. Many properties of absolute values remain intact using the above definitions. Most importantly |z1 * z2| = |z1| * |z2|. It's one of those irregular verbs, isn't it? I have an independent mind, you are an eccentric, he is round the twist - Bernard Woolley in Yes, Prime Minister Diadem Posts: 4103 Joined: Wed Jun 11, 2008 11:03 am UTC Location: The Netherlands ### Re: Crazy numbers Diadem wrote:This does mean that |1 + c| = 0. That's rather strange That is what I was conjecturing. Thanks for the proof, though, I like your definition of these. Mike_Bson Posts: 252 Joined: Mon Jul 12, 2010 12:00 pm UTC ### Re: Crazy numbers What about complex-crazy numbers of the form a+bc+di where a is the real part, b the crazy part and d the imaginary part. Would they have any interesting properties? What would their absolute value be, or their square (generally of course)? Gear wrote:I'm not sure if it would be possible to constantly eat enough chocolate to maintain raptor toxicity without killing oneself. eSOANEM Posts: 2088 Joined: Sun Apr 12, 2009 9:39 pm UTC Location: Grantebrycge ### Re: Crazy numbers Diadem wrote:<snip> What you've described here is the split complex numbers, mentioned earlier by ++$_.

eSOANEM wrote:What about complex-crazy numbers of the form a+bc+di where a is the real part, b the crazy part and d the imaginary part. Would they have any interesting properties? What would their absolute value be, or their square (generally of course)?

You'd need to describe what c*i and i*c would be. The easiest way to get it all happening is to let

1=\begin{pmatrix} 1 & 0 \\ 0 &1 \end{pmatrix}, \ c=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \ \text{ and } i = \begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}.

If you take the closure of that set under products and sums, you get the 2x2 real matrices. Non commutative, zero divisors, multiple idempotents etc. They're a fun set of things, and well worth familiarising yourself with if you ever feel inclined. Not a gigantic leap forward though, just stuff other people have had a good long look at.
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