Crazy numbers

[mcmesher]

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?

[gmalivuk]

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.

[Arariel]

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.

[++$_]

There is an algebra called the split-complex numbers that you might be interested in.

[GyRo567]

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.

[Yakk]

Negative "distances" show up in relativity:
https://secure.wikimedia.org/wikipedia/ ... wski_space

[dhokarena56]

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 log_ecℝ.

[nicomon]

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...)

[skeptical scientist]

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.

[gmalivuk]

Between my writing this and your reading it, we all have gone forward in time without having to do any math at all!

[nicomon]

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.)

[SunAvatar]

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.

[Qaanol]

if |-c|=1 than -c would equal -1.

This does not follow.

[Mike_Bson]

What is |1 + c|?

[Diadem]

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|.

[Mike_Bson]

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.

[eSOANEM]

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)?

[jestingrabbit]

<snip>

What you've described here is the split complex numbers, mentioned earlier by ++$_.

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.