We recently got our hands on the symbol font, activated using the [sym] and [/sym] tags, so now we can do this:
e[sup]i[sym]p[/sym][/sup] + 1 = 0
Anyway, the key to imaginary numbers is to realize that different sorts of numbers have different uses. If I’m a shepherd counting my flock, I just need positive integers: I have 100 sheep. If I find myself with 99 and a half, then that means that the wolves got to one of them, and I only have 99 live sheep. Now, maybe, I settle down and start growing grain. I’m not about to count all the seeds, so I measure them in bushels: Now, 99.5 bushels is worse than 100 bushels, but it’s still better than 99. Fractions are now useful. Well, civilization advances some more, and I need to go to a banker to get a loan to plant my first crop. The banker approves me for a loan of 100 sheckles… Which means that I’m now in debt. Negative numbers are suddenly useful. Civilization advances some more, and I decide that since I was a failure as a farmer, I’ll take up physics or engineering. And all of a sudden, I’ve got imaginary impedences or wave amplitudes or spacetime intervals. An imaginary number of sheep is still meaningless, of course, but then, so is a negative or fractional number of sheep.
I think the real question is, if complex numbers are represented in two dimensions with real being the x-axis and imaginary being the y-axis. What could the z-axis represent?
I realize that the z-axis doesn’t need to represent something. But, if the y-axis represents a set of numbers that cannot be represented by the x-axis, can a third set of numbers be found that cannot be represented by either the x or y axes be found? I.E. Does there exist (or can there be found) a set of numbers that is not a subset of the complex numbers?
Yeah, sure. There’s lots of 'em. Problem is, you lose some nice feature of arithmetic if you attempt to move beyond the complex numbers–the most common problem is that multiplication becomes non-commutative (i.e., a * b isn’t necessarily equal to b * a).
There is a slight technicality here, but I’m not going to explain it unless someone asks about it.
There’s also not as much incentive to use other number systems, since the complex numbers (unlike the reals) are complete. That means that any equation you can write with complex numbers, you can solve with complex numbers. In the reals, equations like ln(x) = -1, sqrt(x) = -1, or sin(x) = 2 don’t have solutions, but in the complex numbers, they all do. No matter what complex numbers you stick in, you’ll always be able to get the equations to work.
GulDan: of course there’s a set of numbers like that. Every set that you think of can be used as a set of numbers. The question is whether it is useful to do so.
Actually, the reals are complete (every Cauchy sequence is convergent). What they aren’t is closed under algebraic operations.
If the complex numbers is truly closed under all algerbraic manipulation, does it bother anyone else that we have finally hit the end of the road numerically? For many years negative and complex numbers weren’t considered (for that matter, so were irrational numbers) as existing in any sense of the word. Perhaps there is an underlying assumption, much like Euclids 5th postulate, that we are working under, preventing us from seeing the next catagory of numbers. And yes, I would like information on why mult. would necessarily be no longer commutative.
But we haven’t hit the end of the road in terms of number systems. Get yourself a good book on abstract algebra and you’ll see all the stuff that’s left to do. FWIW, the proof that any algebraic extension of C (that’s the field of complex numbers) loses some properties of arithmetic is fairly advanced, so don’t hold your breath on that one. And if some kind mathematician wanted to give an exact statement of that theorem, I’d appreciate it.
On the other hand, there are a couple other directions with new numbers. Do a google search on any of the following: transfinite cardinals, transfinite ordinals, surreal numbers, hyperreal numbers, supernatural numbers. Just be prepared to invest some time–some of these are very difficult to get your mind around.
Not sure if this is what you’re thinking of, but I would word the theorem as “There is no algebraic extension field of the complex numbers”.
This is an immediate consequence of the complex numbers being algebraically closed.
On the other hand, there are transcendental extension fields of the complex numbers; one example would be the set of rational functions with complex coefficients.
Well, I don’t think we should get hung up on the semantics. A lot of numbers have names that aren’t particularly complimentary; number can be unnatural, negative, irrational, or imaginary. Fractions can be improper. Let’s just not try to apply some social standards of politeness here. Imaginary numbers are not real. They may be relevant, useful, logical, and accepted, but they cannot be real. The set of real numbers is defined, quite specifically, as excluding all imaginary numbers.
Let’s just no go thru too many definitions of the word “real” in a single discussion.
By definition they are not Real. Again, I’d like to keep to one definition per word. If you’d like to apply synonyms of real in its non-mathematical senses (like genuine, true, or actual) to the Set of Imaginary Numbers, that is fine by me. I really have nothing against these numbers. They do, however, lie entirely outside the Set of Real Numbers, and I’d like to keep to the mathematical definition of the word “real” at least on this thread.
OK, I see what you’re saying, and yes, complex numbers aren’t real numbers. FWIW, the reals aren’t defined to exclude imaginaries, but it’s pretty easy to show that they do (i.e., there is no real number whose square is -1). I was thinking in a completely different sense.
