If you think of real numbers being represented by the x axis and complex numbers being represented by the (x,y) plane. Is the a set of numbers that are represented (x,y,z) ?
There are Quaternions as an extension of complex numbers, but they are four-dimensional.
I’ve never used them, but you might like complex manifolds such as Riemann spheres.
No, because on the complex plane, numbers have both a real and a complex component. Your (x, y, z) lists real numbers twice, but on the complex plane, there is only one real component, not two.
Well, there are 3-D vectors, just like 4-D vectors or any other number of dimensions you want.
But they’re not really that much like complex numbers. The main idea behind complex numbers is that mathematicians realized that real numbers were kind of broken. Using just simple operations like multiplying and subtracting, you could write things like X*X+1 = 0, for which there’s no real X that works. That bothers mathematicians, so they created rules for complex numbers that would fix this problem. And it turns out using these complex numbers makes it much easier to solve a lot of math problems, so they’re used a lot.
With complex numbers, there always is an X that works for any equation of this kind. (A mathematician would say that complex numbers are ‘complete’ where reals numbers aren’t). Since complex numbers aren’t broken in the way that reals are, there’s no reason to create rules for any kind of ‘extra-complex’ number involving triplets.
There’s nothing stopping you from coming up with some kind of ‘hyper-complex’ number system involving triplets (and I’m sure people have), but it wouldn’t be either particularly useful, or theoretically interesting to mathematicians, so nobody would use it.
While “hypercomplex” may not be useful, I don’t really see how complex numbers are more complete than reals. Using nothing more than multiplication and addition I can write things like X*X + i = 0, for which there’s no complex X that works.
X = 0.5^0.5 - (0.5^0.5 * i)
or = - 0.5^0.5 + (0.5^0.5 * i)
i.e.,
approximately 0.707107 - 0.707107 i
or approximately - 0.707107 + 0.707107 i
Thanks for posting this. Despite studying complex numbers in school, none of my teachers ever explained the rationale behind them. But what you say makes perfect sense.
Don’t feel bad, IMO almost all teachers and texts teach it traditionally in a very obfuscated way. The problem starts early in high school algebra where it gets taught as some crazy exception or “just 'cause” process to accept now and worry about later.
It took me until late into grad school in EE before I had my “aha!” moment. I had a prof in one of my complex analysis classes dedicate a 3-hour lecture at the beginning of the class term and restart at ground zero fundamentals, reteaching just what the hell an imaginary number was.
The kicker was that it was his own lecture, literally starting at “Now what does it mean when we say 1+1=2?” and going from there using matrix representations and linear algebra reworkings, that I had never seen from any teacher or textbook to that point (and also wasn’t To date, all of my previous math, physics, engineering coursework used the traditional “x-y plane” approach and it just never clicked for me.
As Giles points out, you can easily solve this equation in the complex numbers; thinking of i as a 90 degree turn, the two possible answers are a 45 degree turn and a 225 degree turn (or, if you insist on breaking it down into parallel and perpendicular components, the answer is (1 + i)/sqrt(2), where sqrt(2) can be taken as either positive or negative).
As for the OP: the main reason for the difficulty in extending complex numbers to three-dimensions, as I see it, is this: in the two dimensional plane, given any vector v and any non-zero vector u, there is a unique combination of rotation and scaling (i.e., a unique complex number) that sends u to v. However, in three-dimensional space, this is no longer true.
I don’t follow. Take the plane defined by u and v (assuming they’re not parallel, but then it’s just scaling without rotation), and then there’s only one rotation within that plane that’ll take u to v. You need a little more sophistication in how you express that rotation, though, and that would probably take something of higher dimension than a “hypercomplex number”.
If by “numbers” you mean entities that obey all the standard laws of algebra the way complex numbers do, the answer is No.
As Giles mentioned, there are the “quaternions” in four dimensions, but you have to give up commutativity of multiplication.
One explanation of this can be found in Chapter 6 of the book Yearning for the Impossible: The Surprising Truths of Mathematics—see especially section 6.3: “Why n-tuples Are Unlike Numbers when n ≥ 3.” I’ve been trying to digest it and boil it down to a brief summary I could contribute here, but I have been unable to (beyond saying that the assumpion that there are three numbers, all 1 unit away from 0 in mutually perpendicular directions, the way 1 and i are in the complex plane, leads to a contradiction).
More specifically: in two-dimensions, combinations of rotation and scaling are closed under addition, as a consequence of the property I mentioned above (for example, scale by 1/sqrt(2) + [scale by 1/sqrt(2) and rotate 90 degrees] = rotate 45 degrees; for any vector in two-dimensional space, carrying out the operation on the left is equal to carrying out the operation on the right). However, in three-dimensions, this falls apart (for example, scale by 1 (i.e., keep unchanged) + rotate 180 degrees around z-axis has the effect of sending a vector <x, y, z> to <0, 0, 2z>; this is not a combination of rotation and scaling; it’s a projection).
You can bite the bullet and move to the full ring of linear operators on three-dimensional space, in all its noncommutative, division-lacking glory. But there aren’t any nice nontrivial subalgebras of this. (In the four-dimensional case, there re-arise nice subalgebras, such as the quaternions, as mentioned above)
Consider: there are infinitely many rotations which take u to u itself, the various rotations around the axis defined by u. Composing these with any rotation taking u to v gives infinitely many rotations from u to v, as well. So there is not a unique rotation sending u to v; indeed, three-dimensional vector space does not arise as a torsor over the multiplicative action of its linear operators, as opposed to the two-dimensional case, where we find a nice calculus of ratios of vectors (i.e., complex numbers).
Ah, true. I can define a privileged rotation from u to v, but the non-privileged rotations would still have to exist, so there has to be some number (or a whole slew of them) representing rotation about the axis of u, and any one of those would act like an identity when multiplied by u. And I don’t remember the details, but multiple identities do cause some significant problems in trying to construct an algebra.
Yup. Greg may have meant “there is no purely imaginary x that works”, which is true, but the amusing thing is that the square root of an imaginary number is partly real.
Now, what’s really interesting is that i[sup]i[/sup] is a purely-real number.
Of course, this is just the observation that turning a vector half of 90 degrees does not move it fully perpendicular to its original starting position.
Yes, although, exponentiation with a complex base and non-integer exponent is not unambiguously defined, so actually, it turns out i[sup]i[/sup] is infinitely many different real numbers.
(Specifically, i[sup]i[/sup] is e[sup]-(z + 1/4)2π[/sup] for any integer z, corresponding directly to how the base i itself can be expressed as a turn through (z + 1/4)2π radians (the integer z therefore specifying how many full revolutions around the circle one takes in addition to the characteristic 90 degree turn))
“Algebraically closed”, right? The reals are complete (the rationals aren’t).