Some of the complex numbers are rational also.
No numbers are more real than others, they are all abstract concepts. Some of them happen to correspond more easily to how we percieve reality.
As for integers, what’s real about 0 or negative numbers. Some mathematician said (memory quote here), “God invented the natural numbers, man the rest.”
Variously translated, as, for example: “God made the integers, all else is the work of man.” Although, “whole numbers” or “natural numbers” would be a more accurate translation than “integers.”
Yeah, the rationals themselves.
Most of the responses are pretty good, but one thing that’s been left out is that i is just a square root of -1. Rigorously, one says “pick a square root of -1 and call it i”. There’s nothing to stop you from adjoining as many square roots of -1 as you want. Whether it’s useful or not is another matter…
A better explanation would be that imaginary numbers are purely a mathematical convenience. That is, they make many types of computations a great deal simpler.
Real numbers, on the other hand, can correspond to real-world objects. You can have two apples, or three apples, or even 3.426 apples. However, you can’t have i apples. That’s why these numbers are called “imaginary.”
More specifically, a 3D vector or a triple.
Tell that to Ahronov and Bohm.
Can you be more specific? As far as I can tell, they use these numbers as mathematical conveniences as well. I doubt that Bohm would every declare, “There are 2i particles here!”
What if you wanted to program a 16-nion matrix?
I’d figure that would provide some interesting insights into M-Theory.
The Ahronov-Bohm effect. The electromagnetic 4-potential is described by a complex 4-vector. The A-B effect shows that the phase has a real physical significance.
Basically, if you run the double-slit experiment and put a solenoid between the slits so that there’s a nonvanishing (complex) 4-potential outside the solenoid, but the e/m fields vanish there, the interference pattern moves to the side.
True. I was simply trying to make my answer apply to n dimensions, though, and to avoid the inevitable: But what about 4 dimensions?
The counter to this argument is that you’re being limited by human “common sense” perspective. It’s like believing that the earth is flat or that we are at the center of the universe with the sun revolving around us. That may be everyday “naive” experience but a more sophisticated view of the universe refutes it.
You should see a doctor immediately.
A note: It’s easy to construct an n-dimensional space with a well-defined definition of addition. The complex numbers are an example in two dimensions, and the ordinary 3-d vector spaces one frequently encounters are a 3-d example. Multiplication, however, is not so straightforward. In general, there will not be a natural multiplication operation to take two n-dimensional vectors and multiply them to get another n-dimensional vector. There is such an operation on the complex numbers (as well as the quaternions), but it’s not completely natural, since it renders the space inherently anisotropic (I presume this is also the case for the octonions, but I never could make heads nor tails of them… It doesn’t seem like there’s enough structure left to figure anything out). For three dimensions, there’s no natural multiplcation at all which gives you back a three-vector (yes, I know about the cross product, but a cross product doesn’t actually live in the same vector space as the original vectors). So a vector space is not quite really a generalization of complex numbers to higher dimension.
The octionions are constructed from the quaternions in exactly the same way that the quaternions are constructed from C. In particular, multiplication is defined in the same manner.
To expand on what Chronos said, it’s actually very easy to come up with multiplication over an arbitrary vector space. For R[sup]3[/sup], define (a[sub]0[/sub], a[sub]1[/sub], a[sub]2[/sub]) * (b[sub]0[/sub], b[sub]1[/sub], b[sub]2[/sub]) as (a[sub]0[/sub]b[sub]0[/sub], a[sub]1[/sub]b[sub]1[/sub], a[sub]2[/sub]b[sub]2[/sub]) and you’re good to go.
The trick is to come up with multiplications that have nice algebraic properties. That one’s associative and commutative, but it’s got a slight flaw, which I’ll leave to the reader to discover. Hint: what’s the product of (1, 2, 0), (1, 0, 3) and (0, 2, 3)?
You can have i apples.
Make an array of apples, i by i. Now add two more. Voila! You have an apple!
I suspect that this is right (save for the “ocasionally”). People earlier must have speculated about expanding the number system in order to solve x^2 + 1 = 0, but the solution inevitably involves complex numbers (that is, sums of real and imaginary). A cubic can one real root and two complex roots or three real roots. There can be multiple roots, but I will ignore that case. If it has just one real root, my recollection is that you can solve for that one without involving complex numbers at all. But the interesting case is that of three real roots. In that case complex numbers inevitably arise in the course of the solution and there is no way to avoid them. Even for a polynomial like x^3 - x = 0, whose roots are obvious by inspection, if you use Cardan’s formula (or any other) to solve it, complex numbers must arise at an intermediate step. So yes it is quite reasonable that that must have been a powerful force pushing the complex numbers into the forefront of mathematics. Since Tartaglia’s formula for the fourth degree polynomial involves solving an auxiliary cubic, that will also require complex numbers.
The original vectors are members of R[sup]3[/sup], as is their cross product. How is that not part of the same vector space? I see the product isn’t orthogonal to the original two vectors, so it wouldn’t be in the subspace spanned by the original vectors, is that what you mean? I don’t see how that makes in not a “natural multiplication”.
Yes, it has real physical significance. Nobody denies that. That’s not the issue at hand, though.
As I said, imaginary numbers do simplify a great many computations. They are used in phasor multiplication, for example, in which they help establish a “phase” factor. This does not mean that they directly represent any real-world quantities, though. Real-world measurements require real numbers, but the subsequent computations frequently employ imaginary numbers – again, as a matter of simplification and convenience.