Originally posted by adam warlock Are you talking about this?: [b]the vector form of a quaternion may also be used. This form assumes that \vec{A} \equiv A_x\mathbf i + A_y\mathbf j + A_z\mathbf k. in that case the i,j,k are a notation to indicate the three coordinate axes and to explain how quaternions can be represented in that way. But don't you confuse a representation of an ob ...[text shortened]... you read the informal introduction, and bare in mind the word informal, you can see that.[/b]
No-the Quaternian was created before vectors, and were used to do what vectors do, just less well.
The i, j and k in vectors nowadays are unit verctors, with the same properties as the quarternions.
And, you're right Adam, the number i is essential in QM. In QM you constantly encounter complex quatities (probabely more often than non-complex quantities.
Originally posted by FabianFnas [b]But if you have say, 2+3i v 4+3i, why isn't the second value larger?
or 2 + 3i vi 2 + 4i, why isn't the second example larger?
Which one is larger? 4+3i or 3+4i? Or this one: 1 or i? In fact, there is no ordering property among the complex numbers. Only equal or non equal.
... could there be further second generation complexities at 90 d ...[text shortened]... y the matematicians because there is other ways to solve those problems with standard methods.
I was thinking about that no complex number being greater than another concept, are you talking about the fact that the complex number line isn't really a line but just another set of numbers tacked on to the number line and there is only that one place, so the complex number point is just that, a point and not a real number line that happens to lie at 90 degrees away?
If it was another number line but going off into another dimension it could extend to infinity + and -. So I guess that points out the complex number as having a unity value. But the numerical part can extend to infinity. I think I am still confused about that.
Originally posted by sonhouse THAT I know: Complex numbers are inherently involved with alternating current and RF, you have to use complex #'s to solve problems in current flow and voltages, say on an RF open wire feedline for instance, the current sine wave and the voltage sin waves are not in sync and require complex #'s to solve the real energy exchange, absorption or emission of RF ...[text shortened]... ameters into Mathcad or other software pacs and don't have to do the math by hand any more.
This interpretation is not correct for the use of complex numbers in electronics. Complex numbers are used in electronics purely as a mathematical simplification, to eliminate the need for more difficult calculations involving the sine and cosine functions.
Originally posted by mtthw i actually appears in Schrödinger's equation, suggesting it's pretty fundamental in QM.
But then again (putting my ignorance on display for all to admire) isn't that just another equation describing sine/cosine wave behavior? I would expect then, that the use of complex numbers is once again just a mathematical nicety.
Originally posted by leisurelysloth But then again (putting my ignorance on display for all to admire) isn't that just another equation describing sine/cosine wave behavior?
It depends on the potential. For some potentials solutions to the schrodinger are standing waves, sines and cosines, but for other potentials we can have more crazy stuff happening.
In quantum mechanics dynamical quantities are represented by operators and the momentum operator (in the coordinate representation) comes with a i on it. Just like that without nobody asking or expecting it. But this is not the final word. A much more compelling and strong argument for the necessity of complex numbers on QM is given on a Sakurai book. And he comes to that conclusion by only looking at sequential Stern-Gerlach types of experiment. The thing is that after a few of those experiments he comes to the conclusion that the set of real numbers isn't enough to describe physically all that is happening. So we need a new set of numbers to fully describe reality and on that set of numbers there is one number that when multiplied by itself must equal -1.
http://www.amazon.com/Modern-Quantum-Mechanics-2nd-Sakurai/dp/0201539292
If you can get this book and just read this part I advice you to do it cause it is very instructive.
Originally posted by adam warlock It depends on the potential. For some potentials solutions to the schrodinger are standing waves, sines and cosines, but for other potentials we can have more crazy stuff happening.
In quantum mechanics dynamical quantities are represented by operators and the momentum operator (in the coordinate representation) comes with a i on it. Just like that wit ...[text shortened]... u can get this book and just read this part I advice you to do it cause it is very instructive.
Square root of minus one
30 Oct '07 05:30
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