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#1




Practical application of imaginary numbers.
Ever since i has been introduced to me, I have been thoubled with the concept of it. It never really stuck with me, and never really made sense to me. Anyway, what is the practical application, in the real world, of i? How can it actually do anything, or be applied to actual life, if it is imaginary? It is a nonexistant number, and yet it is used to prove math equations.
Huh? 
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#2




Practical uses of complex numbers is wide and it simplifies many things :
1> Evaluating complex Integrals. 2> Fourier transforms (Big daddy of anything to do with waves , music MP3s, Pictures and digital movies, Cat Scans, NMR, anything u can imagine of pretty much on those lines) 3> Geometry to an extent 4> All kinds of transforms 5> Control system design (even Artificial Intelligence) hope that helps 
#3




They're used in electrical engineering to descrive how circuits operate inside. Without them, we couldn't design even an AM radio. Do a Googlesearch on "impedance".
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#4




Stephen Hawking thinks that there may be an imaginary dimension to time! From his public lecture The Beginning of Time there is this:
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#5




Imaginary numbers are also used in regular, mainstream quantum mechanics: The Schrodinger wave function has imaginary numbers in it.
Imaginary numbers aren't imaginary like the Invisible Pink Unicorn, any more than irrational numbers are ditsy and confused. They're as metaphysically real as any other number. 
#6




Quote:
Anyway, imaginary/complex numbers are also used to derive a lot of trig equations (remember sin^2 + cos^2 = 1?, that's derived using De Moivre's Formula which links sin, cos and the exponential function). Basiclly, it's useful if a proof involves something squared becoming a negative (i^2 = 1). 
#7




Cardanos/Ferrari's method uses complex numbers to find (real) roots of 3th and 4th degree equations.

#8




Quote:
To wit, your example can actually be proven cleverly with geometry or as an appendage to various real calculus arguments. This is not as clean as the De Moivre way, but we should be honest. Imaginary numbers are incredibly useful as computational shortcuts. There are often occasions when use of imaginary numbers is the only way to solve a problem, but in terms of physical measurements imaginary numbers don't necessarily need to be appealed to. It's really only in the modelling, theory, and explanations that complex mathematics is used. 
#9




Quote:
x^{2} + 1 = 0. The equation and others like it popped up from time to time and were said to have no "real" solution, only an "imaginary" one. 


#10




Just a general philosophical observation:
Negative numbers are EVIL, we must resist learning about them. And anyway, what possible use could such a thing have? Imaginary numbers are EVIL, we must resist learning about them. And anyway, what possible use could there be? NonEuclidian geometry is EVIL, etc., etc. My point: why not say: "WOW, COOOOOL!!!!" and then go looking for new applications? All too often our response is to accuse the new idea of being a blasphemous evil which needs stamping out, rather than to see it as a brand new toy.
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(((((((((((((((((((((( ( (O) ) )))))))))))))))))))))) BILL BEATY _ _ _ _ _ _ _ _ billb,amasci.com http://amasci.com _ _ _ SCIENCE HOBBYIST beaty, chem.washington.edu _ Seattle, WA USA 
#11




Re: Practical application of imaginary numbers.
Quote:
It's worth noting however, that it's basically a matter of convenience. The concept of imaginary numbers lets you pair up real numbers and achieve certain results when those numbers are multiplied, squared, or whatever. These functions are useful in electrical engineering. However, at the end of the day, you are just pairing up numbers and defining functions of those pairs in a useful way. 
#12




I use imaginary numbers all the time. A bunch of friends will be out drinking. One of them asks me "How many girls have you slept with?"
I say, "Oh, at least a hundred." There you go. That's an imaginary number. 
#13




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#14




Yeah, it's possible to do everything without imaginary numbers. It's also possible to do everything without the number 7  you could just use 3+4 instead. So is 7 useless?



#15




In rereading this thread, it is obvious that imaginary numbers are very useful, but the question remains do they ever relate to anything physical. From the responses, it appears not? Possibly the Schrodinger wave function? Could that be restated without the use of imaginary numbers?
It seems that perhaps it is only Hawking that is proposing a physical reality to imaginary numbers. And at the moment, I can't recall why they would be necessary for his hypothesis rather than just another time dimension in real numbers. I'll have to review his idea again unless someone is familiar with his idea and can clue me in. 
#16




Seven is a particularly useless number, you are absolutely correct. Have I mentioned that Thursday is a useless day, too? It has all the feel of the Friday without the bonus of being followed by Saturday.

#17




Also, I guess I should mention that in a lot of cases, it's simpler to work with the complex numbers, because they're algebraically closedi.e., any polynomial with complex coefficients has a complex root (the socalled fundamental theorem of algebra). Contrast this with the reals, where x^{2} + 1 has no real roots, even though all of its coefficients are real.

#18




Imaginary numbers are definitely "real" in electronics. Impedance is of the form a+bi which is the form of all complex numbers. It is often evaluated as the norm and direction of this vector which is given in (usually) magnitude + angle (implying quandrant of the cartesian plane). So it is as real as vectors are. Not much else to say there.

#19




Regarding the imaginary time hypothesis, I think this may be more mainstream than it seems, although I don't know exactly what Hawking is talking about. Very often in Special Relativity it is useful to speak of things like:
x^{2} + y^{2} + z^{2}  t^{2} See? The spatial dimensions squared have positive sign, and the time dimension squared has negative sign. If we think of time as imaginary to begin with, then this seems more natural. 


#20




eris, technically, you can do impedance without appealing to imaginary numbers. I don't know WHY one would want to do such a thing, but it is possible. The norm and the argument are both real numbers, the imaginary part comes in when getting together a relation between the two. Technically, this relation can be acheived without appealing to the squareroot of negative one, but it's a whole lot simpler if you do.
I'd have to say that imaginary numbers do have physical significance, though they are never something which is directly measurable (only mathematically implied through manipulations of real numbers), but for things to have physical significance they do NOT have to be measurable. To make a horrendous analogy, pi is not technically "measurable" (as you can never get it exactly right), but it DEFINITELY has physical importance. 
#21




I recall (vaguely) in college taking a class called Complex Variables, which involved basically calculus using complex numbers (with both a real and imaginary part). One of the problems we solved had to do with the transfer of heat. We were able to describe how heat travelled through a substance over time when one side was adjacent to a heat source.

#22




Fractals, many of them are plotting with the imaginary part being used for the vertical axis.
Fractals have all sorts of real life applications, the least (greatest?) of which is that they look awfully cool. 
#23




I use imaginary numbers to balance my checkbook. My accountant has recommended dropping that practice...

#24




Quote:
We tend to think of the positive integers as more 'real' than imaginary numbers because they're more directly useful and familiar. But the fact that you can't count something with a number doesn't make it useless. Heck, forget the square root of 1; just look at 1. My sixyearold son could tell you that there's no such number. It's seven minus six, right? But you can't take away a bigger number from a smaller number! (Then he'd say 'Duh!' I'm trying to fix that little habit of his ...) Maybe sobut try doing accounting without it. Pain in the neck, if you ask me. I suppose it's possible to do electrical engineering without using i. But I strongly doubt that the equations used to describe AC circuits could have been developed without the use of that handy little number. And that's as 'real' as I need my numbers to be, thank you! (If you can find it, check out the late Isaac Asimov's anecdote on this subject in On Numbers. As is typical for him, he delivers a rhetorically entertaining but logically empty 'zinger'. But entertainment is always worthwhile, I think ...)
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#25




Quote:
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Christian "You won't like me when I'm angry. Because I always back up my rage with facts and documented sources."  The Credible Hulk 
#26




Ok, SCSimmons but consider a spatial analogy. One could plot the position of a car along a road as a function of its past location (zero, say) and how far it has gone forward (+) or in reverse ().
Which is to say that although I cannot imagine a negative amount of apples, I can certainly imagine an object moving along a demarcated line. Or grid, for that matter. OTOH, as I am not a physicist or electrical engineer, the usefulness of imaginary numbers still escapes me. Square root of negative one? Huh? 
#27




Alright, here's a spatial analogy. A generalized complex number can identify a point in a planesomething which requires two real numbers to do. (The x axis is the real numbers, the y axis the imaginaries.) This is the ultimate reason behind the use of complex numbers in engineering and most of physics, actually. Any simple harmonic motion can be expressed as the real portion of rotational motion in the complex plane. The equation of a circular path in rectangular coordinates in this complex plane (r cos theta + i * r sin theta) become both easier to express and easier to deal with in calculus when you convert it to polar coordinates (r * e ^ [i * theta]). It's almost impossible to explain how much easier life becomes with this model ... Integrating combinations of sines and cosines is, frankly, a pain in the necknot to mention how hairylooking the equations become. Integrating exponentials is trivialthe integral of e^x is e^x plus a constant. And multiplying pairs of binomials with sin and cos components makes for very complicated equations; multiplying exponentials involves adding the exponents. Representing simple harmonic motion (or an AC current, for another example) as a rotation in a complex plane makes the math easier to do, and the situation easier to visualize.
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Christian "You won't like me when I'm angry. Because I always back up my rage with facts and documented sources."  The Credible Hulk 
#28




Quote:
OTOH, if you really want to see where some complex sequence converges (like Newton's method on the complex plane), fractals are pretty useful. I suppose in some numerical regimes the forms may become important. However, their actual shapes are pretty much impractical in that they "wow" us and don't exactly teach us anything practical about what they're picturing. This is just my subjectiveself talking though, so take it with a grain of salt. 
#29




Quote:
OTOH, if you really want to see where some complex sequence converges (like Newton's method on the complex plane), fractals are pretty useful. I suppose in some numerical regimes the forms may become important. However, their actual shapes are pretty much impractical in that they "wow" us and don't exactly teach us anything practical about what they're picturing. This is just my subjectiveself talking though, so take it with a grain of salt. 
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