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  #1  
Old 06-16-2002, 12:22 PM
Chekmate Chekmate is offline
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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 non-existant number, and yet it is used to prove math equations.

Huh?
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  #2  
Old 06-16-2002, 12:27 PM
andy_fl andy_fl is offline
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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
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  #3  
Old 06-16-2002, 12:39 PM
Sunspace Sunspace is offline
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They're used in electrical engineering to descrive how circuits operate inside. Without them, we couldn't design even an AM radio. Do a Google-search on "impedance".
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  #4  
Old 06-16-2002, 12:42 PM
rsa rsa is offline
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Stephen Hawking thinks that there may be an imaginary dimension to time! From his public lecture The Beginning of Time there is this:
Quote:
Quantum theory introduces a new idea, that of imaginary time. Imaginary time may sound like science fiction, and it has been brought into Doctor Who. But nevertheless, it is a genuine scientific concept. One can picture it in the following way. One can think of ordinary, real, time as a horizontal line. On the left, one has the past, and on the right, the future. But there's another kind of time in the vertical direction. This is called imaginary time, because it is not the kind of time we normally experience. But in a sense, it is just as real, as what we call real time.

The three directions in space, and the one direction of imaginary time, make up what is called a Euclidean space-time. I don't think anyone can picture a four dimensional curve space. But it is not too difficult to visualise a two dimensional surface, like a saddle, or the surface of a football.

In fact, James Hartle of the University of California Santa Barbara, and I have proposed that space and imaginary time together, are indeed finite in extent, but without boundary. They would be like the surface of the Earth, but with two more dimensions. The surface of the Earth is finite in extent, but it doesn't have any boundaries or edges. I have been round the world, and I didn't fall off.

If space and imaginary time are indeed like the surface of the Earth, there wouldn't be any singularities in the imaginary time direction, at which the laws of physics would break down. And there wouldn't be any boundaries, to the imaginary time space-time, just as there aren't any boundaries to the surface of the Earth. This absence of boundaries means that the laws of physics would determine the state of the universe uniquely, in imaginary time. But if one knows the state of the universe in imaginary time, one can calculate the state of the universe in real time. One would still expect some sort of Big Bang singularity in real time. So real time would still have a beginning. But one wouldn't have to appeal to something outside the universe, to determine how the universe began. Instead, the way the universe started out at the Big Bang would be determined by the state of the universe in imaginary time. Thus, the universe would be a completely self-contained system. It would not be determined by anything outside the physical universe, that we observe.
http://www.hawking.org.uk/text/public/bot.html
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  #5  
Old 06-16-2002, 01:13 PM
coffeecat coffeecat is offline
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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.
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  #6  
Old 06-16-2002, 01:31 PM
funkynige funkynige is offline
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Quote:
Imaginary numbers aren't imaginary like the Invisible Pink Unicorn
If they were invented today, they would probably have been called virtual numbers , but as they were invented years ago, before the virtual world was a concept, they're called imaginary.

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).
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  #7  
Old 06-16-2002, 05:17 PM
Zweistein Zweistein is offline
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Cardanos/Ferrari's method uses complex numbers to find (real) roots of 3th and 4th degree equations.
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  #8  
Old 06-16-2002, 05:49 PM
JS Princeton JS Princeton is offline
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Quote:
Originally posted by funkynige

(remember sin^2 + cos^2 = 1?, that's derived using De Moivre's Formula which links sin, cos and the exponential function).
You can actually derive all trigonometric relations of real numbers without appealing to imaginary numbers. Imaginary numbers just happen to be a lot more convenient often.

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.
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  #9  
Old 06-16-2002, 05:51 PM
David Simmons David Simmons is offline
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Quote:
Originally posted by funkynige


If they were invented today, they would probably have been called virtual numbers , but as they were invented years ago, before the virtual world was a concept, they're called imaginary.

I think that imaginary numbers are called that because early mathematics had no solution for the equation
x2 + 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.
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  #10  
Old 06-16-2002, 06:19 PM
bbeaty bbeaty is offline
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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?

Non-Euclidian 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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  #11  
Old 06-16-2002, 06:29 PM
lucwarm lucwarm is offline
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Re: Practical application of imaginary numbers.

Quote:
Originally posted by Chekmate
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 non-existant number, and yet it is used to prove math equations.

Huh?
As another poster pointed out, imaginary numbers are useful in electrical engineering.

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.
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  #12  
Old 06-16-2002, 10:02 PM
stuyguy stuyguy is offline
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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.
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  #13  
Old 06-17-2002, 09:07 AM
ultrafilter ultrafilter is offline
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Quote:
Originally posted by bbeaty
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?

Non-Euclidian 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.
That's the way it's done today. Modern mathematicians, being somewhat less concerned with practicalities than those from 200 years ago, generally have no reluctance to accept such things.
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  #14  
Old 06-17-2002, 09:43 AM
Achernar Achernar is offline
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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?
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  #15  
Old 06-17-2002, 10:44 AM
rsa rsa is offline
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In re-reading 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.
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  #16  
Old 06-17-2002, 10:45 AM
erislover erislover is offline
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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.
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  #17  
Old 06-17-2002, 10:45 AM
ultrafilter ultrafilter is offline
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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 closed--i.e., any polynomial with complex coefficients has a complex root (the so-called fundamental theorem of algebra). Contrast this with the reals, where x2 + 1 has no real roots, even though all of its coefficients are real.
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  #18  
Old 06-17-2002, 10:52 AM
erislover erislover is offline
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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.
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  #19  
Old 06-17-2002, 11:06 AM
Achernar Achernar is offline
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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:

x2 + y2 + z2 - t2

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.
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  #20  
Old 06-17-2002, 05:54 PM
JS Princeton JS Princeton is offline
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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 square-root 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.
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  #21  
Old 06-17-2002, 06:41 PM
Cheesesteak Cheesesteak is offline
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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.
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  #22  
Old 06-17-2002, 09:03 PM
Com2Kid Com2Kid is offline
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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.
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  #23  
Old 06-17-2002, 09:34 PM
Kirkland1244 Kirkland1244 is offline
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I use imaginary numbers to balance my checkbook. My accountant has recommended dropping that practice...
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  #24  
Old 06-17-2002, 10:18 PM
SCSimmons SCSimmons is offline
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Quote:
Originally posted by rsa
In re-reading 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?
Of course. But that's true of all numbers, including your friendly neighborhood positive integers. Anything that you need imaginary numbers for, can be expressed instead in terms of functions of real numbers; anything you need real numbers for can be expressed instead in terms of (possibly infinite) sequences of (functions of) rational numbers; anything you need rational numbers for can be expressed instead as ratios of integers; anything you need negative integers for can be expressed instead in terms of positive integers and addition/subtraction; and anything you need positive integers for can be expressed in terms of generalized set theory and logic. So really, there aren't any such things as numbers-they're all abstractions of one sort or another. Naturally-who's ever seen the number one?

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 six-year-old 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 so-but 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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Old 06-17-2002, 10:32 PM
SCSimmons SCSimmons is offline
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Quote:
Originally posted by SCSimmons
Heck, forget the square root of -1; just look at -1. My six-year-old 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 ...)
OK-everyone knows I meant to type 'six minus seven'. Right?

Preview-preview-preview ...
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Old 06-17-2002, 11:31 PM
Measure for Measure Measure for Measure is offline
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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?
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  #27  
Old 06-18-2002, 07:34 AM
SCSimmons SCSimmons is offline
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Alright, here's a spatial analogy. A generalized complex number can identify a point in a plane-something 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 neck-not to mention how hairy-looking the equations become. Integrating exponentials is trivial-the 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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  #28  
Old 06-18-2002, 11:01 AM
JS Princeton JS Princeton is offline
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Quote:
Originally posted by Com2Kid
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.
Well, real life applications, yes... but they're much more of mathematical abstractions than anything truly practical. Don't get me wrong, I love fractals, I just don't necessarily see them as being particularly useful in the real-world in the same sense that the complex numbers themselves are useful. There are plenty of real-world situations where complex numbers are useful: in doing conformal mappings, calculating residues (for duing some otherwise nasty integrals), or proofs of basic theorems about harmonic, elliptical, and other fascinating functions. Certain chaos theories and attempts at probing randomness have what I would term oblique references to fractals, but as far as I know they really are just succint geometrical ways of illustrating one's point rather than particularly useful mappings. Of course, teaching a computer to do a fractal is a useful exercize in programming, but I don't know that it has much meaning beyond that.

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 subjective-self talking though, so take it with a grain of salt.
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Old 06-18-2002, 11:08 AM
JS Princeton JS Princeton is offline
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Quote:
Originally posted by Com2Kid
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.
Well, real life applications, yes... but they're much more of mathematical abstractions than anything truly practical. Don't get me wrong, I love fractals, I just don't necessarily see them as being particularly useful in the real-world in the same sense that the complex numbers themselves are useful. There are plenty of real-world situations where complex numbers are useful: in doing conformal mappings, calculating residues (for duing some otherwise nasty integrals), or proofs of basic theorems about harmonic, elliptical, and other fascinating functions. Certain chaos theories and attempts at probing randomness have what I would term oblique references to fractals, but as far as I know they really are just succint geometrical ways of illustrating one's point rather than particularly useful mappings. Of course, teaching a computer to do a fractal is a useful exercize in programming, but I don't know that it has much meaning beyond that.

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 subjective-self talking though, so take it with a grain of salt.
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