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?
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
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”.
Stephen Hawking thinks that there may be an imaginary dimension to time! From his public lecture The Beginning of Time there is this:
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.
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).
Cardanos/Ferrari’s method uses complex numbers to find (real) roots of 3th and 4th degree equations.
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.
I think that imaginary numbers are called that because early mathematics had no solution for the equation
x[sup]2[/sup] + 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.
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.
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.
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.
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.
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?
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.
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.
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 x[sup]2[/sup] + 1 has no real roots, even though all of its coefficients are real.
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.
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[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - t[sup]2[/sup]
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.
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.