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.
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).
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.
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.
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:
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.