Application of Complex Numbers

[danne89]

Hi! I'd a look at complex numbers and can't understand how they can be applied to "the real world". Can anyone give me some concrete examples, please. Or a site that does.

Danne

 

[danne89]

Oh, woop! I now saw the thread a bitter down. But I think it question why, and this thread "what can I do with it". By the way, you can't delete threads anymore, or?

 

[robert Ihnot]

Well, maybe this will help. Gauss proved that every equation of nth degree has n roots. This means the equation X^2+1 has two roots. However, it does not cross the X-axis. Thus the roots, +i and -i represent extensions of the number system. A reference on this is: http://www.uncwil.edu/courses/mat111hb/Izs/complex/complex.html

 

[Hurkyl]

Complex numbers can be interpreted as being the combination of a phase (aka angle) and a magnitude. Thus, they're useful for describing things that are well described by a phase and magnitude. They're useful even when you only care about phase!

 

[daster]

Complex numbers sometimes provide a quicker way to solve certain questions, which is always a plus.

 

[MiGUi]

Complex numbers aid to solve certain integrals that seems impossible like this one:

$$\int_{-\infty}^{\infty} \frac{1 + x^2}{1 + x^4} \, dx$$

Complex numbers also appear in very differential equations, like the wave equation or the heat equation...

The problem is that we can't imagine it easily.

 

[NeutronStar]

A Concrete Example

This is from an old post I made a while back that gives a concrete example of a complex quantity.

To begin with forget about the terms "real" and "imaginary". Think instead in terms of "quantities" and ask whether complex numbers can be thought of as representing quantitative properties. I think that you'll find that complex numbers are just that,.. complex. They have what is called a "real" component and an "imaginary" component. Then what you need to ask is what do these different components represent quantitatively.

The best concrete example I can think of is an electric circuit that contains an inductor. A dynamic current flowing in such a circuit can be described by a complex number. The real part of that description refers to the electron current flow (or hole current flow if you're a semiconductor nut). The imaginary part of that description refers to the magnetic field associated with the inductor. What you lose in electron current you gain in magnetic field and vice versa.

So both the real part and the imaginary part of the complex number represent "real" quantities in the "real" world.

Think of it this way,… electron current in a circuit can only flow in two directions. They are described by the sign of the number that represents the quantity of current flow (the real part of the complex number). Current flow can never be less than zero (no current flow at all), but it can have a magnitude in either of two directions (+ or -) . So the real part of the complex number always represent a quantity of electron current greater than or equal to zero and the sign represents the direction that the current is moving.

However, when some of the electron current gets converted into magnetic field energy by the inductor we can't represent that by either positive or negative. We need a new "direction". The new direction is called i for "imaginary". This new direction can also be positive or negative. In other words, the magnetic field can either be growing (the positive imaginary direction) or collapsing (the negative imaginary direction). So the imaginary part of the complex number represents the quantity of magnetic field present as well as its dynamical state (positive or negative).

In this case the complex number isn't any more mysterious than any other number. It's really just a shorthand way of combining various coordinate systems and quantitative ideas.

I've found that no matter how abstractly we take this idea of complex numbers, if we stop to think about what we are doing, we can break down very abstract concepts into their intuitive counterparts. Assuming, of course, that we have any intuitive understanding to begin with of the concepts that we are working with.

It does appear that some mathematicians have absolutely no intuitive clue concerning the objects that they are working with. This can usually be revealed by simply asking them to offer an intuitive explanation of their project. When they start talking in axiomatic circles and can't reduce their idea to a simple intuitive explanation then you can rest assured that they, themselves, have absolutely no intuitive clue about what the heck they are doing.

Many mathematicians do indeed work entirely from this axiomatic approach. I personally like to keep the intuitive insight alive along the way. But then, I'm a scientist.

 

[Mark44]

If I'm remembering my mathematics history correctly, complex numbers gave rise to the concept of vectors. It's no coincidence that complex numbers in rectangular form can be added and subtracted in exactly the same way as vectors in the plane.