# Nonsense Numbers

1. Dec 5, 2006

Thesedays I am trying to understand the role and working of complex numbers. Previously, I posted my question here but I could not get the required answer. Rightnow I am reading about a complex numbers in Toronto University's website and there I read a very confusing and irritating statment which as follows:

The strength of an electromagnetic field. This is a directly measurable quantity that is measured by a complex number. That number will be purely real if the field is all electric with no magnetic component, purely imaginary if the field is all magnetic with no electric component, and in other cases will have a non-zero real part and a non-zero imaginary part.

So according to this statement in the absence of electric field we have an imaginary magnetic component.
So, what is meant by an imaginary magnetic component?
Let's consider a laboratory where we have an Electromagnetic field. Now let's remove the electrical component from field so that only magnetic component remains.
So, will this magnetic component have no effect on compass?
Secondly, if it has the effect on the compass (since I believe it should be the case). Then, the magnetic field should have the strength which is measured or represented by imaginary number (according to the scientists).
So, how an imaginary number can give the answer (the real valued answer)?

2. Dec 5, 2006

### arildno

Not at all!
A complex number z can be written as : z=x+iy, where x and y are REAL numbers, and i the imaginary unit.

I'm not familiar with the actual situation, but it is evident than when it is said that z is purely imaginary, it means that x=0. y is then the REAL magnetic component.

3. Dec 5, 2006

### chroot

Staff Emeritus
They're just representing the electric and magnetic fields with a two dimensional quantity, a "vector field" with a different vector defined at each point in space. One component of this vector represents the E field, the other represents the M field. There is no particular reason why imaginary numbers need to be used at all; any 2D vector field would work just as well. As it happens, complex numbers have some useful arithmetic properties that make them easy to use in this context, but there's nothing "imaginary" about the magnetic field.

Keep in mind that when you go through a complete calculation and finally solve for, say, the force on a compass's needle, you will always get a real number.

- Warren

4. Dec 5, 2006

### arildno

Well, but I would like to add that the MEASURED quantities are not complex or imaginary (as it seems to say in OP's book); it is the MATHEMATICAL REPRESENTATION that can be cast in the convenient language of complex numbers.

5. Dec 6, 2006

### kesh

don't confuse mathematical meaning of "imaginary", "real", etc. with other meanings of those words.

don't confuse a mathematical model of a situation with that situation itself

6. Dec 6, 2006

### Hurkyl

Staff Emeritus
The role of complex numbers is to describe things well described by complex numbers.

Real numbers are often used to describe things, because real numbers have nice properties. For example, they form a continuum, and they have an ordering. So, one possible use of real numbers is to describe something whose possible values are ordered and form a continuum.

The complex numbers also have nice properties. For example, they are also a continuum, and they can be viewed as having a magnitude and a phase. So, one possible use of complex numbers is to describe something whose possible values form a continuum, and they have magnitudes and phases.

For some purposes, it is useful to consider the field

G = E + i B.

Since E and B are real vectors, we have

Re G = E
Im G = B

It depends on your measurement. :tongue: If you are measuring a complex quantity, such as G, then your measurement had better be complex when appropriate!