In what field are imaginary numbers applied to?

Question

An imaginary number is any number with the square root of -1.

Answer 1

First of all, you (along with some other people here, seem to be confusing imaginary and complex numbers). Complex numbers make a big set. Imaginary numbers make a smaller set inside the complex numbers and real numbers make a smaller set inside the complex numbers. These two sets, however, do not make the entire set of complex numbers. In other words,
1.All real numbers are complex numbers.
2.All imaginary numbers are complex numbers.
3.There are complex numbers that are not real number and they are not imaginary numbers either.

Draw a big circle and draw two smaller circles inside the big circle.

So from now on, everybody, be careful when you use the words imaginary and complex. They are not the same things.

Now for some clarifications:
1.A complex number can always be written as a+bi where i is the imaginary unit (square root of -1).
2.In any complex number a+bi, a is called the real part.
3.And b is called the imaginary part.

So the number 5 for example, is a real number AND it is also complex because 5 can be written as 5=5+0i.

The number 2i for example, is an imaginary number AND it is also a complex number because 2i=0+2i.

So now the light bulb goes off and everyone realizes that a real number is just a complex number with the imaginary part being zero and an imaginary number is a complex number with the real part being zero.

The number 1+2i is the last example. It is a number that is not real not imaginary.

Here is an interesting thing. On the real numbers, we have this notion of ordering, meaning if I give you any two real numbers, you can always put <, >, or = between them, meaning you can always compare them. The real numbers are ordered, in fact, they are "well-ordered". What about the complex numbers that? How in the world do you compare 1+i to 1-i? Is it bigger than, less than, or even equal that goes between them? Well, the answer is that they are both equal to each other. But dig in, and see if you can find out why.

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Now for the question you asked, pretty much all the engineering fields make extensive use of complex numbers but electrical engineers live, eat, drink, and breathe them.

But no worries, they were not invented only for them. In math, if you go far enough, it turns out that it actually is a lot easier to work with complex numbers than with real numbers only.

Here is an example you should be familiar with, when we want to find roots of polynomials.
If we consider a polynomial, as a polynomial with real numbers, then we know for a fact that you can't always find all of the zeros. You might not find any roots at all. x^2+1=0 has all real coefficients but no roots which is a problem.
Now, if we consider the same polynomial as a polynomial with complex numbers, we are ALWAYS guaranteed ALL the roots. Because remember, all real numbers are complex numbers as well. So x^2+1=0 is a polynomial with complex coefficients.

Answer 2

As is often the case, wikipedia said it best. See the source reference below...

Imaginary numbers are rarely used alone, but when "crossed" with real numbers form a very useful field of numbers called the "complex" numbers.

The applications in which I personally have used complex (and hence imaginary) numbers include AC electrical circuit analysis (Laplace and Fourier transforms), quantum mechanics (Schrödinger wave equation and its consequences), and relativity (time is an imaginary-valued fourth dimension to complement the three real-valued spatial dimensions).

ASIDE TO trance:
I'd also like to examine your secondary statement for a second. You said, "an imaginary number is any number with the square root of -1." I am guessing that you are fighting a language barrier, but nonetheless the statement needs to be corrected. An imaginary number is any:
-- non-zero
-- real-number
-- multiple
-- of the square root of -1.
It is also possible to prove that every imaginary number is the square root of some negative real number, so that works as a definition as well.

ASIDE TO prince:
Yes, yes, yes. Imaginary is not the same as complex. However, the question is not about sets; it is about applications. When one is working with complex numbers one is by definition also working with imaginary numbers, and let me explain why. The complex numbers are a two-dimensional vector space that forms an algebraic field. The universally accepted basis vectors for that vector space are the real unit "1" and the imaginary unit "i". So any cartesian reference to a complex number will involve the real multiples of i, which, of course, are the imaginary numbers.

Now in general one need not refer to basis vectors when using POLAR coordinates. However, in this one case THAT argument does not let us off the hook for refering to imaginary numbers. The universally accepted polar form of a complex number is the Euler complex representation:
. . . . . re^(θi)
Here the exponent is a real multiple of i, an imaginary number.

So in general, while the set of imaginary numbers is certainly not the same as the set of complex numbers, any mathematical application that makes use complex numbers must also make use of imaginary numbers by direct construction. There are **NO** applications of complex numbers that do not also use imaginary numbers.

Again, any application that uses complex numbers IS an application that uses imaginary numbers.

Answer 3

Electrical Engineering has a lot of stuff to do with imaginary numbers. My First year course of 'Electric Circuits 1' had to do so much of complex numbers in power calculation in a/c circuits.

Though still to study (maybe my next course), Magnetic Field problems too use a lot of complex numbers.

Answer 4

Formulas involving the magnetic field around an electric current use planar (or complex) numbers, if I'm not mistaken.

Answer 5

they are used heavily in electrical engineering, especially for working with AC signals. in general, any field that uses what is called 'complex algebra'.

Answer 6

Electrical eng.

Answer 7

The field of mathematics!!

^_^