What do we use imaginary numbers in a real world application?

Question

anonymous · Answer

I'm just thinking about graphs of functions and how some of them can approximately model what's going with something like the efficiency of an engine, but then imaginary numbers don't fit on that coordinate plane, I know you have the Complex plane, but what would you do with that?

kainui · Answer

You can just use the complex plane all by itself to model a plane. Basically anything that involves vectors in two dimensions.

For example, a complex number with a length of 1 can have an angle. So if you multiply this form of 1 by another complex number, the only thing that will change will be that the complex number gets rotated by the same angle as 1. I'll show you:

$$u=\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}$$ The length of that is 1, check it. Also notice that it represent 45 degrees. So now look at this number$$3+3i$$ which is in the same direction, 45 degrees, but longer. So if what I said was right, that means that multiplying (3+3i) by u above should rotate the number 45 degrees more, but have the same length. This also means that you have a purely complex number as well since 90 degrees is in the "i" direction.

So basically you can do everything you did with x and y axes with complex numbers, it's just extending the number line into a number plane and it makes rotation as simple as just FOIL by distributing numbers which is powerful because we didn't have to use sine or cosine! Call the directions North, South, East and West if you like or Up, Down, Forwards, and Backwards for an object being thrown in the air. There are many more things you can do.

The main thing you should take away though is that complex/imaginary numbers aren't really any more complicated or fake than the "real" numbers. All numbers exist in your mind and imaginary, don't let the name fool you. =P

anonymous · Answer

you use exponents to describe a larger amount of money than one can describe using numbers

primeralph · Answer

Err, look up circuit theory.

primeralph · Answer

....and Fourier......

kenljw · Answer

Basically the complex plane completes the real number line.  In electronics the real values are voltage, current and resistance, it is completed with reactance's either inductive or capacitive which are functions of frequency which along with resistance effect voltage and current.  This is a basic understanding but as you progress your introduced to Laplace Transform for filter design and control systems.