What is the role of complex numbers in physics?

[johann1301]

In my math class, were having presentations about any topic from our curriculum. I want to talk about Complex numbers role in physics, but i don't know anything about its role. Can anyone tell me some areas were its important/used;) I know Feynman used them, but i don't know why. Anybody know something?

Thanks;)

 

[jedishrfu]

complex numbers are used in electrical impedance problems and obviate the need to use differential eqns:

http://en.wikipedia.org/wiki/Electrical_impedance

 

[Bob S]

Look at the solution of an underdamped harmonic oscillator in (see page 2):

http://www.brynmawr.edu/physics/DJCross/docs/misc/dho.pdf

The general form of the solution is an exponential with a complex number argument:
$$x(t)=Ae^{\left(-a\pm i\omega \right)t}$$

 

[euquila]

Complex numbers are numbers in the plane and can be a useful mathematical tool in many physics and engineering problems and theories (electrical, control theory, and quantum theory). In fact, you can have complex number in higher dimensions - an infinite number in fact (ex: 3d quaternion) where i^2 = j^2 = k^2 = -1. However, you do not combine them: a + bi + cj + dk != a + (b + c + d)i and the multiplication of two quaternions is noncommutative. Quaternions can be useful in making 3d video games because of the way they work.

All in all, putting complex numbers on the complex plane was the biggest discovery I think when someone found that i = 1 /_ 90-degrees. As such, complex numbers can embody angles/trigonometry and can be employed in problems involving oscillations/frequency. Often, in engineering, the laplace transform is taken of your mass-spring-damper system (or other type of system) to change t (time) into i*w (frequency). This is a neat trick that greatly simplifies the mathematics.

 

[e^(i Pi)+1=0]

i = 1 /_ 90-degrees

What does the underscore represent?

 

[euquila]

Sorry I was trying to draw an angle symbol, i = 1 "angle" 90°

In other words, the imaginary axis is perpendicular to the real axis.

 

[Rap]

In my math class, were having presentations about any topic from our curriculum. I want to talk about Complex numbers role in physics, but i don't know anything about its role. Can anyone tell me some areas were its important/used;) I know Feynman used them, but i don't know why. Anybody know something?

Thanks;)

Complex numbers are really good with oscillating systems, like waves or oscillators. Quantum mechanics would be really messy without complex numbers. Electromagnetic radiation (light, radio waves, etc) uses complex numbers.

The main thing about complex numbers is "closure". For example, the square root is not "closed" in the real number system. That means that there are some real numbers that you cannot take the square root of, like -1. In the complex number system, the square root is "closed". Every complex number has a square root.

More generally, if x^y are real numbers, there are some real numbers for which this expression has no answer. In the complex number system, every expression of the form x^y gives a complex number. This makes the math really simple, every time you run into x^y, you don't have to worry about whether it exists or not. It always does.

Thats why it works so well in many physics problems. If you do the physics in complex numbers, you can do the simplified math, and when you get your result, if you did it right, that result will be a real number.

 

[euquila]

Very well said Rap!

Oh, I want to add one thing... when you transform t --> i*w, you will often see this as "s". So do not be confused because s = i*w

t: time domain
s: frequency domain (much simpler mathematics for problems that involve oscillations like alternating current or quantum wave functions)

 

[euquila]

It was Wessel who discovered that √-1 is the unit length 90° to the real axis.

The proof goes like this:

Since I do not have an "angle" symbol, I will use "<" to mean angle.

Imagine (no pun intended) that there is a line segment with length L and direction θ that represents √-1.

In other words, √-1 = L < θ

If you square both sides, you get: -1 = L*L < 2θ

Now, we know from the unit circle that: -1 = 1 < 180°

The LHS of both these equations are the same! L*L < 2θ = 1 < 180°

Therefore, L = 1 and 2θ = 180° --> θ = 90°

Plug this back into our first equation and we find √-1 = 1 < 90° !

So i = √-1 is the "unit" of length that is perpendicular to the real axis.