# Complex numbers physical interpretation

Hi guys.
What is really the need for complex numbers? Is there any physical meaning associated with it?

Landau
Complex number is very useful,for example,we have the Fundamental Theorem of Algebra,we can relate Exponential Function to Trigonometric Function(Euler's Equation),in Differential Equations,it is useful when the characteristic equation has complex roots.As far as I know,in physics the Schrödinger Equation is a complex equation.Also complex number is applied in the computing of electroengineering.

HallsofIvy
Homework Helper
There does not have to be, nor is there, a single "physical interpretation" of any mathematical concept.

Historically, the motivation for defining complex numbers was Cardano's formula for solving cubic equations. That involves taking a square root, followed by other calculations. There exist cubic equations which have only real roots but applying Cardano's formula requires taking the square root of a negative number and so requires knowing how to manipulate complex numbers.

For Example, the differencial equalation ax''[t]+bx'[t]+cx=0, at the discrimant is negative, roots are [A]\pm[/Bi] solution is [e][/At]([C][/1]Cos[Bt] + [C][/2]Sin[Bt]).

lavinia
Gold Member
Hi guys.
What is really the need for complex numbers? Is there any physical meaning associated with it?

There are many physical processes that are described in terms of complex numbers. In quantum mechanics, the wave function is a complex distribution that evolves according to a complex version of the heat equation. Quantum mechanical things have phases which are complex numbers.

A major problem with using simple trigonometry in real world problems is that the trigonometric functions are 4 times redundant over all of the possible angles around a point. Complex numbers solve that problem by giving a unique value for each angle and a unique value for the trigonometric function for all angles 0 to 2 PI.

Pythagorean
Gold Member
I always thought i was a geometric transform operator for two-dimensional systems. I'm still not quite sure what the difference between A 2D vector and complex number is, especially if the vectors are normalized so that their inner product is 1.

lavinia
Gold Member
I always thought i was a geometric transform operator for two-dimensional systems. I'm still not quite sure what the difference between A 2D vector and complex number is, especially if the vectors are normalized so that their inner product is 1.

Complex numbers can be multiplied as well as added. Adding them is the same as 2D vector addition.

I'm still doing school maths, but I can see how complex numbers can be used to represent force vectors if you know what I mean. So I guess it really useful in physics

lavinia
Gold Member
I'm still doing school maths, but I can see how complex numbers can be used to represent force vectors if you know what I mean. So I guess it really useful in physics

The modeling of quantum mechanical systems as complex distributions gives a picture of the world that is fundamentally composed of complex number values.

Historically they came about as a way of solving cubic equations. At the time there was a solution by radicals to cubic equations and sometimes a square root of a negative number would arise, but go away by the time the answer was simplified. It was quite mysterious at the time as you could 'allow' these weird items in order to get real answers.

Then I believe (and I forget who did it first) there was a movement to see complex numbers as vectors. Because, as noted above, you could add them and multiply them (and they follow the parallelogram law for force addition). It was from this point that Hamilton tried finding a 3D version of the complex numbers, after all that's what you'd want. The complex numbers just don't cut in a 3D world. However he discovered the quaternions, which are 4D and have units of 1, i, j, and k. Those eventually were hijacked by Gibbs and Heaviside to form what we think of as vector analysis today.

I'm not fully certain of the question so I'll posit another possibility by posing a question myself. What is the physical significance of the real numbers? When I teach mathematics that involves complex numbers (I dislike the term 'imaginary') I always spend a few minutes on this question.

As for 'need' that depends. For example, do we need real numbers? I'm not sure, we could probably work out all of our results on the rationals but it would be ridiculuous, supposing its even possible. The real numbers are a construction that make analysis much nicer. We can take limits (and the limit points exist) and so voila: calculus. In the same way the complex plane makes many things very nice and very convenient. Could we work without them? Maybe, but its probably not a good idea.

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As a engineer, I'm learning that they have really important applications in fluid mechanics to describe potential (ideal) flow, they are the backbone for solving many complicated integrals that define engineering and scientific processes related to Laplace transforms, Fourier series. They are very important, especially the fundamental theorem of calculus, which presents a really cool way to evaluate a complicated integral.

Alot of applications, and interests.

I think that you already know that some equations are solved only in C (complex)...but interpretation,we have to study philosophy here...I think it is abstract idea,but bright so we can deal with x^2=-1