discussion: are there any places that complex numbers are necessary?

[RisingFury]

You don't get far in quantum physics without complex numbers.

 

[malisle]

There was a similar thread recently.

 

[perseus]

Complex numbers are used in electronic engineering and other fields for an adequate description of the variable periodic signals ( see Fourier analysis ) . In an expression of

type , we can think of r \ , as the amplitude and \ phi \ , as the phase of a sine wave of a given frequency. When we represent a current or AC voltage ( and therefore with sinusoidal behavior) as the real part of a complex variable function of the form

, where ω represents the angular frequency and the complex number z gives the phase and amplitude , the treatment of all the formulas governing resistors, capacitors and inductors can be unified by introducing imaginary resistances for the latter two ( see power grids ) . Engineers use electrical and physical point for the imaginary unit j instead of i that is typically intended for the current.

The complex field is equally important in mathematics underlying quantum mechanics which uses Hilbert spaces of infinite dimension over C ( ℂ ) .

In special relativity and general relativity , some formulas for the metric of space-time are much simpler if we take the time as an imaginary variable.

---------- Post added at 08:23 PM ---------- Previous post was at 08:00 PM ----------

Curious expression that relates the unit in the real numbers with the imaginary unit, the number [FONT=&quot]π[/FONT] and number e

e ^ (i *[FONT=&quot]π[/FONT]) +1 = 0

Eulers formula, you kow that e^(i*[FONT=&quot]π[/FONT])= cos([FONT=&quot]π[/FONT])+ i*sin([FONT=&quot]π[/FONT]i)
and we know that cos([FONT=&quot]π[/FONT]) = -1
and sin([FONT=&quot]π[/FONT]) = 0
Therefore, e^(i*[FONT=&quot]π[/FONT])= -1 + (0*i)
= -1
therefore e^(i*[FONT=&quot]π[/FONT]i) + 1 = -1+1 = 0

Does have a physical or geometric meaning?

There is a very trivial geo metric explanation, if two unit vectors whose angle phase is [FONT=&quot]π[/FONT] (opposite directions),cancel, course
but maybe there's another explanation .....

 

[Linguofreak]

Complex numbers are fairly important in relativity.

For three dimensional Euclidean space, the length, l, of a vector with x, y, and z components labeled a, b, and c, is given by the equation:

l = sqrt(x^2+y^2+z^2)

Relativity adds time to the equation as a fourth dimension, but rather than having four dimensional Euclidean space, with the length of a vector determined by:

l = sqrt(t^2+x^2+y^2+z^2)

we give the time component and the three space components opposite signs, so that we either have:

l = sqrt(-t^2+x^2+y^2+z^2)

or:

l = sqrt(t^2-x^2-y^2-z^2)

The physics is the same with either way of writing the equations: The first treats a time difference as an imaginary spatial distance (as the square root of a negative number is imaginary), the second treats a spatial distance as an imaginary time difference. Either way, all the "odd" features of relativity fall out of the fact that one sign is different from the other three in the equations above (actually, if they were the same, we would have another set of odd features, and time travel would be possible. Newtonian physics leaves time out of the equation entirely).

 

[Fabri91]

As Perseus pointed out, you have a fair chance at finding complex numbers anytime there is some sort of periodicity involved in the phenomenon you're studying, be it AC or, for example, structural dynamics. :hide:

 

[Andy44]

It's used in antenna and microwave circuit engineering.