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="]π[/FONT] and number e
e ^ (i *[FONT="]π[/FONT]) +1 = 0
Eulers formula, you kow that e^(i*[FONT="]π[/FONT])= cos([FONT="]π[/FONT])+ i*sin([FONT="]π[/FONT]i)
and we know that cos([FONT="]π[/FONT]) = -1
and sin([FONT="]π[/FONT]) = 0
Therefore, e^(i*[FONT="]π[/FONT])= -1 + (0*i)
= -1
therefore e^(i*[FONT="]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="]π[/FONT] (opposite directions),cancel, course
but maybe there's another explanation .....