the phase of a complex number

eggshell

in cartesian form, a+ ib you can find the phase by doing arctan(b/a).. my question concerns the phase of a purely imaginary number. during a lecture my professor said that the phase of i*2pi= pi/2, he rationalized this by saying that the number lies on the y-axis so the angle between the real axis and the imaginary axis is pi/2. but if you do arctan(2pi/0) you will get an error.. how is he right?

pwsnafu

Complex numbers don't have a "phase" they have an "argument" which is defined case by case: if z = x + iy we define [itex]\phi = Arg(z)[/itex] as
arctan(y/x) when x > 0
arctan(y/x)+π when x < 0 and y ≥ 0
arctan(y/x)-π when x < 0 and y < 0
π/2 when x = 0 and y > 0
-π/2 when x = 0 and y < 0
indeterminate when x = 0 and y = 0.

Similar Threads for: the phase of a complex number
Thread	Forum	Replies
taking a number to a complex number power	General Math	3
Complex amplitude of E-field and phase changes	Classical Physics	1
Phase of a complex number	Precalculus Mathematics Homework	2
Complex Number (Modulus/Phase)	Calculus & Beyond Homework	1
phase difference and number of wavelengths	Introductory Physics Homework	5

pwsnafu	Complex numbers don't have a "phase" they have an "argument" which is defined case by case: if z = x + iy we define [itex]\phi = Arg(z)[/itex] as arctan(y/x) when x > 0 arctan(y/x)+π when x < 0 and y ≥ 0 arctan(y/x)-π when x < 0 and y < 0 π/2 when x = 0 and y > 0 -π/2 when x = 0 and y < 0 indeterminate when x = 0 and y = 0.