Complex Numbers Continued: Amplitude

This article assumes you have read our article on Polar Representation of a Complex Number

Resuming from there, we had determined t = tan-1(b/a), which is called the amplitude or argument of the complex number a+ib and it is denoted by amp(a+ib).

Principal amplitude:

The value of t which lies in the interval (-π, π), is called the principal amplitude of the complex number a+ib. Principal amplitude of a complex number can be found as follows.

First find t = tan-1|b/a|

Then the principal amplitude is,

a)      t, if both a and b are positive

b)      π-t, if a is negative and b  is positive

c)      –( π-t), if both a and b are negative

d)      –t, if a is positive and b is negative

Properties of amplitude:

For any two complex numbers Z1 and Z2,

(i)                Amp(Z1 Z2) = Amp (Z1) + Amp (Z2)

(ii)              Amp(Z1 /Z2) = Amp (Z1) – Amp (Z2)

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