Complex Number as a Point in The Plane
2026-02-21 20:42 Diff
https://brightchamps.com/en-us/math/numbers/complex-number-as-point-in-the-plane

The argument of a complex number is the angle between the positive real axis and the line joining the origin to the point (a, b) in the complex plane.


For a complex number, z = a + bi, the argument of z: 
 = θ = tan⁻¹\(({b \over a})\)
 

Let us understand how the argument of a complex number z = a + bi is determined in different quadrants.


1. First quadrant (a > 0, b > 0):  = tan-1\(({b \over a})\)

2. Second quadrant (a < 0, b > 0): π - tan-1\(({b \over a})\)

3. Third quadrant (a < 0, b < 0): θ = - π + tan⁻¹\(({b \over a})\)


4. Fourth quadrant (a > 0, b < 0): tan⁻¹\(({b \over a})\)

The principal argument of a complex number lies in -π < θ <π.

The main value of a complex number is restricted to a specific range here. The principal argument is also known as amplitude.

The \(\theta\) is such that: 
The principal argument of z lies in -π < Arg(z) ≤ π (in radians), or equivalently -180° < Arg(z) ≤ 180°.


It is the unique angle formed between the positive real axis and the line representing the complex number in the complex plane. 
 

Example: For z = 1+i
 = tan-1\(({1 \over 1})\) = 4
Thus, the principal argument of z = 1+ i is \(\pi \over 4\)

Let us now understand the principal argument of Z = a =+ ib in different quadrants.

  1. First quadrant (a > 0, b > 0): Arg Z = tan⁻¹\(({b \over a})\)

2. Second quadrant (a < 0, b > 0) : Arg Z = π + tan⁻¹\(({b \over a})\)

3. Third quadrant (a < 0, b < 0): Arg Z = - π + tan⁻¹\(({b \over a})\)
 

4. Fourth quadrant (a > 0, b < 0)