Angle (Argument) of Complex Numbers
Calculation of \(\arg(z)\) - the polar angle in the complex plane
Angle Calculator
Angle (Argument) of a Complex Number
The angle \(\phi = \arg(z)\) is the polar angle to the positive real axis, measured counterclockwise. It is calculated using \(\phi = \arctan(b/a)\), where the quadrant must be taken into account.
Graphical Representation
Angle as Vector
The angle φ is the polar angle from the positive real axis to the vector, measured counterclockwise.
Formulas for the Angle of Complex Numbers
The angle (argument) \(\phi = \arg(z)\) of a complex number \(z = a + bi\) is the polar angle to the positive real axis in the Gaussian number plane.
Basic Formula
⚠️ Quadrant must be considered!
Better Formula
Automatically considers all quadrants
Quadrant Rules
Angle Calculation Based on Quadrant
Quadrant I (a>0, b>0)
\(\phi = \arctan(b/a)\)
Range: 0° to 90°
Example: z = 3+4i → φ ≈ 53.13°
Quadrant II (a<0, b>0)
\(\phi = 180° + \arctan(b/a)\)
Range: 90° to 180°
Example: z = -3+4i → φ ≈ 126.87°
Quadrant III (a<0, b<0)
\(\phi = -180° + \arctan(b/a)\)
Range: -180° to -90°
Example: z = -3-4i → φ ≈ -126.87°
Quadrant IV (a>0, b<0)
\(\phi = \arctan(b/a)\)
Range: -90° to 0°
Example: z = 3-4i → φ ≈ -53.13°
Special Cases (on the axes)
φ = 0°
φ = 90°
φ = ±180°
φ = -90°
Calculation Examples
Example: \(\arg(4+3i)\)
Step 1: Determine Quadrant
a = 4 > 0
b = 3 > 0
→ Quadrant I
Step 2: Calculate Arctan
\(\phi = \arctan\left(\frac{3}{4}\right)\)
\(= \arctan(0.75)\)
\(\approx 36.87°\)
Step 3: Convert to Radians
\(\phi = 36.87° \times \frac{\pi}{180°}\)
\(\approx 0.6435\) rad
\(\approx 0.64\) rad
Step 4: Verification
Polar form: \(z = 5e^{i\cdot 0.64}\)
Reverse calculation:
a = 5·cos(36.87°) ≈ 4 ✓
b = 5·sin(36.87°) ≈ 3 ✓
Example 2: \(\arg(1+i)\)
\(\arctan(1/1) = \arctan(1)\)
\(= 45° = \frac{\pi}{4}\)
Example 3: \(\arg(-1+i)\)
Quadrant II
\(180° + \arctan(1/-1)\)
\(= 135° = \frac{3\pi}{4}\)
Example 4: \(\arg(i)\)
z = 0+1i
On positive imaginary axis
\(= 90° = \frac{\pi}{2}\)
Angle of Complex Numbers - Detailed Description
Geometric Meaning
Every complex number can be represented as a vector in the Gaussian number plane.
• Length: Magnitude \(|z|\)
• Direction: Angle \(\phi = \arg(z)\)
• Measurement: From positive real axis
• Direction of rotation: Counterclockwise (positive)
Angle Ranges
Principal Value:
Positive angles: Counterclockwise
Negative angles: Clockwise
atan2 Function
The atan2 function automatically considers all quadrants:
\[\text{atan2}(b, a) = \begin{cases} \arctan(b/a) & a > 0\\ \arctan(b/a) + \pi & a < 0, b \geq 0\\ \arctan(b/a) - \pi & a < 0, b < 0\\ +\pi/2 & a = 0, b > 0\\ -\pi/2 & a = 0, b < 0\\ \text{undefined} & a = 0, b = 0 \end{cases}\]
Important Properties
- \(\arg(z_1 \cdot z_2) = \arg(z_1) + \arg(z_2)\) (Addition)
- \(\arg(z_1 / z_2) = \arg(z_1) - \arg(z_2)\) (Subtraction)
- \(\arg(z^n) = n \cdot \arg(z)\) (Multiplication by n)
- \(\arg(\overline{z}) = -\arg(z)\) (Conjugate)
- \(\arg(1/z) = -\arg(z)\) (Reciprocal)
Important Note
Multivalued:
The angle is unique only up to multiples of \(2\pi\) (360°)!
\(\arg(z) = \phi + 2\pi k\) with \(k \in \mathbb{Z}\)
The principal value chooses \(k=0\).
Practical Applications
• Electrical Engineering: Phase shift
• Signal Processing: Frequency analysis
• Control Engineering: Nyquist diagram
• Navigation: Direction determination
• Polar Coordinates: Angle transformation
• Complex Functions: Argument principle
• Fourier Transform: Phase
• Vector Calculus: Angle calculation
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