Polar Form of Complex Numbers

Conversion from standard form to polar form - compute magnitude and angle

Polar Form Calculator

Polar form of a complex number

The polar form represents a complex number by its magnitude \(r = |z|\) and angle \(\phi = \arg(z)\): \(z = r(\cos\phi + i\sin\phi) = re^{i\phi}\)

Complex number z = a + bi (standard form)
+
i
Calculation Result (Polar form)
Magnitude r =
Angle (Argument) φ =

Graphical Representation

Polar form as a vector

The vector has length r and forms the angle φ with the positive real axis.

Length r 2
Angle φ 45°
Formula r·e^(iφ)
Trigonometric r(cos φ + i sin φ)


Formulas for the Polar Form of Complex Numbers

The polar form represents a complex number by magnitude and angle instead of real and imaginary parts. This is especially useful for multiplication and division.

Magnitude
\[r = |z| = \sqrt{a^2 + b^2}\]

Length of the vector (distance from the origin)

Argument (Angle)
\[\phi = \arg(z) = \arctan\left(\frac{b}{a}\right)\]

Angle to the positive real axis

Polar Form Representations

Exponential form
\[z = r \cdot e^{i\phi}\]

Most compact representation using Euler's formula

Trigonometric form
\[z = r(\cos\phi + i\sin\phi)\]

Detailed representation with real and imaginary parts

Pair notation
\[z = (r, \phi)\]

Unique specification by magnitude and angle

Conversion between forms
From standard form to polar form:
\(z = a + bi\)
\(r = \sqrt{a^2+b^2}\)
\(\phi = \arctan(b/a)\) (watch quadrant!)
\(z = re^{i\phi}\)
From polar form to standard form:
\(z = re^{i\phi}\)
\(a = r\cos\phi\) (real part)
\(b = r\sin\phi\) (imaginary part)
\(z = a + bi\)

Computation Examples

Example 1: z = 3 + 4i

Magnitude:

\(r = \sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5\)

Angle:

\(\phi = \arctan(4/3) \approx 53.13°\)

Polar form: \(5e^{i\cdot53.13°}\) or \(5(\cos 53.13° + i\sin 53.13°)\)

Example 2: z = 1 + i

Magnitude:

\(r = \sqrt{1^2+1^2} = \sqrt{2} \approx 1.414\)

Angle:

\(\phi = \arctan(1/1) = 45°\)

Polar form: \(\sqrt{2}e^{i\cdot45°}\)

Example 3: z = -1 (negative real number)

Magnitude:

\(r = |-1| = 1\)

Angle:

\(\phi = 180° = \pi\) rad

Polar form: \(1e^{i\pi}\) (Euler's identity!)

Example 4: z = 2i (imaginary number)

Magnitude:

\(r = |2i| = 2\)

Angle:

\(\phi = 90° = \frac{\pi}{2}\) rad

Polar form: \(2e^{i\pi/2}\)

Advantages of the Polar Form

Multiplication (very easy!)

\[z_1 \cdot z_2 = r_1e^{i\phi_1} \cdot r_2e^{i\phi_2} = r_1r_2 \cdot e^{i(\phi_1+\phi_2)}\]

Rule: multiply magnitudes, add angles

Division (very easy!)

\[\frac{z_1}{z_2} = \frac{r_1e^{i\phi_1}}{r_2e^{i\phi_2}} = \frac{r_1}{r_2} \cdot e^{i(\phi_1-\phi_2)}\]

Rule: divide magnitudes, subtract angles

Exponentiation (very easy!)

\[z^n = (re^{i\phi})^n = r^n \cdot e^{in\phi}\]

Rule: raise magnitude to n, multiply angle by n

Root extraction (easy!)

\[\sqrt[n]{z} = \sqrt[n]{r} \cdot e^{i\phi/n}\]

Rule: n-th root of magnitude, divide angle by n
Note: n different solutions!

Comparison: Standard form vs. Polar form
Standard form is better for:
  • Addition and subtraction
  • Direct reading of real and imaginary parts
  • Simple representation
Polar form is better for:
  • Multiplication and division
  • Exponentiation and root extraction
  • Geometric interpretation (rotation)

Polar Form - Detailed Description

Geometric interpretation

Every complex number can be represented in the Gaussian plane as a vector. This vector can also be interpreted as a pointer.

Pointer representation:
• Length r: distance from the origin (magnitude)
• Angle φ: rotation counterclockwise from the positive real axis
• Positive angles: counterclockwise
• Negative angles: clockwise

Computing the angle

The angle φ is computed with arctan, but the quadrant must be considered:

Mind the quadrants!
  • Q I (a>0, b>0): φ = arctan(b/a)
  • Q II (a<0, b>0): φ = 180° + arctan(b/a)
  • Q III (a<0, b<0): φ = 180° + arctan(b/a)
  • Q IV (a>0, b<0): φ = arctan(b/a)

Better: use atan2(b, a)!

Practical applications

The polar form is used in many technical fields:

Applications:
• Electrical engineering: AC calculations, impedance
• Signal processing: Fourier transform
• Mechanics: rotational motion
• Quantum mechanics: wave functions

Euler's formula

The connection between exponential and trigonometric form:

\[e^{i\phi} = \cos\phi + i\sin\phi\]

Euler's formula is the basis of the polar form

Special cases

  • Positive real numbers: φ = 0°
  • Negative real numbers: φ = 180° = π
  • Positive imaginary numbers: φ = 90° = π/2
  • Negative imaginary numbers: φ = -90° = -π/2
More examples in the tutorial

Detailed step-by-step explanations for converting between standard and polar form

Go to tutorial
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