Unleashing the Power of Polar Form in Complex Number Calculations

Table of contents

Rectangular form (standard form):

  • z = 5 + 8.660254i
  • Angle notation (phasor):
  • z = 10 ∠ 60°
  • Polar form:
  • z = 10 × (cos 60° + i sin 60°)
  • Exponential form:
  • z = 10 × ei 1.0471976 = 10 × ei π/3
  • Polar coordinates:
  • r = |z| = 10 ... size (modulus, absolute value)

θ = arg z = 1.0471976 rad = 60° = 0.3333333π = π/3 rad ... direction (argument or phase)

Cartesian coordinates:

  • Cartesian form of complex number:
  • z = 5 + 8.660254i
  • Real part:
  • x = Re z = 5
  • Imaginary part:
  • y = Im z = 8.66025404
  • Steps for Calculation

  • Angle notation (phasor):
  • 10 L 60 = 5 + 8.660254i
  • 10 ∠ 60° = 10 L 60° = 10 cis 60° = 10 * (cos 60° + i sin 60°) = 5 + 8.660254i
  • We assume the trigonometric angle argument in degrees.
  • This calculator performs basic arithmetic operations on complex numbers and evaluates expressions in the set of complex numbers.

As an imaginary unit, i or j (in electrical engineering) is used, which satisfies the basic equation i2 = −1 or j2 = −1. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expressions with complex numbers like 5*(1+i)(-2-5i)^2.

Complex numbers in the angle notation or phasor (polar coordinates r, θ) can be written as rLθ where r is the size/amplitude/radius, and θ is the direction (phase) in degrees, for instance, 5L65 which is equivalent to 5*cis(65°). An example of multiplication of two imaginary numbers in the angle/polar/phasor notation is 10L45 * 3L90.

For educational purposes (such as calculations of alternating currents at high school), a quick and accurate complex number calculator is required.

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