Complex Number Division Calculator

Perform division operations on complex numbers (a+bi)/(c+di) with this calculator

Input Parameters

Divide by:

Calculation Results

Calculation Formula

Result = (a+bi)/(c+di) = [(ac+bd)/(c²+d²)] + [(bc-ad)/(c²+d²)]i

Where:
a = real part of numerator
b = imaginary part of numerator
c = real part of denominator
d = imaginary part of denominator

Calculation Result

Result will be displayed here

Complex Number Division Calculator Usage Guide

Learn how to use the Complex Number Division Calculator and understand the mathematical principles behind complex division

How to Use This Calculator

  1. Enter the real part (a) and imaginary part (b) of the numerator in the top section
  2. Enter the real part (c) and imaginary part (d) of the denominator in the bottom section
  3. Click the "Calculate" button to perform the division
  4. The result will be displayed in the form a+bi, showing both the real and imaginary components

Mathematical Principles

Complex number division is performed by multiplying both numerator and denominator by the conjugate of the denominator:

(a+bi)/(c+di) = [(a+bi)(c-di)]/(c+di)(c-di) = [(ac+bd)/(c²+d²)] + [(bc-ad)/(c²+d²)]i

Where:

  • c²+d² is the denominator of the result (it's the square of the magnitude of the denominator)
  • ac+bd is the real part of the result
  • bc-ad is the imaginary part of the result

Examples

Example 1: (1+2i)/(3+4i) = (3-2)/(25) + (4-6)/(25)i = 1/25 - 2/25i = 0.04 - 0.08i

Example 2: (2+3i)/(2-3i) = (4+9)/(4+9) + (6-6)/(4+9)i = 13/13 + 0/13i = 1

Notes

Remember that division by zero is undefined in complex numbers, so be careful with denominators that could be zero (when c=0 and d=0).