Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
This calculator is also known as Complex Division Calculator.
Read the complete guideDivision by Conjugate Multiplication
To compute (a+bi)/(c+di), multiply top and bottom by the conjugate (c−di): the denominator becomes c²+d² (a real number). The result is [(ac+bd) + (bc−ad)i] / (c²+d²). In polar form: z₁/z₂ = (r₁/r₂)·cis(θ₁−θ₂).
Examples
Divide (3+4i) by (1−2i)
Compute (3+4i)/(1−2i).
Multiply by conjugate: (3+4i)(1+2i) / (1+4) = (3+6i+4i+8i²) / 5 = (−5+10i)/5 = −1+2i.
Key takeaway: The key insight is that (c+di)(c−di) = c²+d², which eliminates i from the denominator.
Dividing Complex Numbers Efficiently
Strategies for complex division:
- Multiply numerator and denominator by the conjugate of the denominator
- The denominator becomes c²+d², which is always a positive real number
- For polar form division, just divide the moduli and subtract the arguments
Frequently Asked Questions about Complex Division Calculator
What if the denominator is zero?
Division by 0 + 0i is undefined, just as division by zero is undefined for real numbers. The calculator will show an error message.
Can I use polar form for division?
Yes. z₁/z₂ = (r₁/r₂)∠(θ₁−θ₂). Polar form is simpler when you only need the modulus and argument of the quotient.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
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