Complex Number Calculator
Perform arithmetic operations on complex numbers with step-by-step solutions and polar form conversion
z = a + bi, where i² = -1
Perform arithmetic operations on complex numbers
z₁ = 0
z₂ = 0
Quick Examples
Invalid Input
Real Part
Imaginary Part
Magnitude |z|
Angle θ
Step-by-Step Solution
Identify the complex numbers
z₁ = , z₂ =
Apply the formula
Final result
z₁ z₂ =
Polar Form
Result in polar form:
(cos ° + i sin °)
Conjugate of Result
z̄ (conjugate):
Complex Number Operations Reference
| Operation | Formula |
|---|---|
| Addition | (a+c) + (b+d)i |
| Subtraction | (a-c) + (b-d)i |
| Multiplication | (ac-bd) + (ad+bc)i |
| Division | [(ac+bd)/(c²+d²)] + [(bc-ad)/(c²+d²)]i |
| Magnitude | |z| = √(a² + b²) |
| Conjugate | z̄ = a - bi |
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About Complex Number Calculator
What are Complex Numbers?
A complex number is a number that can be expressed in the form a + bi, where:
- a is the real part
- b is the imaginary part
- i is the imaginary unit, defined as i² = -1
Basic Operations
Addition
z₁ + z₂ = (a + c) + (b + d)i
Subtraction
z₁ - z₂ = (a - c) + (b - d)i
Multiplication
z₁ × z₂ = (ac - bd) + (ad + bc)i
Division
z₁ ÷ z₂ = [(ac + bd) + (bc - ad)i] / (c² + d²)
Important Concepts
Magnitude (Modulus)
|z| = √(a² + b²)
Conjugate
z̄ = a - bi
Polar Form
z = r(cos θ + i sin θ)
Where r = |z| and θ = atan2(b, a)