Complex Number Calculator
Perform operations on complex numbers with step-by-step solutions and visualizations
Quick examples:
z1 (First Number)
z1 = 3 + 4i = 5.0000 * cis(53.13°)
z2 (Second Number)
z2 = 1 + 2i = 2.2361 * cis(63.43°)
Result
Rectangular Form
-5 + 10i
11.1803
Magnitude (r)
116.57°
Argument (θ)
Calculation Steps
(3 + 4i) * (1 + 2i)
= (3*1 - 4*2) + (3*2 + 4*1)i
= -5 + 10i
Complex Plane
z1 z2 Result
z1 Properties
Magnitude (r):5.0000
Argument (θ):53.13°
Conjugate:3 - 4i
|z1|²:25.0000
About Complex Numbers
Complex numbers extend the real number system by introducing the imaginary unit i, where i² = -1. They are essential in mathematics, physics, and engineering for representing quantities with both magnitude and direction.
Key Formulas
- Rectangular: z = a + bi
- Polar: z = r(cos(θ) + i*sin(θ)) = r*cis(θ)
- r = √(a² + b²), θ = atan2(b, a)
- Multiplication: r1*r2 * cis(θ1 + θ2)
- Division: (r1/r2) * cis(θ1 - θ2)
- De Moivre: z^n = r^n * cis(n*θ)
- Roots: z^(1/n) = r^(1/n) * cis((θ + 2πk)/n)
Key Concepts
- De Moivre's theorem simplifies powers and roots using polar form
- The n-th roots of any complex number are equally spaced on a circle
- Multiplication rotates and scales; division rotates backward and scales inversely
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