Complex Number Calculator
Perform operations on complex numbers. Convert between rectangular and polar form, find roots and powers.
z1 (First Number)

z1 = 3 + 4i = 5 * cis(53.13deg)

z2 (Second Number)

z2 = 1 + 2i = 2.236068 * cis(63.43deg)

Result
-5 + 10i
Rectangular Form
11.1803
Magnitude (r)
116.57deg
Argument (theta)
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:5.0000
Argument:53.13deg
Conjugate:3 - 4i
|z1|^2:25.0000

Key Formulas

Rectangular: z = a + bi
Polar: z = r(cos(theta) + i*sin(theta)) = r*cis(theta)
r = sqrt(a^2 + b^2), theta = atan2(b, a)
Multiplication: r1*r2 * cis(theta1 + theta2)
Division: (r1/r2) * cis(theta1 - theta2)
De Moivre: z^n = r^n * cis(n*theta)
Roots: z^(1/n) = r^(1/n) * cis((theta + 2*pi*k)/n)

Tips:

  • 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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