The Complex Number Calculator handles all operations with complex numbers in the form a + bi. Perform arithmetic, find the modulus and conjugate, convert between rectangular and polar forms, and compute powers using De Moivre's theorem.
Enter the real and imaginary parts of your complex numbers to get instant results with detailed step-by-step working.
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A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i^2 = -1.
The modulus of z = a + bi is |z| = sqrt(a^2 + b^2). The conjugate is z-bar = a - bi.
Every complex number can be written in polar form as r(cos(theta) + i*sin(theta)), where r is the modulus and theta is the argument.
De Moivre's theorem: [r*cis(theta)]^n = r^n * cis(n*theta).
Problema: (3 + 2i)(1 - 4i)
Solucion: 3 - 12i + 2i - 8i^2 = 3 - 10i + 8
Respuesta: 11 - 10i
Problema: Find |3 + 4i|
Solucion: sqrt(9 + 16) = sqrt(25)
Respuesta: 5
Problema: Convert 1 + i to polar
Solucion: r = sqrt(2), theta = 45 degrees
Respuesta: sqrt(2) * cis(45 degrees)
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