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Complex Number Calculator

Enter two complex numbers and choose an operation. The Argand diagram shows both inputs and the result as vectors. Step-by-step KaTeX shows every calculation.

Input

z1 = a + bi

z2 = c + di

Results

Result (rectangular)
4 + 2i
Modulus |z|
4.472136
Argument (radians)
0.463648
Argument (degrees)
26.57°
Polar form
4.4721 ∠ 26.57°

Argand Diagram

ReImz1z2Result

Step-by-Step

1. Formula

z1+z2=(a+c)+(b+d)iz_1 + z_2 = (a+c) + (b+d)i

2. Substitute

(3+4i)+(1−2i)(3 + 4i) + (1 - 2i)

3. Result

=4+2i= 4 + 2i

Reference Guide

Addition and Subtraction

Add or subtract the real and imaginary parts separately.

(a+bi)+(c+di)=(a+c)+(b+d)i(a+bi) + (c+di) = (a+c) + (b+d)i

Multiplication

Use FOIL and remember that i squared equals -1.

(a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di) = (ac-bd) + (ad+bc)i

Division

Multiply numerator and denominator by the conjugate of the denominator to eliminate i from the bottom.

z1z2=z1⋅z2ˉ∣z2∣2\frac{z_1}{z_2} = \frac{z_1 \cdot \bar{z_2}}{|z_2|^2}

Polar Form

Every complex number can be written in polar form using its modulus and argument.

z=r(cos⁡θ+isin⁡θ)=reiθz = r(\cos\theta + i\sin\theta) = re^{i\theta}