🔢 Complex Analysis & Euler's Formula
Complex Number & Polar Form Calculator
Perform complex arithmetic ($z_1 \pm z_2$, $z_1 \times z_2$, $z_1 / z_2$), convert between Cartesian ($a + bi$) and Polar / Exponential form ($r e^{i\theta}$), and compute modulus and phase arguments.
Complex Inputs
Re ($a_1$)
Im ($b_1$)
Re ($a_2$)
Im ($b_2$)
Result (z₁ + z₂) in Rectangular Form
4.000 + 2.000i
Modulus ($|z| = r$)
4.472
Argument ($\theta$)
26.57° (0.464 rad)
Conjugate ($\bar{z}$)
4.000 - 2.000i
💡 Individual Coordinates:
$z_1 = 3 + 4i \implies r_1 = 5.000, \theta_1 = 53.13^\circ$ | $z_2 = 1 - 2i \implies r_2 = 2.236, \theta_2 = -63.43^\circ$
Complex Number Mathematics
Euler's formula connects trigonometry and exponential growth in the complex plane:
- Euler's Form: $$z = a + bi = r(\cos\theta + i\sin\theta) = r e^{i\theta}$$
- Modulus: $$|z| = r = \sqrt{a^2 + b^2}$$
- Argument (Phase Angle): $$\theta = \operatorname{atan2}(b, a)$$
- Multiplication & Division: $$z_1 z_2 = (r_1 r_2) e^{i(\theta_1 + \theta_2)}, \quad \frac{z_1}{z_2} = \left(\frac{r_1}{r_2}\right) e^{i(\theta_1 - \theta_2)}$$