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Arithmetic, functions, and Argand diagram visualization for complex numbers
Perform arithmetic and functions on complex numbers
Format: a+bi, a-bi, 3i, -2.5+1.5i
= 3 + 4i
Format: a+bi, a-bi, 3i, -2.5+1.5i
= 1 - 2i
Forms
Key Identities
Arithmetic Rules
Conjugate & Modulus
(3+4i) × (1−2i)
= 11−2i
|z| = √125 ≈ 11.18 | arg ≈ −10.3°
(1+i) ÷ (1−i)
= i
|z| = 1 | arg = 90°
e^(iπ) using exp on 0+πi
= −1 + 0i
The most beautiful equation in mathematics
√(−1) = i
= 0+i
Polar: 1 ∠ 90°
Input Format: Enter rectangular numbers as 3+4i, -2-i, or 5i
Polar Input: Switch to r∠θ mode and enter magnitude and angle in degrees
Mode Conversion: Click a+bi or r∠θ to switch mode — values are automatically converted
Power: Integer powers use De Moivre's theorem; fractional/complex powers use the general formula e^(z₂ ln z₁)
Results: Click the copy icon on any form to copy the plain-text value to clipboard