Complex Number Calculator — Add, Multiply & Divide
Quick answer: Enter two complex numbers to compute all four arithmetic operations, modulus, argument, and conjugate.
Calculate complex number operations for A = a + bi and B = c + di. Computes sum, difference, product, quotient, modulus (|z|), argument (angle), and complex conjugate. Essential for electrical engineering, physics, and advanced mathematics.
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Last reviewed: April 2026Report an error
Complex A = a + bi
Complex B = c + di
A + B
4 + 2i
Sum: 4 + 2i. Product: 11 − 2i. |A| = 5, |B| = 2.236068.
A + B
4 + 2i
A − B
2 + 6i
A × B
11 − 2i
A ÷ B
-1 + 2i
|A| Modulus
5
arg(A)
53.1301°
Ā (conjugate)
3 − 4i
|B| Modulus
2.236068
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How to Use This Complex Numbers Calculator
- 1Enter the real and imaginary parts of complex number A.
- 2Enter the real and imaginary parts of complex number B.
- 3Read sum, difference, product, quotient, modulus, argument, and conjugate.
Frequently Asked Questions
- A number of the form a + bi, where a is the real part, b is the imaginary part, and i = √(−1). Example: 3 + 4i.
- (a+bi)(c+di) = (ac−bd) + (ad+bc)i. Use FOIL, remembering that i² = −1.
- |a + bi| = √(a² + b²). It is the distance from the origin in the complex plane. |3+4i| = √(9+16) = 5.
- Multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = (a+bi)(c−di)/(c²+d²).
- The conjugate of a+bi is a−bi (flip the sign of the imaginary part). (a+bi)(a−bi) = a²+b² (always real).
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