Complex Number Arithmetic
(a + bi) ± (c + di), ×, ÷. Real part, imaginary part, modulus, argument.
Result
Result
4.0000 + 6.0000i
|z| = 7.2111, arg = 56.31°.
- z₁3.0000 + 4.0000i
- z₂1.0000 + 2.0000i
- Operationadd
- Real part4.000000
- Imag part6.000000
- Modulus |z|7.211103
- Argument56.3099°
- Polar form7.211 ∠ 56.31°
Step-by-step
- (3 + 4i) + (1 + 2i) = (3+1) + (4+2)i = 4.0000 + 6.0000i.
- Modulus = √(re² + im²) = 7.2111.
- Argument = atan2(im, re) = 56.31°.
How to use this calculator
- Pick operation.
- Enter z₁ = a + bi and z₂ = c + di.
- Read result + modulus + argument.
About this calculator
Complex numbers extend reals with i = √−1. Addition/subtraction: combine real and imaginary parts. Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i. Division: multiply numerator and denominator by conjugate. Modulus |z| = √(a²+b²); argument θ = atan2(b, a). Polar form: r∠θ. Used in electrical engineering (impedance), signal processing (FFT), and quantum mechanics.
Frequently asked
Imaginary unit, defined by i² = −1. Allows solving equations like x² + 1 = 0 that have no real solution.
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