Complex Number Calculator

Frequently Asked Questions

How do you multiply complex numbers?

(a+bi)(c+di) = (ac − bd) + (ad + bc)i, using i² = −1. Division multiplies by the conjugate of the denominator to clear the i.

What is the modulus and argument?

For a+bi, modulus |z| = √(a²+b²) (distance from origin) and argument = atan2(b, a) (angle). Together they give the polar form r(cosθ + i sinθ).

What is the complex conjugate?

The conjugate of a+bi is a−bi. Multiplying a number by its conjugate gives a real number, a²+b² = |z|² - the key trick for division.

Why are complex numbers useful?

They model AC circuits, signal processing (Fourier), control systems, and quantum mechanics, and they make polynomial roots and rotations elegant.

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How This Calculator Works

A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1. Addition and subtraction act component-wise: (a+bi) ± (c+di) = (a±c) + (b±d)i. Multiplication uses the distributive law with i² = −1: (a+bi)(c+di) = (ac − bd) + (ad + bc)i. Division multiplies top and bottom by the conjugate of the denominator to make it real: (a+bi)÷(c+di) = [(ac+bd) + (bc−ad)i] ÷ (c²+d²). The modulus |z| = √(a²+b²) is the distance from the origin in the complex plane; the argument θ = atan2(b, a) is the angle from the positive real axis. The conjugate z̅ = a − bi mirrors across the real axis. Polar form writes z = r(cosθ + i·sinθ) = re^iθ, which makes multiplication a matter of multiplying moduli and adding arguments.

Worked Example: (3 + 2i) and (1 − 4i)

Sum: (3+1) + (2−4)i = 4 − 2i. Difference: (3−1) + (2−(−4))i = 2 + 6i. Product: (3·1 − 2·(−4)) + (3·(−4) + 2·1)i = (3 + 8) + (−12 + 2)i = 11 − 10i. Quotient: denominator c²+d² = 1 + 16 = 17; numerator real = 3·1 + 2·(−4) = 3 − 8 = −5; numerator imag = 2·1 − 3·(−4) = 2 + 12 = 14, so (3+2i)÷(1−4i) = −5÷17 + (14÷17)i ≈ −0.2941 + 0.8235i. Modulus of 3+2i = √13 ≈ 3.6056; argument = atan2(2, 3) ≈ 33.69°. Conjugate = 3 − 2i.

Where Complex Numbers Are Used

Electrical engineering: AC circuit analysis - impedance is a complex number (phasors) combining resistance and reactance.

Signal processing: the Fourier transform represents signals as sums of complex exponentials e^iωt.

Control theory & dynamics: system stability is read from the location of complex poles in the s-plane.

Quantum mechanics: probability amplitudes are intrinsically complex; the wavefunction lives in ℂ.

Common Mistakes & Edge Cases

Forgetting i² = −1 when multiplying - the cross term sign is the most common slip.

Dividing without the conjugate trick - you cannot just divide real and imaginary parts separately.

Using atan(b÷a) for the argument: it loses the quadrant. Always use atan2(b, a), as this tool does.

Division by 0 + 0i is undefined - reported explicitly here.

Confusing modulus (a length, always ≥ 0) with the real part, or argument (an angle) with the imaginary part.

Related Concepts & Limits

Euler's formula e^iθ = cosθ + i·sinθ unifies the rectangular and polar views and yields the celebrated identity e^iπ + 1 = 0. De Moivre's theorem extends polar form to powers and roots: zⁿ has modulus rⁿ and argument nθ, and every non-zero complex number has exactly n distinct n-th roots equally spaced on a circle. The fundamental theorem of algebra guarantees every degree-n polynomial has n complex roots - the basis of the Polynomial Roots Calculator. The argument here is reported in degrees in (−180°, 180°] from atan2; add 360° for a positive convention if needed. Computations use double precision (≈ 15 significant digits).

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