Complex Number Addition

Sum of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i

You add two complex numbers one component at a time. The real parts join up, the imaginary parts join up, and that's it: (a + bi) + (c + di) = (a + c) + (b + d)i. So (3 + 2i) + (1 − 4i) gives 4 − 2i. There's a nice picture behind this. Treat each complex number z = a + bi as a vector in the Argand plane, with the real axis running horizontally and the imaginary axis vertically. The sum then obeys the parallelogram rule you already know from vectors: drop the second vector's tail onto the first vector's head, and the result runs from the origin out to that new endpoint. Adding complex numbers is commutative (z₁ + z₂ = z₂ + z₁) and associative, with 0 as the identity element. Subtraction behaves identically: (a + bi) − (c + di) = (a − c) + (b − d)i.

Applications

This shows up all over the place. In electrical engineering, voltages and currents become AC phasors, and a series circuit just adds them. Digital signal processing leans on it too, since the FFT breaks a signal into complex components that get summed back together when you synthesize it. Quantum mechanics uses the superposition principle to add probability amplitudes, which are themselves complex numbers. And it turns up in control systems and in plain 2D vector math, where working on the complex plane makes the geometry easier.

FAQ

Why don't the real and imaginary parts mix? Because i, the imaginary unit, is linearly independent from 1. Real numbers and imaginary numbers live on different axes, so you can't merge them, any more than you'd add together the x and y components of a 2D vector.

What if one of the numbers is purely real or purely imaginary? That's fine. Just treat the missing part as zero. For instance, 5 + (2 + 3i) = (5 + 2) + 3i = 7 + 3i, since the 5 is really 5 + 0i.

Is it the same as adding 2D vectors? Pretty much, yes. A complex number is mathematically the same thing as a 2D vector (a, b), so the sum follows that same parallelogram rule.