adding complex numbers

Adding complex numbers is as easy as keeping two separate tallies: add the real parts together, and add the imaginary parts together, and never mix the two. Real and imaginary parts are like apples and oranges — you count each kind on its own.

In symbols, (a + bi) + (c + di) = (a + c) + (b + d)i. Subtraction works the same way, subtracting matched parts: (a + bi) - (c + di) = (a - c) + (b - d)i. This is exactly combining like terms, treating i the way you would treat an unknown letter.

Geometrically, addition is the parallelogram or tip-to-tail rule for arrows on the complex plane: slide one arrow so its tail meets the other's tip, and the sum reaches the final tip. This is identical to how you add vectors, which is why complex numbers and two-dimensional vectors share so much intuition.