Complex Number Arithmetic

Complex Analysis · Axiom Academy

LESSON Complex Number Arithmetic Add and subtract complex numbers part by part, then multiply with FOIL — where i^2=-1 does the magic. 1. Addition: Combine Each Part Separately To add two complex numbers, combine the real parts with each other and the imaginary parts with each other — just like collecting like terms in algebra. Geometrically, every complex number is an arrow in the plane (right = real, up = imaginary), and adding them is laying the second arrow tip-to-tail on the first. Real with real, imaginary with imaginary. 2. Subtraction: The Same Idea, Part by Part Subtraction works the very same way — subtract the real parts , then subtract the imaginary parts . Here the two parts live on separate tracks: watch the real bar and the imaginary bar each settle to their own difference, never mixing. Subtract the real parts on their own number line: 3 - 1 = 2 . Subtract the imaginary parts on their own number line: 2 - 5 = -3 . A real part can only combine with a real part; i behaves like a variable you cannot cross with a constant. Pair the two results back together to get the answer 2 - 3i . (3 + 2i) - (1 + 5i) = (3 - 1) + (2 - 5)i = 2 - 3i. The imaginary part is the coefficient of i , so here it is -3 , not -3i . 3. Multiplication: FOIL, Then i^2 = -1

This is the written version of the interactive lesson above. See the full Complex Analysis course.