complex addition and multiplication

Adding and multiplying complex numbers is exactly the school algebra you already know, with one extra house rule: whenever you see i^2, replace it by -1. To add, combine the real parts and the imaginary parts separately, just like collecting like terms: (a + i b) + (c + i d) = (a + c) + i (b + d). Geometrically this is nothing but vector addition — slide the arrow for one number to the tip of the other.

To multiply, expand the brackets the usual way (FOIL) and then apply i^2 = -1: (a + i b)(c + i d) = ac + i ad + i bc + i^2 bd = (ac - bd) + i (ad + bc). The minus sign in ac - bd is the only thing to remember; it is i^2 = -1 doing its work. A tiny example: (2 + i)(3 + 4i) = 6 + 8i + 3i + 4i^2 = 6 + 11i - 4 = 2 + 11i. No new rules of algebra are needed — only that one substitution.

These operations make the complex numbers a 'field', meaning addition and multiplication behave just as nicely as for real numbers: both are commutative and associative, multiplication distributes over addition, there is a 0 and a 1, and every nonzero number has an inverse. That is what lets you do calculus, solve equations, and build the whole edifice of complex analysis without ever worrying that the algebra might misbehave.