Complex Numbers

ACT Math · Axiom Academy

The ACT occasionally asks you to simplify i raised to a big power or multiply two complex numbers together — both come down to one rule: i^2 = -1 . 1. The Imaginary Unit — A Power That Loops Forever Define , so that i^2 = -1 . Every higher power of i is built from that one fact — and because i^4 = 1 , the powers don't grow, they just cycle through four values forever . The cycle repeats every 4 powers 2. Complex Numbers — Two Coordinates, One Number A complex number a+bi has a real part a and an imaginary part b (the coefficient of i , not bi itself). Plot it as a point (a,b) on the complex plane, and adding or subtracting two complex numbers works exactly like adding or subtracting vectors — combine the real parts, then combine the imaginary parts, each on its own axis. The horizontal coordinate — how far along the real axis. The vertical coordinate — how far up the imaginary axis (note: it's b , not bi ). (3+2i)+(1+5i) = (3+1)+(2+5)i = 4+7i . (5+3i)-(2+i) = (5-2)+(3-1)i = 3+2i . Real and imaginary parts never mix during addition or subtraction — an ACT distractor answer often comes from adding a real part to an imaginary part by mistake. Keep the two axes separate and the arithmetic is just like combining like terms. 3. Multiplying, Conjugates, and Dividing

This is the written version of the interactive lesson above. See the full ACT Math course.