The Complex Plane

Complex Analysis · Axiom Academy

Every complex number has a home — a single point on a 2D map called the Argand diagram. A real number like 3 or −7 sits on a number line — a single position, one piece of information. But a complex number a + bi carries two pieces: a real part and an imaginary part. One line isn't enough room, so we add a second axis and turn the line into a plane. Watch 3 + 4i find its spot. It starts as plain 3 on the horizontal real axis . Then the + 4i carries it straight up four units along the vertical imaginary axis — landing on the point (3, 4). The number and the point are the same object: 3 + 4i is (3, 4). Drop a point, read its complex number Drag the point anywhere on the plane. Wherever it lands, read it off the two axes: how far right or left gives the real part a , how far up or down gives the imaginary part b . Together they name the complex number a + bi . Drag onto an axis and one part goes to 0 — that's a purely real or purely imaginary number. Build a number, watch where it lands Run it backwards. Set the real part a and the imaginary part b , and the point flies to (a, b). Every complex number you can write down has exactly one home on this plane — and every point names exactly one number. Same point, read two ways: the number a + bi and the location (a, b). A picture you can compute with

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