Complex Numbers in the Plane

A complex number a + bi is not a single value on a line, it's two numbers glued together: a real part and an imaginary part. That naturally makes it a point with two coordinates, (a, b), on a plane where the horizontal axis measures the real part and the vertical axis measures the imaginary part. Because it has both a location and a direction from the origin, it behaves like an arrow, or vector, rather than a dot on a number line.

That arrow has a length and a direction, which gives the polar form. The length is the modulus, r = √(a² + b²), the distance from the origin. The direction is the argument, theta, the angle the arrow makes with the positive real axis, found from tan(theta) = b/a with attention to which quadrant (a, b) sits in. Once you know r and theta, a + bi rewrites as r(cos theta + i sin theta), the same point described by distance and rotation instead of across and up.

This connects directly to the unit's bigger theme of circles and rotation. Multiplying complex numbers in polar form multiplies their moduli and adds their arguments, so multiplication becomes rotation and scaling instead of messy algebra. That's why multiplying by i, which has modulus 1 and argument 90 degrees, always spins a complex number a quarter turn counterclockwise without changing its distance from the origin. This idea sets up polar coordinates and vectors later in the unit, where position is described by r and theta instead of x and y.