The Complex Plane

Electrical engineers picture an alternating voltage as an arrow spinning in a plane, and radar and audio engineers plot their signals the same way. That picture — turning a complex number into a point you can actually see — is the complex plane, and it makes the whole subject suddenly visual.

A complex number z = a + bi carries two real numbers, a and b. Two numbers — that is exactly what a point in the plane needs. So picture z as the point with coordinates (a, b): the real part runs along the horizontal axis, the imaginary part up the vertical one. This picture is the complex plane, or Argand diagram.

Suddenly complex numbers are geometry. The reals are the horizontal axis, i sits one unit up at (0, 1), and every complex number is an arrow from the origin to its point.

It's easy to swap the coordinates: z = a + bi plots as (a, b) — real part first (horizontal), imaginary part second (vertical) — never (b, a). For example z = 2 + 5i sits two units right and five units up, not five right and two up.

And don't let the name "imaginary axis" fool you: it isn't measuring something fake or lesser — it's simply the second, equally real coordinate needed to pin the point down, exactly as the y-axis pins down a point in the ordinary plane.

How far out? The modulus

The most natural measurement of a point is its distance from the origin. For z = a + bi that distance is the modulus |z|, and the Pythagorean theorem hands it to us directly.

Step 1 — drop the right triangle. From the point (a, b), the horizontal leg has length a and the vertical leg has length b; the hypotenuse is the arrow to z.

Step 2 — apply Pythagoras. The squared hypotenuse is the sum of the squared legs:

|z|^2 = a^2 + b^2.

Step 3 — take the root (the modulus is a length, so non-negative):

|z| = \sqrt{a^2 + b^2}.

Notice this is exactly the z\bar{z} = a^2 + b^2 from the conjugate — so |z| = \sqrt{z\bar{z}}. For example z = 3 + 4i has |z| = \sqrt{9 + 16} = \sqrt{25} = 5.

Which direction? The argument

Distance alone doesn't pin down a point — you also need a direction. The argument \arg z is the angle \theta the arrow makes with the positive real axis, measured anticlockwise. Together, the pair (r, \theta) — modulus and argument — locates z just as well as (a, b) does.

Step 1 — read the legs from the angle. In the right triangle, the adjacent leg is r\cos\theta and the opposite leg is r\sin\theta, where r = |z|. That is just trigonometry on the triangle:

a = r\cos\theta, \qquad b = r\sin\theta.

Step 2 — rebuild z from (r, \theta). Substitute into z = a + bi:

z = r\cos\theta + (r\sin\theta)\,i = r(\cos\theta + i\sin\theta).

This is the polar form. Going the other way — from (a, b) back to (r, \theta) — use

r = \sqrt{a^2 + b^2}, \qquad \theta = \operatorname{atan2}(b, a).

(The \operatorname{atan2}(b, a) function is just \arctan(b/a) done carefully, so it returns the correct angle in all four quadrants rather than getting confused by signs.)

Represent z = a + bi as the point (a, b) in the plane. Then:

If z = a + bi is the point (a, b), then a complex number is essentially a two-dimensional vector — and addition agrees perfectly. Adding componentwise,

(a + bi) + (c + di) = (a + c) + (b + d)i,

is exactly the tip-to-tail vector addition of (a, b) and (c, d). The modulus |z| is the vector's length, and the argument is its direction. So everything you know about adding arrows transfers straight across.

What complex numbers add beyond vectors is a genuine multiplication — and as the next page shows, that multiplication has a beautiful geometric meaning that plain vectors lack.

Plot a point and read off r and θ

Move z around with the sliders. The arrow from the origin is z = a + bi; the diagram shows the legs a and b, the modulus r = \sqrt{a^2 + b^2}, and the argument \theta — all updating live. Watch how the same point can be named two ways: by (a, b) or by (r, \theta).

See it explained