The complex plane — every z as a 2D point
A complex number $z = a + bi$ isn't just a calculation result. It's a point on a plane — with a position, a distance from the origin, and a direction. Dragging that point is the fastest way to feel its geometry.
You know what a complex number is. But do you know where it lives?
You've met $i$ — the square root of $-1$. You can write $z = 3 + 4i$ and follow the algebra. But here's what most textbooks skip: a complex number isn't just an algebraic expression. It's a location. It lives somewhere.
Real numbers live on a line — the number line you've used since childhood. Where does $3 + 4i$ live? It can't go on that line. $i$ points in a direction perpendicular to real numbers, which means complex numbers live in a plane — a flat 2D surface with two axes.
Think of an address in a city on a perfect grid.
Imagine a perfectly rectangular city. Every block is numbered: East/West streets carry the number $a$, North/South streets carry the number $b$. To reach any intersection, you need exactly two numbers: go $a$ blocks East, then $b$ blocks North. Two numbers, two directions, one location.
That's the complex plane. The East/West axis is the real axis — it carries the part $a$. The North/South axis is the imaginary axis — it carries the part $b$. The complex number $z = a + bi$ is the intersection at address $(a, b)$.
Notice what this analogy reveals: knowing $a = 3$ and $b = 4$ tells you the address. But the diagonal distance from the origin — the modulus — tells you something different: how far away the point is, regardless of direction. Those two pieces of information together (address and distance) fully characterise any point on the plane.
The complex plane: two axes, one point per number.
The complex plane (also called the Argand plane) is a standard 2D coordinate system with a specific assignment:
Vertical axis = the imaginary axis. Points on this axis have $a = 0$: they are pure imaginary numbers like $2i$, $-i$.
Every other point has both $a \neq 0$ and $b \neq 0$: these are genuinely complex numbers like $3 + 4i$, $-1 + 2i$.
For any complex number $z = a + bi$, the symbols are defined precisely as follows:
- $z = a + bi$ The complete complex number in Cartesian form. $a$ and $b$ are both real numbers. $i$ is the imaginary unit ($i^2 = -1$).
- $\text{Re}(z) = a$ The real part of $z$. This is the horizontal coordinate — how far right (positive) or left (negative) the point is from the origin.
- $\text{Im}(z) = b$ The imaginary part of $z$. This is the vertical coordinate — how far up (positive) or down (negative) the point is from the origin. Note: $\text{Im}(z) = b$, not $bi$ — the imaginary part is the real number $b$.
The modulus is the Pythagorean distance. The argument is the angle.
Two quantities fully capture everything about a point's position on the complex plane — one for how far, one for which direction:
The Modulus $|z|$
The modulus of $z = a + bi$ is its distance from the origin $(0,0)$. By the Pythagorean theorem applied to the right triangle formed by $a$ (horizontal), $b$ (vertical), and the diagonal $|z|$:
$$|z| = \sqrt{a^2 + b^2}$$
where $a = \text{Re}(z)$ and $b = \text{Im}(z)$. Every step is explicit: $a$ is the horizontal leg, $b$ is the vertical leg, and $\sqrt{a^2 + b^2}$ is the hypotenuse — the straight-line distance from the origin to the point.
The Argument $\theta$
The argument of $z$ (written $\arg(z)$ or $\theta$) is the angle the line from the origin to $z$ makes with the positive real axis. Measured counter-clockwise in radians (or degrees):
$$\theta = \arg(z) = \arctan\!\left(\frac{b}{a}\right)$$
You'll handle the argument in full detail in M06 (Polar Form). For now, it's enough to see it in the simulator below — drag the point and watch $\theta$ change as the arrow rotates.
Let $z = 3 + 4i$. Before dragging anything — predict $|z|$ in your head. Use $|z| = \sqrt{a^2 + b^2}$ with $a = 3$ and $b = 4$.
Given $z = 3 + 4i$, we identify $a = \text{Re}(z) = 3$ and $b = \text{Im}(z) = 4$.
Apply the modulus formula:
$|z| = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
The point $(3, 4)$ is exactly 5 units from the origin. This is a 3-4-5 right triangle — one of the most famous Pythagorean triples. Place $z = 3 + 4i$ in the simulator below and confirm the readout shows $|z| = 5.00$.
What the complex plane gives you — in three precise statements
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$z = a + bi$ is a 2D point at $(a, b)$The real part $a = \text{Re}(z)$ gives the horizontal coordinate. The imaginary part $b = \text{Im}(z)$ gives the vertical coordinate. Every complex number has a unique location on the plane — no two numbers share the same point.
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The modulus $|z| = \sqrt{a^2 + b^2}$ is Pythagorean distanceThe distance from the origin to the point $(a, b)$ is exactly $\sqrt{a^2 + b^2}$ — the Pythagorean theorem applied to the right triangle with legs $a$ and $b$. In quantum mechanics, $|z|^2$ is what probabilities are built from.
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The argument $\theta$ is the angle — and it's continuousYou can rotate the point around the origin (keeping $|z|$ fixed) and $\theta$ changes continuously through all angles from $0°$ to $360°$. This is the full circle of phases that M03 showed real numbers can't provide. The complex plane is what makes arbitrary phase possible.
You can now place any complex number on a plane.
You can measure its distance from the origin.
But there's another way to describe a point —
by its distance and angle alone, no coordinates needed.
- Nielsen, M. A. & Chuang, I. L. — Quantum Computation and Quantum Information, Cambridge University Press, 2000. §2.1.1: State spaces and amplitudes. The modulus arises directly from the Born rule.
- Preskill, J. — Lecture Notes for Physics 229, Caltech, 1998. Chapter 2: Foundations — complex amplitudes and the Hilbert space formalism. Available online
- Strang, G. — Linear Algebra and Its Applications, 4th ed. The geometric interpretation of complex numbers as 2D vectors; the connection between modulus and Euclidean norm.