The Complex Plane

Every complex number $z = a + bi$ can be thought of as a point or vector in a two-dimensional plane called the Argand diagram (or complex plane). The horizontal axis is the real axis carrying the value $a = \operatorname{Re}(z)$ , and the vertical axis is the imaginary axis carrying $b = \operatorname{Im}(z)$ .

Coordinates and the Cartesian form

Writing $z = a + bi$ is the Cartesian form. The point is located $a$ units to the right of the origin and $b$ units upward. For example:

$z = 3 + 2i$ sits at $(3, 2)$ — in the first quadrant.
$z = -1 + 0i = -1$ lies on the negative real axis.
$z = 0 + i$ is one unit straight up from the origin.
$z = 2 - 3i$ is in the fourth quadrant at $(2, -3)$ .

The vector picture makes addition transparent: adding $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$ is the same as ordinary vector addition in the plane,

$z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i,$

equivalent to placing the two arrows tip-to-tail.

Complex conjugate — reflection across the real axis

The complex conjugate of $z = a + bi$ is

$z^* = a - bi.$

Geometrically this is a reflection across the real axis: the imaginary part flips sign while the real part stays fixed. Conjugation shows up constantly in quantum mechanics because $\langle\psi|\phi\rangle$ involves the conjugate of the bra components.

The unit circle and pure phases

The set of all complex numbers satisfying $|z| = 1$ forms the unit circle in the Argand diagram. Points on this circle have the form $e^{i\theta} = \cos\theta + i\sin\theta$ for some angle $\theta \in [0, 2\pi)$ . This is Euler's formula, and it turns angle into a complex number.

Multiplying any complex number $z$ by $e^{i\theta}$ rotates it counterclockwise by angle $\theta$ without changing its distance from the origin. This rotation interpretation is essential in quantum mechanics: a global phase factor $e^{i\theta}|\psi\rangle$ is physically unobservable, because the Born-rule probability $|e^{i\theta}\alpha|^2 = |\alpha|^2$ is unchanged.

Four special directions

The positive real axis ( $b = 0,\, a > 0$ ), the positive imaginary axis ( $a = 0,\, b > 0$ ), the negative real axis, and the negative imaginary axis divide the plane into four quadrants. The powers of $i$ cycle through those four special directions:

$i^0 = 1,\quad i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1, \ldots$

Each multiplication by $i$ is a $90^\circ$ counterclockwise rotation — a fact that is immediate from the Argand picture and laborious to verify purely algebraically.

Why this matters for quantum mechanics

Quantum states are vectors in a complex vector space. The amplitudes assigned to measurement outcomes are complex numbers, and only their squared moduli give probabilities. The full two-dimensional freedom of the complex plane — both the distance from the origin and the angle — carries physical content: the angle (phase) governs interference between paths, while the squared distance gives the probability. Developing a geometric intuition for the complex plane now will make those ideas concrete rather than abstract when they appear throughout the rest of the course.