Complex plane

1. Plotting $a + bi$ on the Argand diagram (real vs. imaginary axes)

Plotting Complex Numbers on the Argand Diagram

A complex number $z = a + bi$ is represented as a point in the Argand diagram (complex plane), where the horizontal axis represents the real part $a$ and the vertical axis represents the imaginary part $b$. This geometric representation transforms algebraic operations into visual transformations.

Each complex number corresponds to exactly one point: move $a$ units along the real axis, then $b$ units parallel to the imaginary axis.

Core plotting rules:

• The real axis is horizontal; purely real numbers ($b = 0$) lie on this axis
• The imaginary axis is vertical; purely imaginary numbers ($a = 0$) lie on this axis
• The origin represents $0 + 0i = 0$
• Negative values of $a$ or $b$ indicate movement in the negative direction along the respective axis

This visualization enables geometric interpretation of complex arithmetic and reveals structural properties invisible in algebraic form.

Example: $z = 3 + 2i$ is plotted at coordinates $(3, 2)$, while $-1 - 4i$ appears at $(-1, -4)$.

2. Calculating the modulus and defining the argument angle

Modulus and Argument of Complex Numbers

The modulus $|z|$ of a complex number $z = a + bi$ is its distance from the origin, computed as $|z| = \sqrt{a^2 + b^2}$. The argument $\arg(z) = \theta$ is the angle measured counterclockwise from the positive real axis to the line segment connecting the origin to $z$.

The modulus quantifies magnitude; the argument specifies direction in the plane.

Core rules:

• $|z| \geq 0$ for all $z$, with $|z| = 0$ if and only if $z = 0$
• $\theta$ is typically measured in radians, with $\theta \in (-\pi, \pi]$ (principal value)
• For $z = a + bi$: $\tan(\theta) = \frac{b}{a}$ when $a \neq 0$
• Quadrant matters: use $\arctan$ carefully or $\text{atan2}(b, a)$ to determine the correct angle

Together, $(r, \theta)$ with $r = |z|$ provide polar coordinates for $z$.

Example: For $z = 3 + 4i$, $|z| = \sqrt{9 + 16} = 5$ and $\arg(z) = \arctan(4/3) \approx 0.927$ radians.

3. Previewing the polar form of complex numbers

Polar Form of Complex Numbers

Any nonzero complex number $z$ can be expressed in polar form as $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ is the modulus and $\theta = \arg(z)$ is the argument. This representation separates magnitude from direction.

Polar form simplifies multiplication and division: magnitudes multiply/divide while arguments add/subtract.

Core conversion rules:

• From rectangular: $r = \sqrt{a^2 + b^2}$ and $\theta = \arg(a + bi)$
• To rectangular: $a = r\cos\theta$ and $b = r\sin\theta$
• Euler's formula (preview): $e^{i\theta} = \cos\theta + i\sin\theta$ leads to $z = re^{i\theta}$
• The argument is multi-valued: $\theta + 2\pi k$ for any integer $k$ represents the same complex number

This form is essential for exponentiation, roots, and oscillatory phenomena.

Example: $z = 1 + i$ has $r = \sqrt{2}$ and $\theta = \pi/4$, so $z = \sqrt{2}(\cos(\pi/4) + i\sin(\pi/4))$.

4. Geometric interpretation of complex addition and complex conjugates

Geometric Operations: Addition and Conjugation

Complex addition $(a + bi) + (c + di) = (a+c) + (b+d)i$ follows the parallelogram rule: the sum is the diagonal of the parallelogram formed by the two complex numbers as adjacent sides from the origin. The complex conjugate $\overline{z} = a - bi$ is the reflection of $z = a + bi$ across the real axis.

These geometric interpretations reveal structural symmetries in the complex plane.

Core geometric rules:

• Addition corresponds to vector addition: place vectors tip-to-tail or use the parallelogram
• Conjugation preserves modulus: $|\overline{z}| = |z|$
• Conjugation negates argument: $\arg(\overline{z}) = -\arg(z)$
• $z \cdot \overline{z} = a^2 + b^2 = |z|^2$ (always real and non-negative)

Conjugation is crucial for division and finding real/imaginary parts.

Example: $(2 + 3i) + (1 - i) = 3 + 2i$ forms a parallelogram; $\overline{2 + 3i} = 2 - 3i$ reflects across the real axis.

5. Applications: Phasors and quantum state vectors

Applications in Signal Processing and Quantum Mechanics

In signal processing, sinusoidal signals are represented as phasors: complex numbers $z = Ae^{i(\omega t + \phi)}$ where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is phase. The complex plane encodes both magnitude and phase, simplifying AC circuit analysis. In quantum mechanics, state vectors are complex-valued, with the complex plane representing probability amplitudes whose squared moduli give probabilities.

Complex representation transforms differential equations into algebraic manipulations.

Core application principles:

• Phasors: Real signal is $\text{Re}(Ae^{i(\omega t + \phi)}) = A\cos(\omega t + \phi)$
• Phase differences correspond to argument differences in the complex plane
• Quantum states: Superposition $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ with $\alpha, \beta \in \mathbb{C}$ and $|\alpha|^2 + |\beta|^2 = 1$
• Rotation in the complex plane models time evolution and interference

Example: A 5-volt AC signal with phase $\pi/3$ is represented as phasor $5e^{i\pi/3}$.