Notes on Complex Numbers - Key Concepts

A complex number is of the form $z = a + ib$ where $a, b \, \in \, \mathbb{R}$ and $i^2 = -1$ ; $i$ is the imaginary unit (iota).
Common notations: $z = a + ib$ ; also represented as the coordinate pair $z = (a, b)$ .
The real part and imaginary part:
- $\Re(z) = a$ (the part not attached to $i$ )
- $\Im(z) = b$ (the part attached to $i$ )
Intuition: the number attached to $i$ is the imaginary part; the other component is the real part.

Let $z1 = a + ib$ and $z2 = c + id$ .
They are equal iff their corresponding parts are equal:
- $z1 = z2 \iff a = c \text{ and } b = d$
In words: two complex numbers are equal when both their real parts are equal and their imaginary parts are equal.

The conjugate of $z = a + ib$ is $\overline{z} = a - ib$ ; equivalently, $\overline{z} = (a, -b)$ .
If a complex number is written as z = (a,b), its conjugate is z̄ = (a, -b).
Key special relations:
- If $z$ is purely real (i.e., its imaginary part is zero, $b = 0$ ), then $z = \overline{z} = a$ .
- If $z$ is purely imaginary (i.e., its real part is zero, $a = 0$ ), then $z = ib$ and $\overline{z} = -ib$ ; hence $z = -\overline{z}$ .
Summary for real/imaginary cases:
- Purely real: $z = a\, (b=0)\quad \Rightarrow\quad z = \overline{z}$
- Purely imaginary: $z = ib\, (a=0)\quad \Rightarrow\quad z = -\overline{z}$

Modulus (or modulus) of $z = a + ib$ is the distance from the origin to the point $(a,b)$ in the complex plane:
- $|z| = \sqrt{a^2 + b^2}$
Geometric interpretation: treat $z$ as a point $P = (a,b)$ ; the origin is $O = (0,0)$ ; the length $OP$ is the modulus.
Right triangle interpretation: if $OS = a$ and $PS = b$ , then by Pythagoras, $OP^2 = OS^2 + PS^2 = a^2 + b^2 = |z|^2.$

Multiply a number by its conjugate:
- $z\overline{z} = (a + ib)(a - ib) = a^2 + b^2 = |z|^2$
Therefore the fundamental relation is:
- $z\overline{z} = |z|^2$
Consequently, $|z| = \sqrt{z\overline{z}}$ and $|z|^2 = z\overline{z} = a^2 + b^2.$
This provides a bridge between algebraic form and geometric modulus.

Conjugation distributes over sums (and differences):
- $\overline{z1 \pm z2} = \overline{z1} \pm \overline{z2}$
This extends to finite sums: for any $z1, z2, \dots, z_n$ ,
- $\overline{z1 \pm z2 \pm \dots \pm zn} = \overline{z1} \pm \overline{z2} \pm \dots \pm \overline{zn}$
The conjugate of a product likewise satisfies: $\overline{z1 z2} = \overline{z1}\,\overline{z2}$ (not explicitly stated in transcript but a standard property; kept for completeness in notes)

Complex number: $z = a + ib, \ a, b \in \mathbb{R}$
Conjugate: $\overline{z} = a - ib$
Real and imaginary parts: $\Re(z) = a, \quad \Im(z) = b$
Equality: $z1 = z2 \iff \Re(z1) = \Re(z2) \text{ and } \Im(z1) = \Im(z2)$
Modulus: $|z| = \sqrt{a^2 + b^2}$
Modulus squared: $|z|^2 = a^2 + b^2$
Product with conjugate: $z\overline{z} = |z|^2 = a^2 + b^2$
Real-imaginary special cases:
- Purely real: $z = \overline{z}$ (Im(z) = 0)
- Purely imaginary: $z = ib, \quad \overline{z} = -ib, \quad z = -\overline{z}$

Complex numbers extend real numbers by including an imaginary axis; representation as z = a + ib parallels the Cartesian plane (a, b).
The real and imaginary parts provide a straightforward way to compare complex numbers term-by-term (like coordinates in a plane).
Conjugation reflects a complex number across the real axis (i.e., flips the sign of the imaginary part).
The modulus provides a radial measure (distance from origin) and connects with the conjugate via z\overline{z} = |z|^2.
Geometric interpretation (point (a,b), right triangle) helps with intuition for the modulus and the Pythagorean theorem.