Complex Numbers Vocabulary

Key vocabulary and fundamental concepts from the Complex Numbers lecture notes.

Discriminant ($b^2 - 4ac$)

The expression $b^2 - 4ac$ in the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, used to determine if an equation has repeated real roots ($b^2 - 4ac = 0$), distinct real roots ($b^2 - 4ac > 0$), or non-real imaginary roots ($b^2 - 4ac < 0$).

Imaginary Unit ($i$)

The mathematical symbol $i$ used to denote $\sqrt{-1}$, such that $i^2 = -1$.

General Complex Number

A number represented in the form $x + iy$, where $x$ and $y$ are real numbers ($x, y \in \mathbb{R}$).

Real Part

The scalar component $x$ in the complex number expression $z = x + iy$.

Imaginary Part

The scalar coefficient $y$ of the imaginary unit $i$ in the complex number expression $z = x + iy$.

Argand Diagram

A Cartesian coordinate system diagram where complex numbers are represented as points, with the $x$-axis representing the real part and the $y$-axis representing the imaginary part.

Modulus of a Complex Number ($|z|$)

The distance from the origin $O$ to the point $P(x, y)$ representing $z = x + iy$ on an Argand diagram, calculated as $|z| = \sqrt{x^2 + y^2}$.

Principal Argument

The angle $\theta$ measured between the line $OP$ representing a complex number and the positive real axis, usually given within the range $(\pi, -\pi)$.

Complex Conjugate ($Z^*$)

For a complex number $Z = x + iy$, its conjugate is $Z^* = x - iy$, which represents a reflection of $Z$ across the real axis on an Argand diagram.

Polar Form

The representation of a complex number $z = x + iy$ as $r(\cos(\theta) + i\sin(\theta))$, where $r = |z|$ is the modulus and $\theta = \text{arg}(z)$ is the argument.

Exponential Form

The shorthand representation of a complex number as $r e^{i\theta}$, where $r = |z|$ is the modulus and $\theta = \text{arg}(z)$ is the argument.

DeMoivre's Theorem

A theorem stating that for any complex number in polar form and integer $n$, $[r(\cos(\theta) + i\sin(\theta))]^n = r^n(\cos(n\theta) + i\sin(n\theta))$.

Locus

A geometric path traced out on an Argand diagram by a point representing a variable complex number that satisfies specific geometric or algebraic constraints.

Cube Roots of Unity

The three solutions to the equation $z^3 = 1$, given by $1$, $\omega$, and $\omega^2$, where $\omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$.