Complex Numbers - Introduction to Complex Numbers

A complex number is defined to be an entity of the form \(x+yi\), where \(x\) and \(y\) are real numbers.

Here \(i\) is taken to be a distinguished symbol with no prior meaning attached to it for the moment.

You can think of a complex number to be just a neat and economical way of combining two real numbers together tightly into one entity, just like a two-dimensional vector \(\begin{pmatrix} x \\ y \end{pmatrix}\).

Usually, we use the letters \(z\) and \(w\) to denote complex numbers, reserving \(x\), \(y\), \(u\) and \(v\) to stand for real numbers.

The set of all complex numbers is denoted by \(\mathbb{C}\), i.e., \(\mathbb{C} := \{x+yi \mid x, y \in \mathbb{R}\}\).

Because each complex number is just a compact way of binding two otherwise unrelated real numbers together, we can identity every complex number to be just a point on the two-dimensional plane \(\mathbb{R}^2 := \{(x, y) \mid x, y \in \mathbb{R}\}\). As such, we refer to \(\mathbb{C}\) geometrically as the complex plane.