Lecture 2

Complex numbers

We describe a set of numbers with form \(a+bi\), where both \(a\) and \(b\) are real numbers, and \(i\) is the square root of \(-1\). These numbers are called the complex numbers, and the set of them is denoted \(\mathbb{C}\). Traditionally, an arbitrary complex number is denoted with the letter \(z\). In symbols: \[\mathbb{C} = \{z=a+bi~|~i=\sqrt{-1};~a,b\in\mathbb{R}\}\]

We can utilize this representation of a complex number in order to define multiplication: \[(a+ib)(c+id) := ac-bd+i(ad+bc)\]

Definition. The complex conjugate of \(z=c+id\) is \(\bar{z}=c-id\).

Definition. For a complex number \(z=a+bi\) we call \(a\) the real part, and \(b\) the imaginary part. In symbols, the real part is denoted \(\Re[z]\) and the imaginary part \(\Im[z]\).

Point representation of complex numbers

You should be familiar with typical graphing of points on a Cartesian plane. As an example, we take two real numbers \((x,y)\) and plot them:

Now for a complex number: \(z=2+3i\). Notice \(z\) is completely described by two real numbers, in this case \((2,3)\). Generally, think of a complex number as being represented by two reals \((x,y)\). We plot our complex number much the same: