Electrical and Computer Engineering #

Analytical Foundations

Analytical Foundations of ECE # Complex Numbers # Definition of a Complex Number # Consider \(x^2-1=0\) . To find the roots: \[\begin{aligned} x^2-1 &= 0\\ x^2 &= 1\\ x &= \sqrt{1}\\ \rArr x &= \plusmn{1} \end{aligned}\] Now consider \(x^2+1=0\) : \[\begin{aligned} x^2 + 1 &= 0\\ x^2 &= -1\\ x &= \sqrt{-1}\\ \rArr x &= \plusmn{j} \end{aligned}\] \(j\) is an imaginary unit introduced to solve this type of equation. It is better known as \(i\) : \[i^2 = -1 \begin{cases} \text{Mathematics} \\ \text{Physics} \\ \text{Chemistry} \end{cases}\\\] \[ j^2 = -1 \begin{cases} \text{Engineering} \end{cases}\] A complex number is represented as \(z = a+jb\) , where: \[\begin{aligned} a &= Re(z)\ &\text{Real part of z}\\ b &= Im(z)\ &\text{Imaginary part of z} \end{aligned}\] \[\] Complex Number Example # Sketch the following complex numbers in the 2D complex plane.