Basics

Complex numbers, $\Complex$, are an extension of the real numbers, $\R$ that include imaginary numbers.

Suppose we define $i=\sqrt{-1}$, then imaginary numbers are an scalar multiple in $\R$ of $i$

A complex number $z\in\Complex$ is a number of the form

$$ z=x+iy~~~~~~~x,y\in\R $$

Complex Conjugation, $z^*$

A special operation for complex numbers is complex conjugation and is denoted by $z^*$. For the complex conjugate we swap the sign of the scalar of the imaginary part of $z$.

$$ z^*=x-iy $$

For instance $z=1+i\rightarrow z^*=1-i$

• Motivation for complex conjugation

Real and Imaginary Parts

Using the complex conjugate we can extract the real and imaginary parts of $z$

$$ \textrm{Re}(z)=\frac{z+z^}{2}\\ \textrm{Im}(z)=\frac{z-z^}{2i} $$

Modulus

We define the modulus as $|z|=\sqrt{x^2+y^2}$

Complex conjugation provides another way

$$ |z|=\sqrt{zz^*}=\sqrt{(x+iy)(x-iy)}=\sqrt{x^2+y^2} $$

Properties: