LEC1

• Numbers systems
• Definition, standard form of complex numbers
• Operations on complex numbers.

Number Systems

• We already have a bunch of number systems.
• $C$ is another one.

Complex Number

$$\begin{array}{r}{x}^2+1=0\\ ⟹x=\sqrt{-1}\end{array}$$

The equation has no read solutions. It does not touch the x-axis.

We define $j$ such that $j^2=-1$.

$$⟹j^2+1=0$$

That equation is literally the definition of $j$–the imaginary number.

Standard Form

$$\begin{array}{r}x+yj\\ \\ C=\{x+yj|x,y\in R,j^2=-1\}\end{array}$$

Every Real number is a Complex number

$$\begin{array}{r}x=x+0\\ 0=0j\\ ⟹x=x+0j\\ ⟹x\in C\end{array}$$

If we set the imaginary part to zero, we get a purely real number. If we set the real part to zero, we can a purely imaginary number (like $2j$).

Notation:

$$\begin{array}{r}z=x+yj\\ Re(z)=x\\ Im(z)=y\end{array}$$

Operations on Complex Numbers

• Add (corresponding terms)

• Sub (corresponding terms)

• Multiply (open the brackets dude)

• Division is same as multiplying with the inverse.

• Inverse can be found by multiplying numerator and denominator by the complex conjugate.