Complex Number System

$z=x+iy$

where:

• x is the real part: 

  $\mathrm{Re}(z)=x$

• y is the imaginary part: 

  $\mathrm{Im}(z)=y$

The set of all complex numbers is denoted by Z

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The Argand Diagram

The Argand Diagram is a way to visualize complex numbers on a coordinate plane:

• The horizontal axis (x-axis) represents the real part

  x).

• The vertical axis (y-axis) represents the imaginary part (

  y).For example, the complex number z = 3 + 4i is plotted at the point (3,4) in the plane.

Introduction to Complex Numbers

• The equation 

  $x^2+1=0$

  has no real solution since 

  $x^2=-1$

  is impossible in the real number system.

• To solve this, we define the imaginary unit

  i as:

  $i=\sqrt{-1}$

• Thus, the solutions to the equation are:

  $x=\pm i$

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Powers of i

The imaginary unit follows a cycle:

1. $i=\sqrt{-1}$
2. $i^2=-1$
3. $i^3=i^2\cdot i=(-1)\cdot i=-i$
4. $i^4=(i^2{)}^2=(-1{)}^2=1$

Since 

$i^4=1$

 the powers of i repeat in cycles of four.