COMPLEX PLANE AND APPLICATIONS : COMPLEX PLANE AND APPLICATIONS
Slide 2 : Polar representation
z = r eiθ Rectangular representation
z = x + iy Complex numbers can be represented in two ways
CARTESIAN COORDINATES : CARTESIAN COORDINATES The representation of a complex number as a sum of a real and imaginary number
z = x + iy
is called its Cartesian form.
The Cartesian form is also referred to as Rectangular form.
Slide 4 : The name “Cartesian” suggests that we can represent a complex number by a point in the real plane or
A complex number can be plotted on a plane with two perpendicular coordinate axes.
The real part x representing the horizontal position
The imaginary part y representing the vertical position. Continues….
Slide 5 : ▪ The horizontal x-axis, called the real axis.
▪ The vertical y-axis, called the imaginary axis. CARETESIAN PLANE
Complex Plane : Complex Plane xy-plane is also known as the Complex Plane.
Example : Complex numbers can be represented by a two dimensional graph.
We see the graph of the complex number ( 3 – 2i ). Example
EXAMPLE : EXAMPLE
Lets Try : 1. Plot the complex number on complex plane
z = 2 + 2i Lets Try
Lets Try : 2. Plot the point on complex plane
z = 4 – 2i Lets Try
Lets Try : 3. Plot the point on complex plane
z = -7 + 2i Lets Try
Lets Try : 4. Plot the point on complex plane
z = 0 – 4i Lets Try
Slide 13 : 5. Plot the point on complex plane
z = -1 – 2i Lets Try
POLAR COORDINATES : POLAR COORDINATES In addition to the Cartesian form, a complex number z may also be represented in Polar form:
z = r eiθ
Z = r (cosθ + i sin θ)
Here, r is a real number representing the magnitude of z, and θ represents the angle of z in the complex plane. Tan θ = y/x
Slide 15 : Multiplication and Division of complex numbers is easier in polar form:
Addition and subtraction of complex numbers is easier in Cartesian form.
Converting Between Forms : Converting Between Forms To convert from the Cartesian form z = x + iy to polar form,
Slide 17 : Convert the form z = 5 + 2i into Polar form. Try this!
Slide 18 : Convert the form z = -5 – 2i into Polar form. Try this!
Slide 19 : Convert the point in to Polar Form
z= - 9 + 2i Try this!
Slide 20 : Convert the point in to Polar Form
z = 4 – 2i Try this!
Slide 21 : De Moivre's Theorem is a relatively simple formula for calculating powers of complex numbers.
De Moivre's formula states that for any real number x and any integer n,
(cosx + isinx)n = cos(nx) + isin(nx). De Moivre's Theorem
Powers of complex numbers
Slide 22 : Find (-2 + i)2.
First, write -2 + i in polar form. Note that its graph is in the
second quadrant of the complex plane.
r = = Arctan
= 2.68
=
The polar form of -2 + i is cos 2.68 + i sin 2.68.
Now use De Moivre’s Theorem to find the second power.
(-2 + i)2 = [(cos 2.68 + i sin 2.68)]2
= (r2 (cos 2(2.68) + i sin 2(2.68) ) De Moivre’s Theorem
= 5(cos 5.36 + i sin 5.36 ) Simplify.
= 5(0.60 + i(-0.80))
= 3 - 4i Write the answer in rectangular form.
Slide 23 :
Slide 24 :