COMPLEX PLANE AND APPLICATIONS : COMPLEX PLANE AND APPLICATIONS Real Numbers a + 0i Imaginary Numbers 0 + bi Complex Numbers a + bi
PowerPoint Presentation : Polar representation z = r e i θ 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 .
PowerPoint Presentation : 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….
PowerPoint Presentation : ▪ 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 ● ● ● Im Re z 1 = + i z 2 = - i z 3 = 2 z 4 = -2 ●
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
PowerPoint Presentation : 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 e i θ Z = r ( cos θ + 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
PowerPoint Presentation : 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,
PowerPoint Presentation : Convert the form z = 5 + 2i into Polar form. Try this!
PowerPoint Presentation : Convert the form z = -5 – 2 i into Polar form. Try this!
PowerPoint Presentation : Convert the point in to Polar Form z= - 9 + 2i Try this!
PowerPoint Presentation : Convert the point in to Polar Form z = 4 – 2i Try this!
PowerPoint Presentation : 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
PowerPoint Presentation : 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 = (r 2 ( 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 - 4 i Write the answer in rectangular form.
PowerPoint Presentation :
PowerPoint Presentation :