Transcript 11.2 Geometric Representation of Complex Numbers

10.4 PART 2 GEOMETRIC
REPRESENTATION OF COMPLEX
NUMBERS
Can be confusing, polar form has an anglewith it, rectangular
form does not, this all takes place in a plane that is not the x
and y axis, but behaves similarly.
• Complex numbers are numbers that involve i.
• They are of the form a+bi.
• We cannot graph these numbers on the Cartesian
Plane because both axis on the Cartesian plane
are representative of REAL numbers (not imaginary)
• We do however have a method for graphing these
complex numbers.
JEAN ROBERT ARGAND
• We can represent complex numbers geometrically thanks to Argand
who made the argument that we can replace the y-axis with an
imaginary axis . The following method will be how we plot complex
numbers. Plot 6+5i.
imaginary axis
This is called the
complex plane
or Argand Diagram.
x axis
This point that is representing a+bi can be represented in rectangular
coordinates (a,b) or in polar coordinates (r,ϴ). Lets see how we can work
through the translation. In general we refer to the point by the name “z”
Now that z=(a,b)=a+bi
But in polar coordinates we know that a=rcosϴ and b=rsinϴ. So now
replace a and b you get … rcosϴ + (rsinϴ)i
Then you get rcos ϴ + r i sin ϴ…factor out r you get … r(cos ϴ + i sin ϴ)
We define (cos ϴ+i sin ϴ) as “cis” so
z=rcis ϴ
z = (a,b)
In addition we can still find r by
pythagoras, r or ||z||= 𝑎2 + 𝑏2
r
ϴ
b
a
r cos   a
r sin   b
• Rewrite the following complex numbers in polar
form.
• 3-2i
• -4+2i
• -4i
• Rewrite each complex number in rectangular form.
• 8 cis 110
• 12 cis 250
PRODUCT OF 2 COMPLEX NUMBERS IN
POLAR FORM
• When we want to multiply two complex numbers we
multiply the radii r and s, and we add the angles.
if
i
z1  rcis
z2  scis 
then
z1 z2  (rcis )  (rcis  )
z1 z2  rscis (   )
x
EXPRESS EACH PRODUCT IN POLAR
FORM
• (4 cis 25o)(6 cis 35o)
z1
Point z1 a
distance of 4
away from the
origin and 25
degrees
counterclockwise
of the polar axis
z2
= 24 cis (25o + 35o)
=24 cis (60o)
Point z2 a
distance of 6
away from the
origin and 35
degrees
counterclockwise
of the polar axis
Point z1z2 a
distance of 24
away from the
origin and 60
degrees
counterclockwise
of the polar axis
• Find z1z2 in rectangular form by multiplying z1 and z2.
• Find z1, z2, and z1z2 in polar form. Show that z1z2 in
polar form agrees with z1z2 in rectangular form.
• Show z1, z2, and z1z2 in an Argand diagram.
z1  2  2i 3
z2  3  i
HOMEWORK
• www.faymathematics.pbworks.com
• #s 1-4, 9-11, 13-15, 17-20