11-2 Geometric and Trigonometric
Representation of Complex Numbers
The above diagram is an Argand diagram. Notice
that the real numbers are on the x-axis and the imaginary numbers are on
the y-axis. Finding imaginary numbers in this plane are as easy as
finding points in the real plane. In the form a + bi, a is the real
part and b is the imaginary part. Move a units right or left ( depending
on + or - ) and b units up or down
(depending on + or -).
Rectangular form: z = a +
Polar form: z = r cos
+ (r sin 0)i (remember a = r cos 0 and
b = r sin 0 substitute these in and presto!)
factor out the r and get:
z = r(cos
0 + i sin 0)
Math shorthand looks like
this: z = r cis
The absolute value of z (the
distance from any point to the origin) =
| z | = \/ a2
+ b2 (good
old Pythagorus again)
1) Express 3 cis 50o
in rectangular form.
using the fact that a = r cos
0 and b = r sin 0
a = 3 cos 50 and b= 3 sin 50. Using your calculator gets us
a = 1.93 and b = 2.30 with both answers rounded to hundredths.
Therefore the answer is: 1.93 + 2.30i
2) Express -1 -2i in polar
Use the fact that r = \/(-1)2
+ (-2)2 r = \/ 5
= 2.24. Now use the fact that the Tan
0 = (y/x)
(see page 1 if you forgot!!). 0 = tan-1(2/1).
Using your calculator gives us 63.4o. Add 180 (why? we
are in the 3rd quadrant!!!) 63.4 + 180 = 243.4
Therefore, the answer is: 2.24 cis 243.4o
To Multiply two complex numbers
in polar form:
1) Multiply their absolute
2) Add their polar angles.
In math terms if z1
= r cis w and z2 =
s cis y then z1z2
= rs cis ( w + y)
1) Express (5cis 30o)(7cis60o)
in polar and rectangular form.
Polar form first: multiply the radii and add the angles.
Answer in polar form: 35 cis 90o
Now change this to get the rectangular form.
Remember, a = r cos
and b = r sin 0
so a = 35 cos 90 = 0 and
b = 35 sin 90 = 35.
Answer in rectangular
form is: 0 + 35i
section involves finding the powers of complex numbers!!