Mathematical Investigations IV
Name:
Mathematical Investigations IV
Complex Concepts-Making “i” contact
Introduction
The search for solutions to equations of the form x 2  a 2 led to the development of the
complex numbers, specifically the number i  1 . Recall the quadratic formula for finding
2
roots of the equation ax  bx  c  0 .... what is it?
2
What is the relationship between a, b, and c whenever the roots of ax  bx  c  0 are complex
(i.e. non-real)?
n
In MI 3, we investigated the equation x  1 as part of our study of polynomials. How many
solutions does this equation have over the complex numbers?
n
Find solutions of x  1 when
n = 2:
n = 4:
n = 8:

Complex 1.1
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Mathematical Investigations IV
Name:
Let’s revisit graphing in the complex plane:
im agina ryax is
z = a + bi
b
r

a
real axis
cartesian (rectangular)
coordinates
(a, b)
polar
coordinates
(r, )
To relate these coordinates, we have:
z = a + bi
(rectangular form)
= x + yi
= r cos  + (r sin ) i
= r(cos  + i sin )
(polar form)
= r cis 
(shortcut cis form or notation)
A complex plane graph is also called an Argand diagram, named after Jean Robert Argand, a
Parisian, who first introduced this graphic representation in 1806.
1.
Let z1 = –3 + 3 3 i and z2 = 2 3 + 2i . Find the product z1· z2.
Now change z1, z2, and z1z2 to cis form. (Use degrees.)
2.
Let z1 = –4 – 4 3 i and z2 = –1 +
3 i. Find the product z1· z2.
Then find z1, z2, and z1z2 in cis form. (Use positive angles in degrees.)
Complex 1.2
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Mathematical Investigations IV
Name:
3.
In each of the preceding examples, what is the relationship between the values of the radii
in z1 and z2 and the radius of the resulting product?
What is the relationship between the values of ?
4.
Use your generalization to write z1z2 in polar form and in rectangular form if
z1 = 3 cis20 and z2 = 5 cis70.
5.
Let z1 = 12i and z2 = 1 +
3 i. Find the quotient z1  z2.
Then find z1, z2, and z1  z2 all in cis form.
What is the relationship between the the radii? between the angles?
6.
If z1 = 15 cis140 and z2 = 5 cis35, find z1  z2 in cis form.
Complex 1.3
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Mathematical Investigations IV
Name:
7.
Let z = 4 + 4 3 i. Multiply this by i to find zi.
Find both z and zi in cis form, using degrees.
Plot z and zi on an Argand diagram.
Describe what happened graphically to z when it was multiplied by i = 1 cis 90
8.
Suppose z = 6 cis 40. Use the results of #7 to find zi in cis form without doing the
actual multiplication.
9.
What happens to the graphic representation of a complex number z if it is:

multiplied by i=cis   ?
 2

multiplied by 1  cis  ?
 3 
multiplied by i  cis   ?
 2
Complex 1.4
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Mathematical Investigations IV
Name:
10.
If z = 5 + 5 3 i, find z2 and then change both z and z2 to cis form.
11.
If z = r cis , find z2 in cis form.
Now find z3 in cis form.
12.
Let z = cis 30. Find z2, z3, z4 and z10 in cis form
Plot all five on an Argand diagram.
Complex 1.5
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