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LC Math 1 Adv – Graphing Complex Numbers (LT 7&8)
In order to increase the acceptance of complex numbers, mathematicians attempted to represent these numbers
using a familiar structure, the coordinate plane. The mathematician who pioneered this idea was Argand (1806),
when he developed what was called the Argand Plane. In particular, this visualization helped "imaginary" and
"complex" numbers become accepted in mainstream mathematics as a natural extension to negative numbers
along the real line. Today, the Argand Plane is more commonly known as the complex plane, and is shown below.
Learning Target 7
Notice that the x-axis is called the real axis and the y-axis is
called the imaginary axis.
Each complex number is comprised of a real component (a)
and an imaginary component (bi). You’ll notice that the point
that represents the complex number a  bi has the coordinate
a, b on the complex plane.
We will often refer to a complex number using notation such
as z  a  bi , so that we can refer to the point on the plane
with a single letter.
Ex 1) Graph each of the following numbers on the complex plane, labeling them with a single letter.
m  2  4i
v  6
r  3i
n  1 7i
q  5  2i
s 5
3
 2i
2
t  4i
p
imaginary
a) What quadrant will the complex number z  a  bi
lie in if a  0 and b  0 ? Explain.
b) Without graphing, describe all points z  a  bi , a  0, b  0 . What type of numbers are these?
c) Without graphing, describe all points z  a  bi , a  0, b  0 . What type of numbers are these?
real
Learning Target 8
The conjugate of a complex number of the form z  a  bi is given by z  a  bi .
Ex 2) Find the complex conjugate of each of the following complex numbers:
a) 5  3i
b)  2  6i
c) 4 
3
i
2
d) Alex is asked to find the conjugate of 4i  2 , and he responds with “ 4i  2" . Why does this answer not
conform to the definition of a conjugate listed above? What is the correct conjugate of 4i  2 ?
e) Find the conjugate of  3i  6
f) Find the conjugate of 2i
g) Find the conjugate of 5
We will now examine the effect of “conjugating” (the process of finding the conjugate of a complex number) on the
complex plane. To analyze the effect of this procedure, we will graph several pairs of complex conjugates.
Ex 3) Graph the following pair of complex conjugates on the coordinate planes below:
a) 6  2i and its conjugate
b) 3  4i and its conjugate
imaginary
imaginary
real
real
c) What transformation describes the effect of conjugating a complex number on the complex plane? Why does
this make sense given the meaning of a complex conjugate?
d) Since conjugating a complex number causes a reflection over the x-axis, is there a process by which you could
reflect any complex number over the y-axis? If so, describe this process and name it in your own way.
The modulus of a complex number of the form z  a  bi is given by z  a 2  b 2 .
The diagram to the left illustrates that the modulus of the complex number
z  8  6i is 10. You can see from the diagram that the modulus is just
the distance between the origin and the point that represents the given
complex number on the complex plane.
The formula used to calculate the modulus is just another form of what
famous theorem? How was it adjusted to find the modulus?
Ex 4) Graph the following points on the complex plane provided and then find the value of the modulus. Use exact,
simplified values for the modulus only (no decimal approximations!)
a) Find z if z  3  5i
b) Find z if z  6  2i
imaginary
imaginary
real
real
Ex 5) Without graphing, find the value of the modulus of each complex number.
a) Find z if z  3 2  7i
b) Find z if z  4 
1
i
2
Practice Problems
Graph each of the following complex numbers on the complex plane, then find the value of the modulus.
1. z  2  4i
z 
2. z  2  4i
3. z  2  4i
z 
z 
Identify the complex number graphed in the plane. Then write and graph its conjugate.
4.
5.
6.
7. True or False: There is no complex number that is equal to its conjugate. Explain or provide a counterexample.
8. True or False: i 44  i 150  i 74  i 109  i 61  1 . Explain or show work.