Download Graphing and Absolute Value of Complex Numbers

Graphing Complex Numbers
AND
Finding the Absolute Value
of Complex Numbers
SPI 3103.2.2
Compute with all real and complex numbers.
Checks for Understanding
3103.2.7 Graph complex numbers in the complex plane and recognize differences
and similarities with the graphical representations of real numbers
graphed on the number line.
3103.2.9 Find and describe geometrically the absolute value of a complex number.
Graphing Complex Numbers
• Complex numbers cannot be graphed on a normal
coordinate axes.
• Complex numbers are graphed in an Argand diagram,
which looks very much like a regular Cartesian coordinate
axes.
• An Argand diagram shows a relationship between the
x-axis (real axis) with real numbers and the y-axis
(imaginary axis) with imaginary numbers.
• In an Argand diagram, a complex number (a + bi) is the
point (a, b) or the vector from the origin to the point (a, b).
Argand Diagram
Imaginary axis
Real axis
Graph 2 + 5i
The graph of 2 + 5i is
represented by the point
(2, 5) OR by the vector
from the origin to the
point (2, 5).
yi
2 + 5i
x
Graph 5 – 6i
yi
The graph of 5 – 6i is
represented by the point
(5, –6) OR by the vector
from the origin to the
point (5, –6).
x
5 – 6i
Graph 3i
The graph of 3i is
represented by the point
(0, 3) OR by the vector
from the origin to the
point (0, 3).
3i is the same as
0 + 3i.
yi
3i
x
Graph –7
The graph of –7 is
represented by the point
(– 7, 0) OR by the vector
from the origin to the
point (– 7, 0).
–7 is the same as
–7 + 0i
yi
–7
x
Try These
1. –2 + 7i
2. –6 – i
3. 2
4. 8i
Absolute Value of Complex Numbers
• The absolute value of a real number is the distance from
zero to the number on the number line.
• The absolute value of a complex number is also the
distance from the number to zero, but the distance is
measured from zero to the number in an Argand diagram
rather than on a number line.
• The most efficient method to find the absolute value of a
complex number is derived from the Pythagorean Theorem.
Absolute Value of Complex Numbers
• The absolute value of a complex number z = a + bi is
written as z .
• The absolute value of a complex number is a nonnegative
real number defined as z = a2  b2 .
• Since a complex number is represented by a point or by
the vector from the origin to the point, the absolute value
is the length of the vector, called the magnitude.
Find the absolute value of 3 + 4i
yi
To find the absolute value
of a complex number,
find the distance from the
number to the origin.
The formula to find the
absolute value of a
complex number is
as z = a2  b2 .
3 + 4i
x
Find the absolute value of 3 + 4i
yi
z = a 2  b2
3 + 4i =
3 + 4i
32 4 2
3 + 4i = 25
3 + 4i = 5
4
3
x
Find the absolute value of 2 – 3i
yi
z = a 2  b2
2 – 3i
2
2
2

(

3)
=
2 – 3i
=
4 9
2
x
3
2 – 3i
= 13
2 – 3i
Try These
1. –4 + 6i
2. –3 + 5i