Graphing Complex Numbers

1
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.
2
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).

3
Argand Diagram
Imaginary axis
Real axis
4
Graph 2 5i
yi
The graph of 2 5i is represented by the
point (2, 5) OR by the vector from the origin to
the point (2, 5).
2 5i
x
5
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
6
Graph 3i
yi
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.
3i
x
7
Graph 7
yi
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
7
x
8
Try These

• 2 7i
• 6 i
• 2
• 8i

9
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.

10
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 .
  
• 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.

11
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 .
3 4i
Absolute Value
x
12
Find the absolute value of 3 4i
yi
z
3 4i
3 4i
4
3 4i
x
3
3 4i
5
13
Find the absolute value of 2 3i
yi
z
2 3i
2
2 3i
x
3
2 3i
2 3i
14
Try These

• 4 6i
• 3 5i