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Absolute
Value
of Complex Numbers |
| Math B |
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The absolute value of a
complex number
 is written as
 .
It is a nonnegative real number defined
as:
 .
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Geometrically, the
absolute value of a complex number is
the number's distance from the origin in the complex plane.
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In the diagram at the left, the
complex number 8 + 6i is plotted in the complex plane on an Argand
diagram (where the vertical axis is the imaginary axis). For
this problem, the distance from the point 8 + 6i to the origin is 10
units. Distance is a positive measure.
Notice the Pythagorean Theorem at
work in this problem.
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A complex number can be
represented by a point, or by a vector from the origin to the point.
When thinking of a complex number as a vector, the absolute value of
the complex number is simply the length of the vector, called the
magnitude. |
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The formula for finding the absolute
value of a complex number
can be derived from the Pythagorean theorem,
 (see
example 2 below).
In the Pythagorean
Theorem,
c
is the hypotenuse and when represented in the coordinate plane, is always positive.
This same idea holds true for the distance from the origin in the complex
plane. Using the absolute value in the formula will always
yield a positive result.
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To find the absolute value
of a complex number
a +
bi:
1. Be sure your number is expressed in
a + bi form
2. Pick out the coefficients for
a and b
3. Substitute into the formula
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Example 1)
Plot z = 8 + 6i on
the complex plane, connect the graph of z to
the origin
(see graph below), then find
| z | by appropriate use of the
definition of the
absolute value
of a complex number.
Example 2)
Find the
| z |
by appropriate use
of the Pythagorean Theorem when z = 2 - 3i.
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You can find the distance | z | by using the
Pythagorean theorem. Consider the graph of
2 - 3i shown at the left. The horizontal side of the triangle has
length | a |, the vertical side has length | b |, and
the hypotenuse has length | z |. By applying the Pythagorean
Theorem, you have, | z |2
= a2 + b2 .
Notice: you can drop the absolute
value symbols for a and b since | a |2
= a2 and
| b |2
= b2. You must keep the absolute value symbol for
z to insure that the final answer will be positive.
Solving this equation for |
z |, you have just derived the formula for the absolute
value of a complex number:
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Example 3)
If z
= - 8 - 15i,
find | z |.
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