Greenwood press

16
CIRCUMFERENCE

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when introduced to current and resistors that have imaginary number compo-
nents. The amount of voltage in a circuit is determined by the product of its cur-
rent and resistance. Without an imaginary number component in both current and
resistance, the voltage reading will remain unaffected. For example, suppose the
current is reading 3 + 2i amps on a circuit with 20 ohms of resistance. The net
voltage would be (3+2i)(20) = 60 + 40i volts. In this case, the voltmeter would
show a reading of 60 volts, because the 40i volts are imaginary. However, if the
resistance was 20 + 4i ohms, then the net voltage would be (3 + 2i)(20 + 4i) =
60 + 12i + 40i + 8i
2
. Since i
2
=
−1, this expression simplifies to 52 + 52i. That
means that the introduction of an imaginary number component in the resistance
of the circuit would result in a voltage drop of 8 volts!
Electromagnetic fields also rely on complex numbers, because there are two
different components in the measurement of their strength, one representing the
intensity of the electric field, and the other the intensity of the magnetic field.
Similar to the electric circuit example, an electromagnetic field can have sudden
variations in its strength if both components contain imaginary components.
Complex numbers also indirectly have applications in business. The profit of
the sales of a product can be modeled by a quadratic function. The company will
start with initial expenses and rely on the sales of their product to transfer out of
debt. Using the quadratic formula, the business can predict the amount of sales
that will be needed to financially break even and ultimately start making a profit.
If complex zeroes arise after applying the formula, then the company will never
break even! On a graph in the real plane, the profit function would represent a
parabola in the fourth quadrant that never touches the horizontal axis that
describes the number of products sold. This means that the business will have to
reevaluate their sales options and generate alternative means for producing a
profit.
To generalize this case, any quadratic model that produces complex solutions
from an equation will likely indicate that something is not possible. For exam-
ple, in the business-sales setting, the company may want to test when the profit
will equal one hundred thousand dollars. When solving the equation, the quad-
ratic equation could ultimately be applied, and the existence of imaginary com-
ponents in the solution would verify that this would not be possible. The same
argument could be applied to determine if the world’s strongest man could throw
a shot put 50 feet in the air. If a person can estimate the throwing height 
h
0
and
the time 
t the ball is in the air, then the quadratic function h = 0.5gt
2
+ v
0
t + h
0
can be applied to determine the initial velocity 
v
0
and whether the ball will reach
a height 
h of 50 feet. (Note that the gravitational constant g on earth is equal to
–9.8 meters per second
2
, or –32 feet per second
2
.)
online sources for further exploration
The relevance of imaginary numbers