Does Anyone Ever Really Use Complex Numbers?
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Surprise!  Complex numbers are not just a "math oddity" that only we crazy mathematicians enjoy. 

Since complex numbers provide a system for finding the roots of polynomials, and polynomials are used as theoretical models in various fields, complex numbers enjoy prominence in several specialized areas.  Among these specialized areas are engineering, electrical engineering and quantum mechanics.  Topics utilizing complex numbers include the investigation of electrical current, wavelength, liquid flow in relation to obstacles, analysis of stress on beams, the movement of shock absorbers in cars, the study of resonance of structures, the design of dynamos and electric motors, and the manipulation of large matrices used in modeling.  While many of these applications are beyond the scope of an Algebra2/Trig curriculum, an introductory glimpse of the application of complex numbers to electrical circuits can be easily understood and manipulated by students.

 

Application to Electrical Engineering:
  • First, set the stage for the discussion and clarify some vocabulary.   Information that expresses a single dimension, such as linear distance, is called a scalar quantity in mathematics.  Scalar numbers are the kind of numbers students use most often.  In relation to science, the voltage produced by a battery, the resistance of a piece of wire (ohms), and current through a wire (amps) are scalar quantities.
  • When electrical engineers analyzed alternating current circuits, they found that quantities of voltage, current and resistance (called impedance in AC) were not the familiar one-dimensional scalar quantities that are used when measuring DC circuits.  These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account.
  • In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities.  In order to accomplish this task, scalar numbers were abandoned and complex numbers were used to express the two dimensions of frequency and phase shift at one time.
  • In mathematics, i is used to represent imaginary numbers.  In the study of electricity and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current.  It is also customary for scientists to write the complex number in the form a + jb.
  • Introduce the formula E = I Z  where E is voltage, I is current, and Z is impedance.
                                                  

Possible Student Questions:

  • The impedance in one part of a series circuit is 2 + j8 ohms, and the impedance in another part of the circuit is 4 - j6 ohms.  Find the total impedance in the circuit.  Answer:  6 + j2 ohms
     
  • The voltage in a circuit is 45 + j10 volts and the impedance is 3 + j4 ohms.  What is the current?
    Answer:
    E = I Z
    45 + j10 = I (3 + j4)

    7 - j6 amps
     
  • The current in a circuit is 4 + j2 amps and the impedance is 5 - j ohms.  What is the voltage?
    Answer:
    E = I Z
    E = (4 + j2) (5 - j)

    22 + j6 volts

 

Hopefully, this will help to answer the question "Does anyone ever really use complex numbers?"