Addition & Division of Complex Numbers

Feb 11th, 2015
DreamIt
Category:
Calculus
Price: $10 USD

Question description

Explain how complex numbers combine algebraically and graphically (solely using the graph, meaning just graphing the result of the algebraic computation is not sufficient) under the following operations:

a.  Addition

b.  Division

  • Hints/Notes Given on Task: 
  • For Addition

    Algebraically: Give the general formula for adding 2 complex numbers  z1 = a + bi  and  z2 = c + di

    Give a simple example demonstrating the addition

    Graphically: Show how to add 2 complex numbers graphically.  It is not necessary to provide any numbers.  This is easily done when you use the complex plane and draw the complex numbers as vectors.   

    Your method should allow someone who is given z1 and z2 to “eye-ball”/sketch where z1 + z2 will be. 

    For Division

    Algebraically: Give the general formula/explanation for dividing 2 complex numbers  z1 = a + bi  and  z2 = c + di

    Give an appropriately worked out example of division. 

    Graphically: Instead of expressing a complex number as  z = a + bi, we can define it in terms of the angle θ  it makes with the x-axis going counter-clockwise and its radius (also called modulus, radius, or distance from the origin)

    r=√(a^2+b^2  )  θ=〖tan〗^(-1) (b/a)

    Division is a very similar operation to multiplication (meaning it involves a rotation and scaling). 

    Search online for a definition of division in polar form.

    Take 2 simple complex vectors in the first quadrant (radii = 1, 2, 3, … and angles such as 90, 60, 45, 30, etc) 

    Divide them algebraically:  z1/z2 (you can use the same example you have for the first part of the problem)

    Convert that answer to polar form to understand what dividing by Z2 does to Z1

    Draw an appropriate picture demonstrating your understanding of division in polar coordinates. 


    Tutor Answer

    (Top Tutor) Daniel C.
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    School: Boston College
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