PART A3A - ADDITION

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

Give a simple example demonstrating the addition

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. Resources: Dave’s Short Course, Wikipedia (look up Complex numbers), Parallelogram method, Head to Tail method.

Your method should allow someone who is given z1 and z2to “eye-ball”/sketch where z1 + z2 will be.

z1

z2

A3B – 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.

A3B GRAPHICAL DIVISION: POLAR FORM

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)

Now z can be expressed as (r, θ)

or z = r(cos θ + i sin θ)

REPRESENTING DIVISION GRAPHICALLY 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.

www.desmos.com/calculator change window setttings to Polar and plot points by typing (1,1) or (-2,3). Use Word to draw the lines with arrows

(2,0) or (2, 0⁰)

(-2,3) or (√13, 124⁰) (0,3) or (3, 90⁰)

(1,1) or (√2, 45⁰)

CONVERTING TO POLAR COORDINATESa) (2,0) or z = 2 + 0i

b) (0,3) or z = 0 + 3i

c) (1,1) or z = 1 + i

d) (-2, 3) or z = -2 + 3i