1. Complex Numbers in Polar Form

2. Imaginary and Complex Numbers
Imaginary Number – Basic definition:
i = √-1 = Sqrt(-1)
i2 = -1
i3 = i*i2 = i*(-1) = -i
i4 = ?
Complex Number – Basic definition:
A number that has both a real and imaginary part:
z = a + bi ( a – bi is called the complex conjugate)
For example: z = 5 + 3i or z = i

3. Basic operations on complex numbers
Addition/subtraction: combine all real parts together and all imaginary parts together
Multiplication: expand first and then combine real and imaginary parts together
Division: to get a real number in the denominator, we multiply the top and bottom of the fraction by the complex conjugate

4. Graphing Complex Numbers
Cartesian Form in the complex plane:
The real part goes on the x-axis
The imaginary part goes on the y-axis
z = a + bi
Polar Form:
r is the distance from the origin to the point
θ is the angle measured up from the x-axis
Examining the diagram, we can see that:
a = r cos θ b = r sin θ
y-axis
x-axis

5. Polar form of a complex number
Plug in the expression for a and b to get:
z = r cis θ
r is the modulus, aka magnitude or length
θ is the argument, aka angle
the absolute value of any complex number is: |z| = r
Examine the right triangle to find:
r2 = a2 + b & θ = tan-1(b/a)
Recap of definitions:
z = a + bi = r cis θ
a = r cos θ & b = r sin θ
r2 = a2 + b & θ = tan-1(b/a)

6. Operations in polar form:
1) multiply
2) reciprocal
3) divide
4) exponents
5) roots

7. Operations in polar form:
1) Multiply two complex numbers together: z1z2
But we see there is a shortcut:
Multiply the moduli, add the arguments
z1z2 = r1r2 cis (θ1 + θ2)

8. Operations in polar form:
2) Find the reciprocal: 1/z
But we see there is a shortcut:
Take the reciprocal of the modulus, and negative θ
1/z = 1/r cis (-θ)

9. Operations in polar form:
3) Divide two complex numbers: z1/z2
Apply the two tricks we just learned
But we see there is a shortcut:
Divide the moduli, subtract the arguments
z1/z2 = r1/r2 cis (θ1 - θ2)

10. Operations in polar form:
4) Raising a complex number to the nth power: zn
First using the tricks we have learned
But we see there is a shortcut:
Raise the modulus to the nth power, multiply θ by n
zn = rn cis (n*θ)
This is known as De Moivre’s Theorem

11. Operations in polar form:
5) Taking an nth root of complex numbers: n√z = z1/n
Here we have to be careful to include all possible results
Result:
An nth root will have n total solutions, evenly spaced around the pole in the complex plane