Complex Numbers in Polar Form

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 = 1.4 + 2.9i

Basic operations on complex numbers

1) Addition/subtraction: combine all real parts together and all imaginary parts together

2) Multiplication: expand first and then combine real and imaginary parts together

3) Division: to get a real number in the denominator, we multiply the top and bottom of the fraction by the complex conjugate

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

x-axisy-axis

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 + b2 & θ = tan-1(b/a)

• Recap of definitions:z = a + bi = r cis θ

a = r cos θ & b = r sin θ

r2 = a2 + b2 & θ = tan-1(b/a)

Operations in polar form:

• 1) multiply

• 2) reciprocal

• 3) divide

• 4) exponents

• 5) roots

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)

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)

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)

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

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