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:
y-axis
x-axis
– 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
Polar form of a complex number
• Plug in the expression for a and b to get:
– z = r cis θ
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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:
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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