April 12, 2013
6.5 Complex Numbers
Objective:
* Complex numbers and graph (review).
* Write the trig. form of a complex number.
* Multiply/Divide complex numbers.
* Powers of complex numbers.
April 12, 2013
Complex numbers.
1. (3 + 2i) + (-1 + 5i)
2. (3 + 2i)(-1 + 5i)
3.
3 + 2i
-1 + 5i
4.
(3 + 2i)12
April 12, 2013
complex number: z = a + bi
real part
imaginary
part
April 12, 2013
Trig. Form of a Complex Number (Polar Form)
Imaginary
axis
(a + bi)
r
b
O
a
Real axis
z = a + bi
z = rcosθ + (rsinθ) i = r(cosθ + i sinθ)
argument
April 12, 2013
Ex. Plot -2 + 5i and find its absolute value.
April 12, 2013
2. Write z = -2 - 3i in trig. form.
April 12, 2013
Write in the complex number in standard form.
z = √8 (cos(
-π
-π
) + i sin(
))
3
3
April 12, 2013
Product and Quotient Rule
z1 = r1(cosθ1 + i sin θ1)
Product:
Quotient:
and
z2 = r2(cosθ2 + i sin θ2)
z1z2 = r1r2 [ cos(θ1 + θ2) + i sin(θ1 + θ2 )]
z1
z2
=
r1
r2
[ cos (θ1 - θ2) + i sin (θ1 - θ2) ]
April 12, 2013
Find the product z1z2.
2π
z1 = 2 (cos 2π + i sin
) and z2 = 8 (cos 11π + i sin 11π)
3
6
6
3
April 12, 2013
DeMoivre's Theorem
If z = r (cos θ + i sin θ) and n is a positive integer, then
zn = rn [ cos nθ + i sin nθ ]
Evaluate.
( -1 + √3 i )12