Download 10.3 The Complex Plane

10.3 The Complex Plane; DeMoivre’s Theorem
In the complex plane, the x-axis becomes the real axis (z = x + oi = x) and the y-axis becomes the
imaginary axis (z = 0 + yi = yi). The magnitude or modulus of z z is the distance from the origin to
the point (x, y); z  x 2  y 2 . If z = x + yi is multiplied by its conjugate z = x – yi, the product is
x2 + y2 and z  zz An equation z = x + yi in rectangular form can be converted to polar coordinates
z = r cos  + r sini = r(cos + i sin), r > 0 and 0 <  < 2. In polar form, the angle  is called the
argument of z and r is the magnitude of z.
Product of complex numbers: z1 z2  r1r2 cos  1  2   i sin  1   2   .
Quotient of complex numbers:
z1 r1
 cos  1  2   i sin  1  2   .
z2 r2 
De Moivre’s Theorem: If z = r(cos + i sin) is a complex number, then zn = rn(cos(n)+ sin(n),
where n > 1 is a positive integer.
Complex Roots: Let w = r(cos +i sin) be a complex number and let n > 2 be an integer. If w  0, there are
n distinct complex roots of w given by zk  n cos  0  2k    i sin  0  2k   , where k = 0, 1, 2,… n-1.
 n
n 
 n
n 
Plot each complex number in the complex plane and write it in polar form; express the argument in degrees:
1. 1  3i
2.  5  i
Write each complex number in rectangular form:
3. 3(cos120 + i sin 120)
13
13 
4. 0.7  cos
 sin
10
10 

Find z w and z/w; leave answers in polar form:
5. z  12  cos

5
5 
13 
 13
 i sin  ; w  6  cos
 i sin
4
4 
16
16 

Write each answer in a + bi form (use DeMoivre’s Theorem):
6.  3  cos 80o  i sin 80o  
3
7. Find the complex fourth roots of 3  i
TRY THESE
10.1
1. Plot the point (-3, 4) in polar coordinates, and find another polar coordinate (r, ) for which r > 0
and 2 ≤  ≤ 4.
2.
Find the rectangular coordinates of the point (8.1, 5.2), which is in polar coordinates.
3.
Find the polar coordinates for the point (-2.3, 0.2), which is in rectangular coordinates.
10.3
4. Plot the complex number 2  3i and write it in polar form; express the argument in degrees.
5.
Find zw and z/w; leave your answer in polar form: z  4  cos 3  i sin 3  ; w  2  cos 9  i sin 9 
6.
Write in standard a + bi form:
7.
Find all the complex cube roots of -8 - 8i; leave your answer in polar form with arguments in
degrees.

 1  5i 
8
8 

16
16 
8