Complex Numbers?
What’s So Complex?
Complex numbers are
vectors
represented in the
complex plane as
the sum of a
Real part and an
Imaginary part:
z = a + bi
Re(z) = a; Im(z) = b
Just like vectors!
|z| = (a2 + b2)1/2 is length or
magnitude, just like vectors.
a = tan-1 (b/a) is direction,
just like vectors!
Just like vectors!
For two complex numbers
a + bi and c + di:
Addition/subtraction combines
separate components,
just like vectors.
Useful identities
Euler:
eix = cos x + i sin x
cos x = (eix + e-ix)/2
sin x = (eix - e-ix)/2i
Things named Euler
Sure, he’s French, but we
must give props:
DeMoivre:
(cos x + i sin x)n =
cos (nx) + i sin (nx)
cos 2x + i sin 2x = ei2x
cos 2x = (1 + cos 2x)/2
sin 2x = (1 - cos 2x)/2
What about
multiplication?
Just FOIL it!
Scalar multiples of a
complex number: a line
Multiplication:
the hard way!
z1z2= r1 (cosa1 + i sina1) r2 (cosa2 + i sina2)
= r1 r2 (cosa1 cosa2 - sina1 sina2) +
i r1 r2 (cosa1 sina2 + cosa2 sina1)
= r1 r2 [cos(a1 + a2) + i sin(a1 + a2)]
Multiplication:
the easy way!
z1 z 2  r1 e
ia 1
r2 e
 r1 r2 e
ia 2
i (a 1  a 2 )
“Neither dot nor cross do you multiply complex numbers by.”
Multiplication:
by i
iz  ir e
ia
 ir (co s a  i sin a )
  r sin a  ir co s a
o
Rotate by 90 and swap Re and Im
i ‘s all over the Unit Circle!
Note i4 = 1 does not
mean that 0 = 4
i ‘s all over the Unit Circle!
Did you see i½?
Square root of i?
Find the square root of 7+24 i.
(Hint: it’s another complex number,
which we’ll call u+vi).
u  vi 
( u  v i)
2
7+24i
 7  2 4i
u
2
 2uvi - v
u
2
- v  7; 2 u v  2 4
2
2
 7  2 4i
Which can be solved
by ordinary means to
yield 4+3i and -4 - 3i.
Complex Conjugates
z  a  bi
z  a  bi
zz  a  b  z
2
2
2
Complex Conjugates
z  a  b i ; a  tan
1
z  a  b i ; a  tan
1
(b / a )
(b / a )
a  a
Complex conjugates reflect in the
Re axis.
Complex Reciprocals
zz  a  b  z
2
1
z

2
2
z
2
z
The reciprocal of a complex
number lies on the same ray as its
conjugate!
Powers
of z
The graph of
f(z)=zn for
|z|<1 is
called an
exponential
spiral.
This shape is at the heart of
the computation of
fractals!
The basic
geometry of
the solar
system!
It shows up in
nature!
And the decorative arts!
The rotation comes from
our old buddy DeMoivre:
Raising a unit z to the nth
power is multiplying its
angle by n.
(cos x + i sin x)n =
cos (nx) + i sin (nx)
How about a slice of p:
Roots of z
If z3 = 3+3i = 4.24eip/4
then
ip / 12
z1 
3
4.24 e
z2 
3
4.24 e
z3 
3
4.24 e
Each successive nth root
is another 2p/n around the circle.
i 9 p / 12
i 17 p / 12
Find the roots of the complex
equation z2 + 2i z + 24 = 0
Sounds like a job for the
quadratic formula!
z
 2i 
(2 i )  4(24)
2
2
 i 
 4  96
2
  i  5i   6 i, 4 i
Was that so complex?
And never forget, epi = -1