18.03 Class 6
, Feb 16, 2010
Complex exponential
[1] Complex roots
[2] Complex exponential, Euler's formula
[3] Euler's formula continued
[1]
More practice with complex numbers:
Magnitudes Multiply:
zw = zw
Arguments Add:
Arg(zw) = Arg(z) + Arg(w)
so:
z^n = z^n
and
Arg(z^n) = n Arg(z)
For example:
Let's take some powers of
z = 1+i .
z = sqrt 2
Arg(z) = pi/4
z^2 = 2
Arg(z^2) = pi/2
z^3 = 2 sqrt 2
Arg(z^3) = 3pi/4
z^4 = 4
Arg(z^4) = pi
Notice that these numbers march out along a spiral. This continues
for all powers of
1+i , even negative ones.
How about fractional powers? i.e. roots: of unity, first.
What are the cube roots of 1? Well
1
is one. If
z
is one, then
z^3 = z^3 = 1 , so it lies on the unit circle.
The argument has to be so that 3 times it is zero  or 2pi, or 4pi, or .
...
Thus the argument is
0
,
giving 1 , or
2pi/3 ,
giving
(1+sqrt(3)i)/2
or
4pi/3 ,
giving
(1sqrt(3)i)/2 .
The nth roots of unity divide the unit circle into
n
equal pieces.
How about cube roots of
8 .
If
z^3 = 8 , then
z^3 = z^3 = 8 .
z
is a positive real number so
z = 2 : all cube roots of
8
lie on the circle with center 0 and radius 2.
The argument of 8
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 Spring '09
 vogan
 Complex Numbers

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