18.03 Class 5
, Feb 15, 2008
Complex Numbers, roots of unity
[1] Complex algebra
[2] Complex conjugation
[3] Polar multiplication
Complex numbers provide a tool for expressing aspects of the real world.
Anything with an amplitude and a phase is secretly a complex number,
and it is worthwhile bringing that secret identity out in the open.
[1] Complex Algebra
We think of the real numbers as filling out a line.
The complex numbers fill out a plane. The point up one unit
from
0
is written
i
(but written
j
by many engineers).
Addition and multiplication by real numbers is as
vectors. The new thing is
i^2 = 1 .
The usual rules of algebra apply.
For example FOIL:
(1 + i)(1 + 2i) = 1 + 2i + i  2 = 1 + 3i.
Every complex number can be written as
a + bi
with
a
and
b
real.
a = Re(a+bi)
the real part
b = Im(a+bi)
the imaginary part:
NB this is a real number.
Maybe complex numbers seem obscure because you are used to imagining
numbers by giving them units: 5 cars, or 3 degrees. Complex numbers do
not accept units. Also, there is no ordering on complex numbers, no "<."
Question 1.
Multiplication by
i
has the following effect
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 Spring '09
 vogan
 Algebra, Multiplication, Complex Numbers, Unit Circle, Complex number

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