18.03 Class 5, Feb 17, 2006
Complex Numbers, complex exponential
Today, or at least 2006, is the 200th anniversary of the birth of
complex numbers. In 1806 papers by Abb\'e Bul\'ee and by JeanRobert
Argand established the planar representation of complex numbers.
They had already been in use for several hundred years, but they were
kept fairly secret and were regarded as perhaps not entirely real.
[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 .
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 miles. Complex numbers do not accept
units.
Also, there is no ordering on complex numbers, no "<."
Question 1.
Multiplication by
i
has the following effect
on a complex number.
1. It rotates the number around the origin by 90 degrees
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 Winter '08
 Staff
 Complex Numbers, Complex number, 90 degrees, JeanRobert Argand

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