Transcript Document

Complex Numbers
The imaginary number i is defined as
i  1
so that
i 2  1
Complex numbers are in the form a + bi
where a is called the real part and bi is the imaginary part.
Complex Numbers
If a + bi is a complex number, its complex conjugate is a – bi.
To add or subtract complex numbers, add or subtract the
real parts and add or subtract the imaginary parts.
To multiply two complex numbers, use FOIL, taking
advantage of the fact that i 2  1 to simplify.
To divide two complex numbers, multiply top and
bottom by the complex conjugate of the bottom.
Complex Numbers
Complex solutions to the Quadratic Formula
When using the Quadratic Formula to solve a quadratic
equation, you may obtain a result like  4 , which you
should rewrite as  4  4  1  2i .
In general  a  a i if a is positive.
Polynomial Roots (zeros)
If f(x) is a polynomial of degree n, then f has
precisely n linear factors:
f  x   an  x  c1  x  c2  x  c3 ... x  cn 
where c1, c2, c3,… cn are complex numbers.
This means that c1, c2, c3,… cn are all roots of f(x), so
that f(c1) = f(c2) = f(c3) = … =f(cn) = 0
Note: some of these roots may be repeated.
Polynomial Roots (zeros)
For polynomial equations with real coefficients, any
complex roots will occur in conjugate pairs.
(If a + bi is a root, then a - bi is also a root)