Lecture Notes in Differential Equations - Bruce E. Shapiro

160 LESSON 18. COMPLEX ROOTS
Why Do Complex Numbers Work?*
We have not just pulled i = √ −1 out of a hat by magic; we can actually
define the Field of real numbers rigorously using the following definition.
Definition 18.7. Let a, b ∈ R. Then a Complex Number is an ordered
pair
z = (a, b) (18.72)
with the following properties:
1. Complex Addition, defined by
2. Complex Multiplication, defined by
z + w = (a + c, b + d) (18.73)
z × w = (ac − bd, ad + bc) (18.74)
where z = (a, b) and w = (c, d) are complex numbers.
Then we can define the real and imaginary parts of z as the components
Rez = Re(a, b) = a and Imz = Im(a, b) = b.
The Real Axis is defined as the set
{z = (x, 0)|x ∈ R} (18.75)
and the imaginary axis is the set of complex numbers
{z = (0, y)|y ∈ R} (18.76)
We can see that there is a one-to-relationship between the real numbers
and the set of complex numbers (x, 0) that we have associated with the
real axis, and there is also a one-to-one relationship between the set of all
complex numbers and the real plane R 2 . We sometimes refer to this plane
as the complex plane or C.
To see that equations 18.73 and 18.74 give us the type of arithmetic that we
expect from imaginary numbers, suppose that a, b, c ∈ R and define scalar
multiplication by
c(a, b) = (ca, cb) (18.77)
To see that this works, let u = (x, 0) be any point on the real axis. Then
uz = (x, 0) × (a, b) = (ax − 0b, bx − 0a) = (ax, bx) = x(a, b) (18.78)