Rotation in Three Dimensions - 3D Math Primer for Graphics and Game Development - page 265

Now let's see if we can create a similar mapping for the set of complex
numbers. You probably already have been introduced to complex numbers;
if so, you should remember that the complex pair (a,b) defines the number
a+bi. The number i is a special number such that i 2 = −1. It's often called
the imaginary number because no ordinary scalar (a “real” number) can
have this property. The word “imaginary” gives one the impression that
the number doesn't really exist; we're going avoid this term and instead
stick with the more descriptive one: “complex.”
Complex numbers can be added, subtracted, and multiplied. All we
need to do is follow the ordinary rules for arithmetic, and replace i 2 with
−1 whenever it appears. This results in the following identities:

(a + bi) + (c + di) = (a + c) + (b + d)i,
(a + bi) − (c + di) = (a − c) + (b − d)i,
(a + bi)(c + di) = ac + adi + bci + bdi 2
= ac + (ad + bc)i + bd(−1)
= (ac − bd) + (ad + bc)i.

Adding, subtracting, and
multiplying complex
numbers

Now, how can we extend our system of embedding numbers in the space
of 2 × 2 matrices to include complex numbers? Before, we only had one
degree of freedom, a, and now we have two, a and b. The mapping we use is

a −b
b a

Mapping each complex
number to a 2 × 2 matrix

a + bi ≡ . (8.9)

We can easily verify that the complex number on the left behaves exactly
the same as the matrix on the right. In a certain sense, they are just two
notations for writing the same quantity:

a −b
b a

c −d
d c

a + c −(b + d)

Addition, subtraction,
and multiplication in
standard notation and
our 2 × 2 form

(a + bi) + (c + di) ≡ + =

b + d a + c

≡ (a + c) + (b + d)i,

a −b
b a

c −d
d c

a − c −(b − d)

(a + bi) − (c + di) ≡ − =

b − d a − c

≡ (a − c) + (b − d)i,

a −b
b a

c −d
d c

ac − bd −(ad + bc)

(a + bi)(c + di) ≡ =

ad + bc ac − bd

≡ (ac − bd) + (ad + bc)i.