æ
ç
è
x
-y
y
x
 ö
÷
ø 		+ æ
ç
è
u
-v
v
u
 ö
÷
ø 		= æ
ç
è
(x+u)
-(y+v)
(y+v)
(x+u)
 ö
÷
ø 		.

æ
ç
è
x
-y
y
x
 ö
÷
ø 		æ
ç
è
u
-v
v
u
 ö
÷
ø 		= æ
ç
è
(xu-yv)
-(xv+yu)
(xv+yu)
(xu-yv)
 ö
÷
ø 		.
Note that for the complex numbers:
(x+iy)(u+iv)=(xu-yv) + i(xv+yu)
and
æ
ç
è
0
-1
1
0
 ö
÷
ø

is a model for the complex number i because:


æ
ç
è
0
-1
1
0
 ö
÷
ø 		2

 
 = æ
ç
è
-1
0
0
-1
 ö
÷
ø 		.
Arithmetic for the set R* is isomorphic to the arithmetic of real numbers R where the matrix
æ
ç
è
x
0
0
x
 ö
÷
ø

corresponds to the real number x

- both sets are closed under addition and multiplication which are commutative

- addition and multiplication are associative

- the additive identity is
æ
ç
è
0
0
0
0
 ö
÷
ø

corresponds to the real number zero

- the multiplicative identity
æ
ç
è
1
0
0
1
 ö
÷
ø

corresponds to the real number 1

- each element has an additive and multiplicative inverse

- the distributive property holds.

Arithmetic for the set C* is isomorphic to the arithmetic of complex numbers C where the matrix
æ
ç
è
x
-y
y
x
 ö
÷
ø

corresponds to the complex number x+i y

- both sets are closed under addition and multiplication which are commutative

- addition and multiplication are associative

- the additive identity is
æ
ç
è
0
0
0
0
 ö
÷
ø

corresponds to the complex number 0+i0

- the multiplicative identity
æ
ç
è
1
0
0
1
 ö
÷
ø

corresponds to the complex number 1+i0

- the additive inverse of
æ
ç
è
x
-y
y
x
 ö
÷
ø 		~ x + iy

is
æ
ç
è
-x
y
-y
-x
 ö
÷
ø 		~ -x - iy

- the multiplicative inverse of
æ
ç
è
x
-y
y
x
 ö
÷
ø 		~ x + i y

is
1
(x2+y2)
 æ
ç
è
x
y
-y
x
 ö
÷
ø 		~ x - iy
x2+y2

- the distributive property holds.

(2)(a)
i2=j2=k2= æ
ç
è
-1
0
0
-1
 ö
÷
ø

(b) i j=k=-j i, (c) j k=i=-k j (d) k i=j=-i k

(3) i j=k, i j k=k2=-1, i j k i=-i, i j k i j=-i j=-k,  i j k i j k=-k2=1 so the sequence i, i j, i j k, i j k i, i j k i j, ... is a cycle of order 6 repeating the terms i, k, -1, -i, -k, 1