John Stillwell - Naive Lie Theory.pdf - Index of

6 1 The geometry of complex numbers and quaternions
• Therefore, the multiplicative property of absolute value, |z 1 z 2 | =
|z 1 ||z 2 |, follows from the multiplicative property of determinants,
det(A 1 A 2 )=det(A 1 )det(A 2 ).
(Take A 1 as the matrix representing z 1 ,andA 2 as the matrix representing
z 2 .)
• The inverse z −1 = a−bi of z = a + bi ≠ 0 corresponds to the inverse
a 2 +b 2
matrix ( ) −1 a −b
= 1 ( ) a b
b a a 2 + b 2 .
−b a
The two-square identity
If we set z 1 = a 1 + ib 1 and z 2 = a 2 + ib 2 , then the multiplicative property
of (squared) absolute value states that
(a 2 1 + b2 1 )(a2 2 + b2 2 )=(a 1a 2 − b 1 b 2 ) 2 +(a 1 b 2 + a 2 b 1 ) 2 ,
as can be checked by working out the product z 1 z 2 and its squared absolute
value. This identity is particularly interesting in the case of integers
a 1 ,b 1 ,a 2 ,b 2 , because it says that
(a sum of two squares) ×(a sum of two squares) = (a sum of two squares).
This fact was noticed nearly 2000 years ago by Diophantus, who mentioned
an instance of it in Book III, Problem 19, of his Arithmetica. However,
Diophantus said nothing about sums of three squares—with good reason,
because there is no such three-square identity. For example
(1 2 + 1 2 + 1 2 )(0 2 + 1 2 + 2 2 )=3 × 5 = 15,
and15isnot a sum of three integer squares.
This is an early warning sign that there are no three-dimensional numbers.
In fact, there are no n-dimensional numbers for any n > 2; however,
there is a “near miss” for n = 4. One can define “addition” and “multiplication”
for quadruples q =(a,b,c,d) of real numbers so as to satisfy all
the basic laws of arithmetic except q 1 q 2 = q 2 q 1 (the commutative law of
multiplication). This system of arithmetic for quadruples is the quaternion
algebra that we introduce in the next section.