Renee Russell
Kim Kesting
Caitlin Hult
SPWM 2011

Sir William Rowan Hamilton
(1805-1865)
Physicist, Astronomer and Mathematician

Contributions to Science and
Mathematics:
•
•
Optics
Classical and Quantum
Mechanics
•
Electromagnetism
“This young man, I do not say will be, but is, the first mathematician of his age”
– Bishop Dr. John Brinkley
•
Algebra:
•
Discovered
Quaternions &
Biquaternions!

A
quaternion
is a number of the form of:
Q = a + bi + cj + dk
where
a, b, c, d

R
, and
i
2
= j
2
= k
2
= ijk =
-1.

Biquaternions
•
A
biquaternion
is a number of the form
B = a + bi + cj + dk
and
i
2
= j
2
= k
2
= ijk =
-1.

Biquaternions
CONFUSING
:
(a+bi) + (c+di)i + (w+xi)j + (y+zi)k
* Notice this
i
is different from the
i
component of the basis, {1,
i, j, k}
for a (bi)quaternion! *
We can avoid this confusion by renaming
i, j,
and
k:
B = (a +bi) + (c+di)e
1
+(w+xi)e
2
+(y+zi)e
3 e
1
2
= e
2
2
= e
3
2
=e
1 e
2 e
3
=
-1.

Biquaternions
B
can also be written as the complex combination of two quaternions:
B = Q + iQ’
where
i =
√-1, and
Q,Q’

H
.
B = (a+bi) + (c+di)e
1
+ (w+xi)e
2
+ (y+zi)e
3
=(a + ce
1
+ we
2
+ye
3
) +i(b + de
3
+ xe
2
+ze
3
)
where
a, b, c, d, w, x, y, z

R

Properties of the Biquarternions
ADDITION:
•
We define addition component-wise:
B = a + be
1
B’ = w + xe
1
+ ce
2
+ ye
2
+ de
3
+ ze
3
where
a, b, c, d
 where
w, x, y, z

C
C
B +B’ =(a+w) + (b+x)e
1
+(c+y)e
2
+(d+z)e
3

Properties of the Biquarternions
ADDITION:
•
•
•
•
•
Closed
Commutative
Associative
Additive Identity
0 = 0 + 0e
1
Additive Inverse:
+ 0e
2
+ 0e
3
-
B = -a + (-b)e
1
+ (-c)e
2
+ (-d)e
3

Properties of the Biquarternions
SCALAR MULTIPLICATION
:
•
hB =ha + hbe
2
+hce
3
+hde
3
where
h

C
or
R
The Biquaternions form a vector space over
C and R
!!

Properties of the Biquarternions
MULTIPLICATION:
•
The formula for the product of two biquaternions is the same as for quaternions:
(a,b)(c,d) = (ac-db*, a*d+cb)
where
a, b, c, d

C.
•
Closed
•
Associative
•
NOT Commutative
•
Identity:
1 = (1+0i) + 0e
1
+ 0e
2
+ 0e
3

C
!
biquaterions

Properties of the Biquarternions
So far, the biquaterions over
C
have all the same properties as the quaternions over
R
.
In other words, does every non-zero element have a multiplicative inverse?

Properties of the Biquarternions
Recall for a quaternion,
Q

H
,
Q
-1
= a – be
1 a
2
+ b
2
– ce
2
+ c
2
– de
3
where
a, b, c, d

+ d
2
R
Does this work for biquaternions?

Biquaternions are NOT a division algebra over
C
!
Vector Space?
Quaternions
(over R)
✔
Biquaternions
(over C)
✔
Algebra?
✔ ✔
Division
Algebra?
Normed
Division
Algebra?
✔
✔
✖
✖

2x2
Define a map
f: B
Q

M
2x2
(C)
by the following:
f(w + xe
1
+ ye
2
+ ze
2
[ ]
-y+zi w-xi
where
w, x, y, z

C.
We can show that
f
is one-to-one, onto, and is a linear transformation. Therefore,
B
Q
is isomorphic to
M
2x2
(C).

Applications of Biquarternions
•
•
•
•
Special Relativity
Physics
Linear Algebra
Electromagnetism