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

C where w, x, y, z

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
– ce
2
– de a 2 + b 2 + c 2 + d 2
3 where a, b, c, d

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