Download The Biquaternions

The Biquaternions
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!
Review of Quaternions, H
A quaternion is a number of the form of:
Q = a + bi + cj + dk
where a, b, c, d  R,
2
2
2
and i = j = k = ijk = -1.
So… what is a biquaternion?
Biquaternions
•
A biquaternion is a number of the form
B = a + bi + cj + dk
where a, b, c, d  C,
and
2
2
i =j =
2
k =
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)e1 +(w+xi)e2 +(y+zi)e3
e12 = e22 = e32 =e1e2e3 = -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)e1 + (w+xi)e2 + (y+zi)e3
=(a + ce1 + we2 +ye3) +i(b + de3 + xe2 +ze3)
where a, b, c, d, w, x, y, z  R
Properties of the Biquarternions
ADDITION:
• We define addition component-wise:
B = a + be1 + ce2 + de3
B’ = w + xe1 + ye2 + ze3
where a, b, c, d  C
where w, x, y, z  C
B +B’ =(a+w) + (b+x)e1 +(c+y)e2 +(d+z)e3
Properties of the Biquarternions
ADDITION:
•
•
•
•
•
Closed
Commutative
Associative
Additive Identity
0 = 0 + 0e1 + 0e2 + 0e3
Additive Inverse:
-B = -a + (-b)e1 + (-c)e2 + (-d)e3
Properties of the Biquarternions
SCALAR MULTIPLICATION:
• hB =ha + hbe2 +hce3 +hde3
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) + 0e1 + 0e2 + 0e3
Biquaternions
are an algebra
over C!
biquaterions
Properties of the Biquarternions
So far, the biquaterions over C have all the same
properties as the quaternions over R.
DIVISION?
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 – be1 – ce2 – de3
2
2
2
2
a +b +c +d
where a, b, c, d  R
Does this work for biquaternions?
Biquaternions are NOT a division algebra
over C!
Quaternions
(over R)
Biquaternions
(over C)
Vector Space?
✔
✔
Algebra?
✔
✔
Division
Algebra?
✔
✖
Normed
Division
Algebra?
✔
✖
Biquaternions are
isomorphic to M2x2(C)
Define a map f: BQ  M2x2(C) by the following:
f(w + xe1 + ye2 + ze2 ) =
[
w+xi
-y+zi
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, BQ is isomorphic to M2x2(C).
Applications of Biquarternions
• Special Relativity
• Physics
• Linear Algebra
• Electromagnetism