the biquaternions renee russell kim kesting caitlin hult spwm 2011
TRANSCRIPT
“This young man, I do not say will be, but is, the first
mathematician of his age” – Bishop Dr. John Brinkley
• Optics• Classical and Quantum Mechanics • Electromagnetism
• Algebra:• Discovered Quaternions & Biquaternions!
Contributions to Science and Mathematics:
Review of Quaternions, H
A quaternion is a number of the form of:
Q = a + bi + cj + dk
where a, b, c, d R, and i2 = j2 = k2 = ijk = -1.
So… what is a biquaternion?
Biquaternions
• A biquaternion is a number of the form
B = a + bi + cj + dk
where ,
and i2 = j2 = k2 = ijk = -1.
a, b, c, d C
CONFUSING:
(a+bi) + (c+di)i + (w+xi)j + (y+zi)k
Biquaternions
We can avoid this confusion by renaming i, j,and k:
B = (a +bi) + (c+di)e1 +(w+xi)e2 +(y+zi)e3
e12 = e2
2 = e32 =e1e2e3 = -1.
* Notice this i is different from the i component of the basis, {1, i, j, k} for a (bi)quaternion! *
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
Biquaternions
Properties of the BiquarternionsADDITION:
• We define addition component-wise:
B = a + be1 + ce2 + de3 where a, b, c, d C B’ = w + xe1 + ye2 + ze3 where w, x, y, z C
B +B’ =(a+w) + (b+x)e1 +(c+y)e2 +(d+z)e3
Properties of the BiquarternionsMULTIPLICATION:
• 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
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 where a, b, c, d R a2 + b2 + c2 + d2
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).
[ ]