The Biquaternions Renee Russell Kim Kesting Caitlin Hult SPWM 2011

Sir William Rowan Hamilton ( ) Physicist, Astronomer and Mathematician

“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 i 2 = j 2 = k 2 = ijk = -1. So… what is a biquaternion?

Biquaternions A biquaternion is a number of the form B = a + bi + cj + dk where, and i 2 = j 2 = k 2 = 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)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. * 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)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 Biquaternions

Properties of the Biquarternions ADDITION: We define addition component-wise: B = a + be 1 + ce 2 + de 3 where a, b, c, d  C B’ = w + xe 1 + ye 2 + ze 3 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 + 0e 2 + 0e 3 Additive Inverse: -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!! Oh yeah!

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

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 – be 1 – ce 2 – de 3 where a, b, c, d  R a 2 + b 2 + c 2 + d 2 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 M 2x2 (C) Define a map f: B Q  M 2x2 (C) by the following: f(w + xe 1 + ye 2 + ze 2 ) = 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, B Q is isomorphic to M 2x2 (C). [ ]

Applications of Biquarternions Special Relativity Physics Linear Algebra Electromagnetism