The Biquaternions Renee Russell Kim Kesting Caitlin Hult

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)e 1 +(w+xi)e 2 +(y+zi)e 3 e 12 = e 22 = e 32 =e 1 e 2 e 3 = -1.

Biquaternions B can also be written as the complex combination of two quaternions: B = Q + i. Q’ 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 + ce 2 + de 3 B’ = w + xe 1 + ye 2 + 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 + 0 e 1 + 0 e 2 + 0 e 3 Additive Inverse: -B = -a + (-b)e 1 + (-c)e 2 + (-d)e 3

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

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+0 i) + 0 e 1 + 0 e 2 + 0 e 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 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 M 2 x 2(C) Define a map f: BQ M 2 x 2(C) by the following: f(w + xe 1 + ye 2 + ze 2 ) = [ w+xi -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 M 2 x 2(C).

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