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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