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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 Algebra:

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The Biquaternions

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  1. The Biquaternions Renee Russell Kim Kesting Caitlin Hult SPWM 2011

  2. Sir William Rowan Hamilton (1805-1865) Physicist, Astronomer and Mathematician

  3. Contributions to Science and Mathematics: • Optics • Classical and Quantum • Mechanics • Electromagnetism • Algebra: • Discovered Quaternions & Biquaternions! “This young man, I do not say will be, but is, the first mathematician of his age” – Bishop Dr. John Brinkley

  4. 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?

  5. 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

  6. 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.

  7. 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

  8. Properties of the Biquarternions • ADDITION: • We define addition component-wise: • B = a + be1 + ce2 + de3where a, b, c, d  C • B’ = w + xe1 + ye2 + ze3where w, x, y, z  C • B +B’ =(a+w) + (b+x)e1 +(c+y)e2 +(d+z)e3

  9. 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

  10. Properties of the Biquarternions • SCALAR MULTIPLICATION: • hB =ha + hbe2 +hce3 +hde3where hC or R • The Biquaternions form a vector space over C and R!! Oh yeah!

  11. 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 • NOTCommutative • Identity: • 1 = (1+0i) + 0e1 + 0e2 + 0e3

  12. Biquaternions are an algebra over C! biquaterions

  13. 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?

  14. 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?

  15. Biquaternions are NOT a division algebra over C!

  16. 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, BQis isomorphic to M2x2(C). [ ]

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

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