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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 Algebra:

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the biquaternions
The Biquaternions

Renee Russell

Kim Kesting

Caitlin Hult

SPWM 2011

slide2

Sir William Rowan Hamilton

(1805-1865)

Physicist, Astronomer and Mathematician

slide3

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

review of quaternions h
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
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

slide6

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.

slide7

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

slide8

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
slide9

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
slide10

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!

slide11

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
slide12

Biquaternions

are an algebra

over C!

biquaterions

slide13

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?

slide14

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?

slide16

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).

[ ]

slide17

Applications of Biquarternions

  • Special Relativity
  • Physics
  • Linear Algebra
  • Electromagnetism
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