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The Biquaternions

Renee RussellKim Kesting

Caitlin Hult

SPWM 2011

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Sir William Rowan Hamilton(1805-1865)

Physicist, Astronomer and Mathematician

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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:

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Review of Quaternions, H

A quaternion is a number of the form of:

Q = a + bi + cj + dk

where a, b, c, dR,

and i2

= j2

= k2

= ijk = -1.

So what is a biquaternion?

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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)e1 +(w+xi)e2 +(y+zi)e3

e12 = e22 = e32 =e1e2e3 = -1.

* Notice this i is different from the i component of the

basis, {1, i, j, k} for a (bi)quaternion! *

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B can also be written as the complex combination

of two quaternions:

B = Q + iQ where i =-1, and Q,QH.

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, zR

Biquaternions

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Properties of the Biquarternions

ADDITION: We define addition component-wise:

B = a + be1 + ce2 + de3 where a, b, c, d C

B = w + xe1 + ye2 + ze3 where w, x, y, zC

B +B =(a+w) + (b+x)e1 +(c+y)e2 +(d+z)e3

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Properties of the Biquarternions

SCALAR MULTIPLICATION:

hB =ha + hbe2 +hce3 +hde3 where h C orR

The Biquaternions form a vectorspace overC and R!!

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Biquaternions

are an algebra

overC!biquaterions

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Properties of the Biquarternions

So far, the biquaterions overC have all the same

properties as the quaternions overR.

DIVISION?

In other words, does every non-zero element have a

multiplicative inverse?

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Properties of the Biquarternions

Recall for a quaternion, Q H,

Q-1 = abe1ce2de3 where a, b, c, dRa2 + b2 + c2 + d2

Does this work for biquaternions?

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Biquaternions are NOT a division algebra

overC!

Quaternions

(over R)

Biquaternions

(over C)

Vector Space?

Algebra?

DivisionAlgebra?

Normed

Division

Algebra?

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Biquaternions are

isomorphic to M2x2

(C)

Define a map f: BQM2x2(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 fis one-to-one, onto, and is a linear

transformation. Therefore,BQis isomorphic toM2x2(C).

[ ]

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Applications of Biquarternions

Special Relativity Physics

Linear Algebra

Electromagnetism