The Biquaternions

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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

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:
• 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
• 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
• 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

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

• 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

Properties of the Biquarternions

• Closed
• Commutative
• Associative
• 0 = 0 + 0e1 + 0e2 + 0e3
• -B = -a + (-b)e1 + (-c)e2 + (-d)e3

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!

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

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 where a, b, c, d  R

a2 + b2 + c2 + d2

Does this work for biquaternions?

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).

[ ]

Applications of Biquarternions

• Special Relativity
• Physics
• Linear Algebra
• Electromagnetism