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

“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

, and

i

2

= j

2

= k

2

= ijk =

-1.

So… what is a biquaternion?

Biquaternions

A

biquaternion

is a number of the form

B = a + bi + cj + dk

and

i

2

= j

2

= k

2

= 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)e

1

+(w+xi)e

2

+(y+zi)e

3 e

1

2

= e

2

2

= e

3

2

=e

1 e

2 e

3

=

-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)e

1

+ (w+xi)e

2

+ (y+zi)e

3

=(a + ce

1

+ we

2

+ye

3

) +i(b + de

3

+ xe

2

+ze

3

)

where

a, b, c, d, w, x, y, z

R

Properties of the Biquarternions

ADDITION:

We define addition component-wise:

B = a + be

1

B’ = w + xe

1

+ ce

2

+ ye

2

+ de

3

+ ze

3

where

a, b, c, d

 where

w, x, y, z

C

C

B +B’ =(a+w) + (b+x)e

1

+(c+y)e

2

+(d+z)e

3

Properties of the Biquarternions

ADDITION:

Closed

Commutative

Associative

Additive Identity

0 = 0 + 0e

1

Additive Inverse:

+ 0e

2

+ 0e

3

-

B = -a + (-b)e

1

+ (-c)e

2

+ (-d)e

3

Properties of the Biquarternions

SCALAR MULTIPLICATION

:

hB =ha + hbe

2

+hce

3

+hde

3

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) + 0e

1

+ 0e

2

+ 0e

3

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 – be

1 a

2

+ b

2

– ce

2

+ c

2

– de

3

where

a, b, c, d

+ d

2

R

Does this work for biquaternions?

Biquaternions are NOT a division algebra over

C

!

Vector Space?

Quaternions

(over R)

Biquaternions

(over C)

Algebra?

✔ ✔

Division

Algebra?

Normed

Division

Algebra?

Biquaternions are isomorphic to M

2x2

(C)

Define a map

f: B

Q

M

2x2

(C)

by the following:

f(w + xe

1

+ ye

2

+ ze

2

[ ]

-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,

B

Q

is isomorphic to

M

2x2

(C).

Applications of Biquarternions

Special Relativity

Physics

Linear Algebra

Electromagnetism

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