# 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

“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

B = a + be

1

B’ = w + xe

1

+ ce

2

+ ye

2

+ de

3

+ ze

3 where a, b, c, d

C where w, x, y, z

C

B +B’ =(a+w) + (b+x)e

1

+(c+y)e

2

+(d+z)e

3

Properties of the Biquarternions

Closed

Commutative

Associative

0 = 0 + 0e

1

+ 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

– ce

2

– de a 2 + b 2 + c 2 + d 2

3 where a, b, c, d

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?

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