# Quaternions Package

This package implements Hamilton's quaternion algebra. Quaternions have the form

where

a,

b,

c, and

d are real numbers. The symbols

i,

j, and

k are multiplied according to the rules

. Quaternions are an extension of the complex numbers, and work much the same except that their multiplication is not commutative. For instance,

.

Because of the similarities between quaternions and complex numbers, this package imitates

*Mathematica*'s treatment of complex numbers in many ways. To provide a clear distinction between quaternions and complex numbers, all quaternions should be entered using the form

Quaternion where

a,

b,

c, and

d are real numbers. Only limited support is offered to the symbolic form

a+Ib+Jc+Kd.

Defining, testing, and transforming quaternions.

Out[2]= | |

Out[3]= | |

The primary purpose of this package is to define arithmetic for quaternions. It adds rules to

Plus,

Minus,

Times,

Divide, and, most importantly,

NonCommutativeMultiply. It is only legal to use

Times when multiplying two nonquaternions or a quaternion by a scalar (i.e., real) number. When multiplying two quaternions, you must use

NonCommutativeMultiply.

Addition is done component by component.

Out[4]= | |

Be sure to use

rather than

when multiplying quaternions.

Out[5]= | |

This multiplication is noncommutative.

Out[6]= | |

Be careful with

Divide since

*Mathematica*'s internal rules quickly turn it into

Times.

Out[7]= | |

Although quaternions are whole algebraic objects, just as with complex numbers, it is sometimes useful to look at their component parts. These objects form a vector space over the real numbers, with their standard basis being

. You may use standard

*Mathematica* techniques for extracting the individual components. However, there are other vector-type parameters such as length and direction which you may want to look at as well.

The projection of a quaternion onto

space, the nonreal part of the quaternion, is called the pure quaternion part. This plays a role similar to the pure imaginary part of a complex number.

Re[w] | the real part Re w |

Conjugate[w] | the quaternion conjugate |

Abs[w] | the absolute value |

AbsIJK[w] | the magnitude of the pure quaternion part of w |

Norm[w] | the sum of the squares of the components of w |

Sign[w] | the sign of the quaternion w |

AdjustedSignIJK[w] | the sign of the pure quaternion part of w, adjusted so its first nonzero component is positive |

Component functions of quaternions.

In the conjugate of a quaternion, all the signs of the nonreal components are reversed.

Out[8]= | |

The sign of a quaternion is defined in the same way as the sign of a complex number. It is the "direction" of the quaternion.

Out[9]= | |

This returns a quaternion with norm 1 and real part 0.

Out[10]= | |

This gives the standard Euclidean length.

Out[11]= | |

A quaternion with a zero

I component will still have a nonzero pure quaternion part.

Out[12]= | |

For a complex number

,

is defined by

. The package defines

in a similar way, using the pure quaternion part of

q instead of the pure imaginary part of a complex number. Indeed, it makes analogous definitions for the following elementary functions:

Exp,

Log,

Cos,

Sin,

Tan,

Sec,

Csc,

Cot,

ArcCos,

ArcSin,

ArcTan,

ArcSec,

ArcCsc,

ArcCot,

Cosh,

Sinh,

Tanh,

Sech,

Csch,

Coth,

ArcCosh,

ArcSinh,

ArcTanh,

ArcSech,

ArcCsch, and

ArcCoth.

The exponential of a quaternion can be quite complicated.

Out[13]= | |

Just as with complex numbers, it is important to beware of branch cuts.

Out[14]= | |

A four-dimensional analog of de Moivre's theorem is used for calculating powers of quaternions.

Out[15]= | |

The functions so far have been intended to work with quaternions whose components are arbitrary real numbers. Just as the integers and Gaussian integers are interesting subsets of the reals and complexes, there is a special subset of the quaternions called the quaternion integers. This subset is a little broader than you might expect. It includes not only those quaternions that have all integer components, but also those quaternions that have all components being odd multiples of

. In this subset there are 24 quaternions that have multiplicative inverses. These are the units of the algebra. They correspond roughly to

in the Gaussian integers.

Round[w] | the closest integer quaternion to w |

OddQ[w] | test whether the quaternion w is odd |

EvenQ[w] | test whether the quaternion w is even |

IntegerQuaternionQ[w] | test whether the quaternion w is an integer quaternion |

UnitQuaternions | the list of 24 units of Hamilton's division algebra |

UnitQuaternionQ[w] | test whether w is a unit quaternion |

Integer quaternion functions.

Round for quaternions returns a

Quaternion in which either all components are integers, or all components are odd multiples of 1/2.

Out[16]= | |

A quaternion is even if its norm is even.

Out[17]= | |

A quaternion integer has components that are either all integers or all halves of odd integers.

Out[18]= | |

Given a quaternion

q and a unit quaternion

e, then

and

are, respectively, right and left associates of

q. It is useful to choose an arbitrary associate and call it the primary associate. This package chooses the associate with the largest real component.

The associates of an integer quaternion.

This is the primary left associate of the quaternion.

Out[19]= | |

The primary right associate is often very similar.

Out[20]= | |

Quaternion multiplication is noncommutative, so there are two greatest common denominators, one for the left side and one for the right. Since this function depends on the value returned by

PrimaryLeftAssociate and

PrimaryRightAssociate, the

RightGCD and

LeftGCD are not unique.

LeftGCD[w,u] | the greatest common left divisor of w and u |

RightGCD[w,u] | the greatest common right divisor of w and u |

Mod[w,u] | w modulo u (remainder on division of w by u) |

Some integer division functions.

The largest quaternion that divides both of these is

Quaternion.

Out[21]= | |

Out[22]= | |

Just as with complex numbers, the quaternion

Mod works.

Out[23]= | |

You can specify a quaternion as the modulus.

Out[24]= | |

PrimeQ has the option

GaussianIntegers->True that checks to see if a number is prime with respect to the Gaussian integers. This package extends

PrimeQ farther to check if a number is prime with respect to the quaternions.

Lagrange proved that every integer can be expressed as a sum of squares of, at most, four integers. Therefore, given an integer

n, there is a quaternion

q with integer components such that

q**Conjugate[q]==n. So no integer is prime with respect to the quaternions. In fact, a quaternion integer is prime if and only if its norm is prime in the usual sense.

An extension of PrimeQ.

19 is a prime with respect to the Gaussian integers.

Out[25]= | |

It is not a prime with respect to the quaternions. It can be factored into

Quaternion and

Quaternion.

Out[26]= | |

Out[27]= | |