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Quaternions

He put him in prison, and delivered him to
four quaternions of soldiers to keep him. Acts 12:4

It is long accepted idea that the real numbers have a geometrical interpretation as points lying on a line, and that the complex numbers have a similar geometrical interpretation as points of a plane.  Therefore it seems natural, to ask whether there might be some form of hyper-complex numbers whose elements may be interpreted as points in three-dimensional space, with a corresponding representation as triples of real numbers.  William Rowan Hamilton (1805-1865), Irish physicist and mathematician, spent years trying to find a three dimensional number systems, but with without success. Hamilton also tried to find a real significance of the complex unit .

Already in 1833 Hamilton pointed out in a paper that the addition sign used in the Cartesian representation of a complex number is misleading, since a real and purely imaginary number cannot be directly added together arithmetically.

An obvious question was that if a rule for multiplying two numbers together was known, what about multiplying three numbers?  For over a decade, this simple question had bothered Hamilton He wrote to his son:

Every morning in the early part of the above-cited month [Oct. 1843] on my coming down to breakfast, your brother William Edwin and yourself used to ask me, 'Well, Papa, can you multiply triplets?' Whereto I was always obliged to reply, with a sad shake of the head, 'No, I can only add and subtract them.'

When he looked in 4 dimensions instead of 3 he found a solution.  On October 16, 1843 he discovered such a hyper-complex number system which he called quaternions.¹ The origin of the name quaternions was motivated by the fact, as we shall see below, that there is a quaternion of their generators.²

Constructions of quaternions

Quaternions form an extension of the field of complex numbers having the property that the commutative law fails for multiplication, despite the fact that every non-zero element has an multiplicative inverse

Quaternions via vector space structure

For introduce the product

where , and

are the standard dot and vector (cross) products of vectors, resp., and denotes a multiplication of vector by a scalar .

Relative to this product and the standard vector space structure, becomes a non-commutative field, denoted by and whose elements are called quaternions.

The following properties can be readily verified

• the identity element in is ,
• the multiplicative inverse of is  ,
• the associativity of the product is implied by the vector triple identity ,
• if we define the norm norm of the quaternion by , then for all ; this property is another form of the Euler four-square identity .
• if we define the absolute value of the quaternion by , then for all ,
• the elements , , , obey the multiplication rules , , , ;  the multiplication of units can be derived from the diagram below: if appear on the circle in the order indicated by the arrows then   :

• the quaternion can be written in the form of  the linear combination  ,
• the dot product in and the product of are connected by the relation , for all .

Quaternions with are called pure quaternions. If we define the conjugation by  , , , , then we get an automorphism of the -algebra . Actually, conjugation is an anti-automorphism . Moreover we have .

Example: Fix and define the map

This mapping is norm preserving and thus orthogonal. Since it leaves the elements invariant, it defines an orthogonal transformation of   which is given by

The determinant of the transformation equals one.

Hamilton used the term biquaternion for a quaternion with complex coefficients.

In 1877 Frobenius proved that the only associative finite dimensional division algebras are the real numbers, the complex numbers and the quaternions.

Notes

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