Linear Algebras

Rise of Quaternions.

Very soon after complex numbers were applied to the study of directed lines in a plane, attempts were made to find numbers for directed lines in space of three dimensions, and as the result of some ten years' study of the problem, in 1843 Sir William Rowan Hamilton (q.v.) announced his discovery of "quaternions." These are ratios of directed seg ments in ordinary space, and so long as one is confined to a single plane they are precisely the same as complex numbers for that plane. In this way the idea of number has been extended to include not only ordinary numbers, but negative numbers, then complex numbers, then quaternions, each extension including as special cases the preceding known numbers.

However, it must be kept well in mind that in all the extensions the fundamental notion was that of finding solutions for equations which could not be solved in terms of known numbers. Hamilton, for instance, created other "roots of unity" discovering thus the tetrahedral, octahedral, and icosahedral groups. In more mathe matical terms, the problem was to find a domain for reducibility of equations that were found to be irreducible in given domains. For instance, 3x= 5 demands the domain of fractions for its solvability; the domain of V7, an irrational number, for its solution; x+8=5, the domain of the negatives; the domain of complex numbers ; and we may add that to factor demands the domain of quaternions. This ability of the human mind to create ever new and wider domains of ideal numbers for the handling of problems, is very important, for it is incontrovertible evidence of the fact that mind transcends sense data, and creates a rational world whose objects cannot be studied by the senses. The bearing of this on philosophical questions is fundamental.

One of the characteristics of numbers as usually understood, namely, that the quotient of a number by another should be a unique number, Hamilton considered to be essential in the exten sions of numbers. In the field of complex numbers this character istic is preserved, for the quotient of c+di by a+bi is the definite number This characteristic is also preserved for the quaternions in which Hamilton was specially interested, namely, those in which the coefficients are ordinary real numbers. By this is meant that in the quaternion Q=a+bi-kj-kdk the numbers a, b, c, d, are real integers, fractions or irrationals. The hypernumbers j, k, are such that their squares are each equal to —I, and such that ijk=-1. For two such quaternions as Q the quotient is unique. This would not be true if the coefficients a, b, c, d were themselves allowed to be complex numbers. Further, this closes the list of number systems, or linear associative algebras, for which division is a unique process, when the coefficients are chosen from the real or complex domains. However, if the coefficients are allowed to be numbers from a "Galois field," there are other "division" algebras. These have

been studied by Leonard Eugene Dickson. There are also other division algebras which are non-associative.

Doubtless the reader has noticed above that ij= —ji, a char acter of quaternions quite different from anything in complex numbers, or real numbers, where we always have xy=yx. In his studies Hamilton tried to preserve this "commutative property," but found that he could not at the same time have this, the property of unique division, and also another property considered highly desirable, "the associative property," which is stated in the equation x.(y.z)-- (x.y).z. (See ASSOCIATIVE LAWS.) The associative property, which permits the multiplication of any two adjacent factors, is almost a necessity in order to have a flexible system, and one that does not produce elaborate, compli cated forms. The property of unique division is also quite con venient, though not so necessary. Consequently Hamilton decided to let the commutative property go. Hence in quaternions we have the first case of a non-commutative algebra. The associative property Hamilton considered so essential that he devotes a great deal of space to the establishment of it, both by algebraic and by geometric considerations.

Hypernumbers.

The hypernumbers j, k are called units, but it must be noticed that they are not the only units. For if we let 1, in, n be any three real numbers such that 1, then a=lid-mj-knk is also a unit, and if 0 is any angle, is a unit. The latter could be called a unit quaternion, the form a without 0 a unit vector. A vector is a quaternion whose square is a negative number. If we draw all the planes in ordinary space through a fixed point, to each plane corresponds a unit vector as the imaginary for that plane. For instance, to a horizontal plane viewed from above we could affix i; to the meridian plane, viewed from the east side, j; and to an east and west plane, viewed from the north side, k. To any other plane would be affixed the unit vector a =li+mj+nk, where 1, m, n would be the direction cosines of the normal to the plane, referred to the three planes just mentioned. In each case the square of the unit vector is - Y.. If we confine all our geometry to a single plane the algebra will involve but a single imaginary unit and we have the ordinary com plex numbers. From the investigations of Hamilton emerged two distinct results. The most important one is that the ordinary real and complex numbers are special cases of quaternion numbers, and naturally the suggestion follows that quaternions themselves are special cases of more general numbers, or hypernumbers if we wish to be precise. The second result is that the problem of the algebraic handling of direction in three-dimensional space has been solved. The first result Hamilton saw was of prime importance, and he began to push his investigations into the most general types of hypernumbers.