Package ca.uwaterloo.alumni.dwharder.Numbers

The Numbers package contains five classes which represent progressively higher number systems using the Cayley-Dickson technique for constructing algebras.

See:
Description

Class Summary

Complex	A class implementing complex numbers using pairs of doubles.
Octonion	A class implementing octonions using octets of doubles.
Quaternion	A class implementing quaternions using quartets of doubles.
Sedenion	A class implementing sedenions using sixteen doubles.
Trigintaduonion	A class implementing trigintaduonions using thirty-two doubles.

Package ca.uwaterloo.alumni.dwharder.Numbers Description

The Numbers package contains five classes which represent progressively higher number systems using the Cayley-Dickson technique for constructing algebras.

We begin by building the complex numbers from the real numbers, and progressively, we define the quaternions based on the complex numbers, the octonions based on the quaternions, the sedenions based on the octonions, and finally the trigintaduonions. At most steps, another property is lost:

• Quaternions are no longer commutative (qr ≠ rq),
• Octonions are no longer associative (i.e., o(pq) ≠ (op)q),
• Sedenions are no longer alternative (i.e., s(st) ≠ s²t), there exists zero divisors (i.e., s ≠ 0 and t ≠ 0 but st = 0), and consequently, |st| ≠ |s||t|,
• For trigintaduonions, I don't think there is anything else you can loose; this one is added for fun.

At each step, we introduce a new symbol, the square of which is -1. These symbols are assumed to commute with real numbers. The product of any new symbol introduces yet another symbol, the square of which is -1. This produces a sequence of spaces, each of which has twice the dimension of the previous space.

Name	Symbol	Dimension	Additonal Symbols
Complex Numbers	i	2
Quaternions	j	4	k = ij
Octonions	e'	8	i' = ie', j' = je', k' = ke'
Sedenions	e''	16	i'' = ie'', j'' = je'', k'' = ke'', e''' = e'e'', i''' = i'e'', j''' = j'e'', k''' = k'e''
Trigintaduonions	e(4)	32	i(4) = ie(4), j(4) = je(4), k(4) = ke(4), e(5) = e'e(4), etc.

References

• Hubert Holin, "The Quaternionic Exponential (and beyond)".
• M. Koecher and R. Remmert, "Hamilton's Quaternions" in J.H. Ewing (ed.), Numbers, Springer-Verlag, New York, 1991, p.189-220.
• J.A.R. Holbrook, class notes, 1990-4, unpublished.
• Jack B. Kuipers, Quaternions and Rotation Sequences, Princeton University Press, Princeton, N.J., 1990.
• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series-55.
• Maple 9.5. Maplesoft, Inc., Waterloo, Ontario.