**Quaternion Grassmann-Hamilton-Clifford algebras: new mathematical tools for classical and relativistic modeling**

by Patrick R. Girard (University of Lyon; Creatis; Insa-Lyon; France)

in O. Doessel and W.C. Schlegel (Eds.): WC 2009, IFMBE Proceedings 25/IV,

pp. 65-68, 2009. www.springerlink.com

**Abstract**

The paper presents new mathematical tools for classical and relativistic modeling based on a quaternion formulation of Clifford algebras which we shall call Grassmann-Hamilton-Clifford algebras. These algebras allow to develop an associative exterior calculus for any metric and any dimension yielding probably the best representations of the covariance groups. As applications, the paper develops these algebras in euclidean three-space and pseudo-euclidean spacetime, and in particular the Frenet frame and the relativistic moving frame.

**Keywords **

Quaternions, biquaternions, tetraquaternions, Grassmann-Hamilton-Clifford algebras, classical and relativistic modeling.

*Source:* Email by P. Girard (21 Oct. 2009), patrick.girard_at_insa-lyon.fr

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Neither Hamilton nor John Graves had the remotest understanding of the Cotes formula cosu + isin u = e^(iu) discovered by Roger Cotes in 1714. It is this Cotes formula which proves that there are only n nth roots of any real, imaginary or complex number.

Why do the roots of +1, -1, +i and -i always add up to zero? The answer is that computing all these roots is a matter of going round the angles of a circle and finishing at the 360 degrees equalling 0 degrees starting point. Incidentally, if you have computed the roots of +i, then the roots of -i are easily obtained by substituting -i for +i.

Further to my previous comments, -1 does not have more than two square roots, nevertheless -1 does have three cube roots, these are cos60+isin60, cos180+isin180 which equals -1, and cos300+isin300. Hamilton seemed to have no understanding of this.

Hamilton’s Quaternions equation i^2=j^2=k^2=ijk=-1 is incorrect because -1 like all real, imaginary and complex numbers only has two square roots, so that -1=k^2 is wrong unless k can equal either i or j. Likewise all real, imaginary and complex numbers can only have three cube roots, four fourth roots, five fifth roots etc.

I can clarify my comments of 25 August 2011 as follows. Hamilton on 16 October 1843 alleged that i^2=j^2=k^2=ijk=-1. This equation is incorrect because every real, imaginary and complex number has only two square roots, so k^2=-1 is wrong unless k can equal either i or j. It can likewise be demonstrated that every real, imaginary and complex number has only three cube roots, four fourth roots, five fifth roots etc. Hamilton asks what are we to do with ij when i and j are the unequal roots of a common square. There is in fact no law of arithmetic which makes ij equal to anything but +1.

Hamilton in his letter to John Graves of 17 October 1843 discovers the fallacious non communatitive multiplication of imaginary numbers through failing to understand imaginary numbers. This has been an educational disaster area ever since. All multiplication whether of imaginary or real numbers is communitative.

I cannot find this on Springer. Any clue how to access it?