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This page believes that Clifford algebras, and their rules, were discovered and not invented. If you don’t believe this try this page.
There are many square roots of −1 called γ0, γ1... .
Of course this is already seen in the quaternions: for any real θ,
(i cos(θ)+ j sin(θ))2 = −1. The γs that Clifford discovered also have
if i≠j then γiγj = −γjγi.
Products of these roots can always be rearranged so that none of the roots appears more than once but possibly with a “−” in front. Thus if you consider only those roots γ0 thru γn−1 then there are 2n products of n these roots, if we ignore minus signs, one product for each subset of the n roots. These products also include 1, which corresponds to the null set. These 2n products span a 2n-D vector space over the reals called Cl(n) or the nth Clifford algebra members of which are called Clifford numbers or CNs here. The sub-space spanned by the n roots alone is very important and is called V(n) and Cl(n) is said to be generated by V(n) because you can get anything in Cl(n) starting from V(n) by multiplies and adds. Note that 1 is a vector in Cl(n) but 1 is not in V(n). All scalars, being a multiple of 1, are also in Cl(n). When we multiply a scalar by a vector to get a vector it doesn’t matter whether we think of the scalar as a vector or not—we get the same answer.
Knowing how to multiply the basis elements together in a vector space allows one to compute the product of any two vectors if one adheres to the rules:
(a+b)c = ac+bc
a(b+c) = ab+ac.
Multiplying arbitrary pairs of Clifford numbers is possible since we know the products of pairs of basis elements of Cl(n). This is a special case of associative algebra. Speaking of rules you might have noticed that like quaternions, most Clifford numbers don’t commute, but they do associate:
a(bc) = (ab)c.
Speaking of quaternions they can be found here too under the following alias:
i = γ0γ1
j = γ1γ2
k = γ2γ0.
The even algebra of Cl(3) is thus like the quaternions.
The following triple is also isomorphic to the quaternions:
i = γ0
j = γ1
k = γ0γ1.
Cl(2) is thus like the quaternions.
The subspace spanned by 1 and some particular γi is isomorphic to the complex numbers.