Real numbers are dual numbers

Well not exactly reals, but rationals, which approximate \mathbb{R}.

Rationals are defined as pairs of integers (p,q)=\frac{p}{q}.

By using a nilpotent versor \epsilon^2=0, we can represent a rational number as

(p,q)=\frac{p}{q}=\epsilon p+q

Then the rule for addition of rational numbers becomes a product of a dual number

(p,q)+(r,s)=\frac{p}{q}+\frac{r}{s}=\frac{ps+rq}{qs}=(\epsilon p+q)(\epsilon r +s)=\epsilon(ps+rq)+qs

This should be a hint on how to generalize geometric algebra, so “real numbers” get treated in the same way as versors squaring to {1,-1,0}.

(the rule for product of rationals needs some investigation)

Besides, integers can also be treated as pairs of natural numbers:


And the product of integers uses the formula for products


This is because, like e_1^2=1, (-1)^2=1

so an integer can be modeled as (a,b)=a+e_1b

And a rational number is

((a,b),(c,d))=\epsilon(a,b)+(c,d)=\epsilon a +\epsilon e_1b+c+e_1d

So \mathbb{Q} has always been some kind of multivector space.

Maybe computations in geometric algebra can be reduced to unsigned integer arithmetic.

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