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:
(a,b)=a-b
And the product of integers uses the formula for products
(a,b)(c,d)=(a-b)(c-d)=(ac+bd)-(ad+bc)=(ac+bd,ad+bc)
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.