Bookkeeping with grades
We gave zero a reciprocal and put its "size" on a shelf called the grade. Now we actually do arithmetic with it — and multiplication turns out to be completely, almost embarrassingly, solved. The whole of it is bookkeeping with a single integer.
Every value here splits cleanly into two pieces: an ordinary coefficient, and a grade that says how many factors of zero it carries. The tidiest way to write that is a coefficient times a power of zero:
The grade is the exponent on the zero; the coefficient is whatever ordinary number rides in front. That is the entire data of a value — a number, and a place for it to stand.
Once values look like that, multiplication writes itself. To multiply two of them, multiply the coefficients the ordinary way and add the grades — because multiplying powers of the same base adds their exponents, which is a rule you already trust.
That one line is the whole multiplication table. Watch the case that used to be forbidden fall straight out of it:
The grades +1 and −1 simply add to 0, the coefficients multiply to 1, and you are left standing on the ordinary numbers holding a plain 1. Taking a reciprocal is just as easy: flip the sign of the grade and invert the coefficient.
In ordinary mathematics 0 · ω — "zero times infinity" — is the archetype of an indeterminate form, a question limits are summoned to argue over because the plain arithmetic refuses to answer. Here there is nothing to argue. It is grade +1 meeting grade −1; they cancel to grade 0; the answer is 1. The single indeterminate form multiplication ever had is closed by one integer of bookkeeping — no limit, no approach, no argument. Multiplication is total, reversible, and exact, with no exceptions left anywhere in it.
A careful reader might stop here. We just said 0·ω = 1 lands at grade 0 — but so does 0^ω = −1 from a few chapters back. Two different values, same grade. Isn't that a contradiction?
No — and seeing why keeps the bookkeeping straight. The grade is only one coordinate of a value; it tracks size and order, not the whole thing. Grade 0 is simply the ordinary numbers, and ordinary numbers of course still differ among themselves — +1 and −1 are both grade 0, told apart by their coefficient, exactly as they always were. The grade never claimed to name the value by itself; it names where the value stands, and the coefficient says which one it is.
Multiplication is finished: one integer per value, one rule, no gaps. It is the cheap, clean half of arithmetic in this world. You might reasonably expect addition to be just as easy — and here the good news stops. Addition, it turns out, is the hard half, and the precise reason it's hard is the hinge the whole second half of this book swings on.