Of Surreal Geometry 3

Sadly, the arctangent function does not preserve a linear rate of growth, and I do not know if Riemann’s sphere retains this property. We want a transformation that, as you squish numbers together, their magnitude becomes proportionally smaller. Start with zero. The distance from zero to one is one. Zero has magnitude zero and one has magnitude one. But let’s squish in one to one half to allow for the number two to fit between zero and one. Then the number one will have magnitude one half, and the number two will have a magnitude of one. So, by mathematical induction, every time we squish in a number, we result in having each previous number divided by a magnitude of the newly squished in number. The question is what happens at infinity.

In theory, no matter how many numbers we squish in, the slope of growth of the magnitude of our natural numbers will always be one. What we know for sure is that, at infinity, or at ω, the distance between each number is infinitesimal, and all the natural numbers are accounted for.

The number n is at a distance of n/ω from zero. If we add n/ω+m/ω, we get (n+m)/ω. Thus preserving linearity.

Now, were we to add ω/ω+ω/ω, we get two. And there it is! The geometric result we were looking after. We can then talk about there being a single natural number ω/2, represented in our method as ω/2ω=1⁄2, which we will call h, h⁄ω in our method, such that h+h=ω.

Sums between these numbers, from h to ω, result in transfinite numbers. Meaning that the natural numbers are not absolutely closed under addition, and that as numbers grow larger and larger, they acquire properties that previous numbers can’t possess. Such properties are left for us to discover as time passes.