Is it possible to unify the number system?

Question

And have a unified set of numbers? Now one can ask what the use of this is but let's leave that aside. I think Sedenions in a sense are the highest we have been up to. What if we were to go beyond that and have one ultimate set of numbers?

Answer 1

Too long to post it as a comment hence posting as an answer. Here is something that I had explored but not found much useful.

Let $d_i$, $i = 1,2,3,\ldots$ be a sequence of positive real numbers such that $\sum_{i}d_i$ is divergent. We define $$ S_n = \sum_{k = 1}^{n}d_n. $$

The set $S = \{S_1, S_2, S_3, \ldots\}$ can be considered as unified system of numbers in some sense. For example, if $d_i = 1$ for all $i$ then we have the system of natural numbers while if if $d_i = 1/i$ then get the system of harmonic numbers; thus we are able to unify the sequence of natural numbers and harmonic number under a single system.

Analytical properties can then be generalized under this definition. For example, the well known Riemann sum that is taught in a starting class of integral calculus

$$ \lim_{n \to \infty}\frac{1}{n}\sum_{r = 1}^{n}f\Big(\frac{r}{n}\Big) = \int_{0}^{1}f(x)dx $$

is generalizes to

$$ \lim_{n \to \infty}\frac{1}{S_n}\sum_{r = 1}^{n}d_r f\Big(\frac{S_r}{S_n}\Big) = \int_{0}^{1}f(x)dx $$

Answer 2

The progression that the typical layperson experiences

$$ \mathrm{naturals} \subseteq \mathrm{integers}\subseteq \mathrm{rationals} \subseteq \mathrm{reals} \subseteq \mathrm{complexes} $$

is really rather misleading; the number systems that come up in mathematics simply aren't a linear progression like that. They branch off in all sorts of different ways for different purposes that aren't really compatible with one another.

For example, number theorists often study number systems that one might call "the integers mod $n$", which is a number system where, if you add one to itself repeatedly the right number of times (e.g. exactly $n$ times), you get zero.

In calculus, one uses the extended real numbers, which adjoins two new numbers $\pm \infty$ that quantify various kinds of geometric behavior, and consequently have unusual arithmetic behavior, such as the fact that $x+y = y$ does not imply $x=0$.

In set theory, one is interested in cardinal numbers and ordinal numbers, which have their own arithmetic with unusual behavior; e.g. if $x,y$ are cardinal numbers and they aren't both finite, then $x+y = \max(x,y)$.

In ancient greece, geometers would do arithmetic with line segments — the idea of quantifying the length of a line segment is, AFAIK, a comparatively recent idea. The closest the ancient greeks got was the study of proportions; e.g. identifying the proportion one line segment might make with another.

Algebraists of various sorts might treat all sorts of strange and mystifying things as numbers, depending on the context. An algebraic geometer might have lots of cause to treat polynomials as numbers. A functional analyst might treat matrices as numbers.

Really, there isn't any sense one can make of the idea of an "ultimate" number system.