This link illustrates how numbers developed
Charles Muse studied numbers to the limit where he realised the Pythagorean base of our number systems.
The general article in wiki on number systems excludes Musean numbers or hyper numbers!
The rise of number Theory obscures the underlying Pythagorean theory of Arithmoi! Supposedly numbers are mathematical objects but no one says what that means. Further they avoid any possible contradictions by saying that these things are axiomatised! Then they define certain combinatorial rules and say proudly: we do not need to relate thes notations to anything in particular, we only need to show that whatever we do relate them to satisfy these axioms!
Well that is all well and good, but Newton found that his axioms for stolid objects extendible to elastic objects by scalar factors were woefully inadequate for fluid substances! It was not a case of not enough infinitesimals, it was because his axioms lacked crucial defining elements!
The Arithmoi are crucial for engaging with space, but the fundamental Arithmos for fluids and gases and the Aether is the sphere!
I always like to refer to Hamilton's papers on the Science of Pure Time or conjugate couples or functions.. In this paper Hamilton demonstrates clearly how number systems arise as counting notation of scales or steps in what we now loosely call a vector space, but which actually are lineal combinations of segments. It is no wonder hr recognised the mastery of Grassmann when he read the Usdehnungslehre 1844 where this concept is developed very fully, and called an extensive arithmetic of Strecken written in algebraic terminology. This algebraic terminology was of its time; that is o say it was imply a symbolic arithmetic. Later, algebraists would twist the otd to defend a new branch of mathematics gist the onslaught of those who like Newton, thought in terms of Analysis and Synthesis..
Hamilton introduced his intensive form of numbers by Fiat. Thus he brings in the rational numbers as notation, and then the real numbers as a type of lambda variable that represented the limit of a rational fraction which was potentially infinite. Grassmann introduced the intensive orm by vertical projections of Strecke onto Strecke. This is precisely the way to introduce the trigonometric ratios into an extensive metric system, and thus to provide rational divisions of the unit Strecken. This also provides a ready means of evaluation using the extant and detailed Sineor trigonometric tables. The tables go beyond the right triangle into logarithmic and exponential tables.
Grassmann was therefore conscious of his general arithmetic having an evaluative basis in the trig tables.
This sounds so pretentious, but trigonometry was what we did extensively at secondary school, and it was classed as arithmetic, with formulas for area being the main algebraic content. What Grassmann did was profoundly simple: he generalised trigonometry using algebraic terminology and notation. By presenting it in this generalised way he was able to demonstrate that it was a group or a ring structure. This was a new way of analysing arithmetics made famous by a few mathematicians who were deriving it from Galois theory..
Grassmanns treatment of a group structure on trigonometry, generalised into all so called applied geometry and mechanics. It's chief and obvious advantage was to cut down on unnecessary intermediary calculations, but it also gave a coherent geometric insight into dynamical systems hich could be modelled by trigonometry. Further it extended easily into graph analysis, and thus polynomials by the use of coordinate notation.
The other fundamental notion Grassmann introduced in the concept of strecke was the allied notion of stretch between points! A Strecke was therefore a complex symbol, not just a line! It is the stretch between points that carries the dynamics of projection( parallel or otherwise) and the concepts of difference as an object. This difference between points could be abstracted and treated as an arithmetic which then could be added back to a point to see hat the result was. The stretch captured many dynamic quantities which otherwise could not be visualised, particularly force and velocity, but the also momentum etc and any magnitude that could be written as a scaled extension of some unit that defined the magnitude.
Suddenly geometry was applicable or extendable almost everywhere!
The final contribution to extending trigonometry was the introduction of the Null parallelogram. This in fact was later shown to be the exact concept to describe a method called the determinant of a 2 x2 matrix. In addition, Caylet generalised it to n x n matrices.
The determinant or so called wedge product was extendible to n x m matrices and spawned or underpinned the notion of a whole class of combinatorial products.
Grassmann's extension of Trigonometry into a general group or ring structure is one of the fundamental turning points in modern mathematics, along with Hamilton 's Quaternions.
The Network of these straight lines criss crossing space are the Arithmoi. The tangled network of curved lines twisting criss cross through space I have named the Shunyasutras.