MVVM: Mathematical Notes and Papers

MATHEMATICAL NOTES AND PAPERS

Quick access links in alphabetical order (apart from a or the):

The concepts and connections between concepts which are to receive special attention in these notes and papers are the following:

SPACE/ CONTEXT	CHONG-PLUS OPERATION	AVERAGE/ MEAN	NUMERAL SYSTEM AND BASE
one-dimensional	chong +0 (addition)	arithmetic	no place-value notation
two-dimensional	chong +1 (multiplication)	geometric	base 2 (binary) or base 4 (quaternary)?
three-dimensional	chong +2 ((power-3) involution)	?	a base other than 2 or 4? (senary or octal?)
four-dimensional	chong +3 (reiterative involution)?	?	a chong-3 system?
five- or more-dimensional	chong +4 or higher?	?	a chong-4 or higher-level system?

Addition is an elementary mathematical operation on the zeroth level of reiteration, multiplication such an operation on the first level, involution one on the second level, reiterative involution one on the third level. In a universal system of ever higher operational levels of reiteration i call these arithmetic operations "chong operations": 'chong +0' for addition, 'chong -0' for subtraction, 'chong +1' for multiplication, 'chong -1' for division, 'chong +2' for involution, and so on and so forth. Addition may be considered to be 'native' to one-dimensional space, whereas multiplication and operations of a higher level of reiteration are not. Similarly, multiplication is, then, native to two-dimensional space, involution in general, or else only involution with a power of 3, native to three-dimensional space.

The current standard notational system for numbers is a multi-place place-value chong-2 numeral system -- 'chong-2', because it does not go further than involution. One might wonder whether a chong-3 system would not represent numbers more adequately in a context where chong-3 operations play a role, or a higher chong system in a context in which even higher operations play a role. Apart from this fundamental issue, there is the more ordinary question of what base number deserves primacy where and when (over base 10 in particular), even if we take the chong-2 numeral system for granted. This issue of the level of reiteration and this question of the primacy of the numeral base used or to be used in different contexts also touches on the relevancy (relevance or irrelevance) of the kind of mean, such as the arithmetic or geometric mean, used or to be used in different contexts. While this is of mathematical interest in itself, it is certainly important in the formulation, interpretation and corroboration of the as discussed in section 2.6.3 of the of the .

At the moment the most innovative, if not 'revolutionary', document in the Math folder at this MVVM-site is no doubt The Chong Operators, in which the whole system of operational levels of reiterations is explained. (If it is not the thriller its title may promise it to be, it is, in a sense, of greater magnitude than any such book so far.) The other mathematical documents publicly available in the same folder deal primarily with the representation of numbers in the base-2 or 'binary' and the base-4 or 'quaternary' system as distinct from their representation in the base-10 or 'denary' system. Altogether the mathematical notes and papers are in reverse chronological order:

The final digits of the 'negative' powers of two
How the pattern in the decimal notations of the negative-exponential powers of two turns out to be impressive and instructive
The powers of two in number theory
You thought they were 2 (parents or children), 4 (grandparents or -children), 8 (great-grandparents or -children), 16 (great-great-grandparents or -children), ... But are they?
The chong operators — a universalization in arithmetic
What would it look like to be fed with a spoon of unlimited dimensions?
A substantive terminology for sequences and cycles
You might as well call a table "a chair" and a chair "a table", but a substantive terminology of things structures, if not reality itself, then at least your thought about things.
The quaternary rectangular representation of product integers
Why not turn numbers into pictures? Especially the pictures of products may stimulate the arithmetic mind.
Final digit information in different numeral systems
This note contains no radical proposals or something but is a preparation for what is to come (or else not to come) — the initial letter, as it were.
The irrelevance of denary arithmetic and the riddle of pi's repetend
About the preposterous undertaking of forever adding more digits to the decimal representation of pi, and the even more preposterous undertaking of trying to remember those more and more useless digits.

Those who have always thought that the verbal expressions ('names') for numbers must take the base-10 system as their frame of reference, almost by definition, should read the poem At Eighty I Should Give Up. That poem reveals the verbal expressions possible for the denary ('decimal') number 80 in a macrobinary system. Binary?! Yes, altogether the elementary binary expressions for numbers (however large) need not be more cumbersome, not even or hardly longer, than in natural languages which use the base-10 system. Well, the phrase for 80 happens to be at least one syllable longer, but that does not prove anything, especially since This Language is not the sole natural language in the world. (The binary word for the denary number 16 can be one syllable shorter than speakers of the present language are used to.)