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Besides systematization and universalization in arithmetic, 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 RADIX |
| one-dimensional |
chong +0
(addition) |
arithmetic |
no place-value notation |
| two-dimensional |
chong +1
(multiplication) |
geometric |
a radix in a micro-macro-binary supersystem |
| three-dimensional |
chong +2 ((power-3) exponentiation) |
? |
a radix of a different supersystem, still with chong-2 notation? |
| four-dimensional |
chong +3 (reiterative exponentiation)? |
? |
a chong-3 notation of numbers? |
| five- or more-dimensional |
chong +4 or higher? |
? |
a chong-4 or higher-level system? |
Of the elementary mathematical operations, addition is an operation on
the zeroth level of iteration, multiplication one on the first level,
exponentiation one on the second level, repeated or (re)iterative
exponentiation one on the third level.
(I may use reiterative as well as iterative on the second
and higher levels of iteration, where an iterative process is repeated
itself.)
In the 67th year after the Second World War i published a universal system
of ever higher operational levels of iteration in which i called the
arithmetic operators involved
"chong operators": 'chong +0' for addition,
'chong -0' for subtraction, 'chong +1' for multiplication, 'chong -1' for
division, 'chong +2' for exponentiation, and so on and so forth.
Should there be any relationship between the iterativity, that is, the
level of iteration, of mathematical operators and the dimensionality of
space, addition (and subtraction) may be considered 'native' to
one-dimensional space, whereas multiplication (and division), and
operations of a higher level of iteration, are not.
Similarly, multiplication is, then, native to two-dimensional space,
exponentiation in general, or else only exponentiation 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 system — 'chong-2', because it does not go
further than exponentiation.
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 numeral radix, or type of radix, deserves primacy where and when
(over radix or 'base' 10 in particular), even if we take the chong-2
numeral notation for granted.
This issue of the level of iteration and this question of the adequacy of
the numeral radix used or to be used for 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 iteration 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.)
In the other mathematical documents publicly available in the same folder
the representation of numbers in the radix-2, -4 and -16 systems, as
distinct from their representation in the radix-10 system, continues to
draw special attention, so long as ten dominates all in practice.
The mathematical notes and papers offered here are in reverse
chronological order:
- Googol &
googolplex(plex(...))
Googleplex arithmetic may impress the little ones enormously, when
you look at it from a higher operational level it will strike you
as the product of a disoriented branch of mathematics
- 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 radix-10 or '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 (decimal-)denary
number 80 in a (micro)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 radix-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 macro-binary morpheme for the denary number 16 is even one syllable
shorter than speakers of the present language are used to.)
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