TRANSDECIMAL CALCULATION OF NUMBER IDENTITY:

A
NOTE ON *INTEGRAL PRODUCT* & RELATED TERMS

The methodology for the mathematical reading of language
employed in places throughout this web is essentially distinguished from the *gematria*
of the ancients and the kabbalah by virtue of the fact that the numerical values of
letters in the ancient languages are for the first time in history treated in an *essentially*
mathematical way, viz., as* elements in a proportion* or *members of a series of
ratios*, and no substitutions are allowed. These and other related significant
technical differences distinguish this methodology from earlier and, in some respects,
similar practice, thereby reflecting the essential novelty of the new world consciousness
set forth in D.G. Leahy, *Novitas Mundi**:** Perception of the History of
Being *(Albany, 1994), *Foundation**:* *Matter the Body
Itself* (Albany, 1996), and *Faith and Philosophy: The
Historical Impact* (Aldershot, 2003).

The letter-numbers of any text are understood as elements
of a ratio or proportion. In any language text the word is taken as the natural unit of
division, so that the beginning of each word is the beginning of a ratio or proportion.
For example, the Hebrew for *cogito ergo sum* is אני חושׁב
ועל בן אני קים,
and is rendered 1 : 50 :: 10 : [1] (::) 8 : 6 :: 300 : 2 (::) 6 : 70 :: 30 : [1] (::) 20 :
50 (::) 1 : 50 :: 10 : [1] (::) 100 : 10 :: 40 : [1], where (::) links words and [1] is
the implicit final value of words with an odd number of letters. There are two primary
values calculated for any text, the rational product, which is the product of the string
of ratios which comprises the text, and the integral product, which is the square product
of the string of the numerators of the ratios (= the square product of the odd-numbered
letter-numbers of the text). In the example chosen the rational product = (2.304E4)/7 and
the integral product = 1.1943936E25. The linear product is the product of the entire
string of letter-numbers. The integral product also = the rational product × the linear
product (always for the letter-numbers of a text; for pure natural numbers only
when there is no zero among the digits composing the number). In the case cited the linear product is 9! × 1E16. The linear sum
in this case is 764. Cf., also, Leahy, *Foundation*,
p. 357, n. 13, et passim.

This
transdecimal method of mathematical computation is fruitfully
employed in the consideration of pure numbers themselves without any reference to
language. Cf. Leahy, *Foundation*, Sections III.6 and III.7, where it is shown, at
the places cited, that this fact is not a matter of mere coincidence but is a function of
a fundamental characteristic of the natural number system. The foundational quality and
prior mathematical essentiality of the methodology applied to the reading of language is
there demonstrated through the related discovery that [9!/(1+2+3+4+5+6+7+8+9)]^{2} = 1 × 784 × 82944, that is, that *the
square of the quotient of factorial 9 divided by the sum of its constituents -- the
primary natural numbers -- equals the product of the only three numbers in the natural
number system which are -- identically -- their integral products*.

For the deep
structure of integral product constitutional hypercubicity relations see, on
this web,
Measure Beyond Beyond Reach. Cf., also,
The Deep Epidermal Surface**,
**
Dali Crucifixion, and
Epidermal
Immunity Structure & the Absolute Dead Center Hypercube
Volume/Boundary Ratio.