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 :  (::) 8 : 6 :: 300 : 2 (::) 6 : 70 :: 30 :  (::) 20 : 50 (::) 1 : 50 :: 10 :  (::) 100 : 10 :: 40 : , where (::) links words and  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.