Deconstructing Base-10

Not my own idea, came in large part from what I read in גנת אגוז of R' Gikatilia, but sunk in today with the right words so I'm writing it down. A little bit different way of looking at it than common perception.

We're use to the concept of Base-10 and it's unit markers: Ones Place, Tens Place, Hundreds Place, etc.... In truth though, in terms of actual real existing units, all you have is 9 digits, so that really what we're computing is Units of 9. When we pass the last final real digit of 9 we have no real digit further so we instead put '10' as if to say we've gone past one full set of 9's and yielded one (1) additional unit of the emergent כולל coming from that full set of '9'. In other words, '10' is 9 units of 9-based singular units (פרט) and '1' unit of כלל.

Another example: 47. On the one hand it's 4 sets of Tens and 7 units of Ones. On the other though, going by pure real digits i.e. 9 (because 10 is only a fantasy), it is 4 sets of 9's , 4 emergent units of כלל thereof, and 7 units of singular-unit Ones. 36+4+7= 47

My chavrusa and friend Shimshon Tabrikian formulated the equation for me:

Take any integer of abc.

In Base-10 it's essentially: a(100) +b(10) +c(1) = abc

What I said essentially subdivides the equation into the following:

[ a(99) +b(9) + c(1) ] + [a +b +c] = abc

The first portion being the calculation of the real units of פרט and the second portion being the emergent כלל.


With this perspective of looking at it it can become easily understood (after working through it somewhat) how an old mathematical "trick" of Digital Roots and '9' works. Essentially, take any integer in the world, calculate the Mispar Katan (Digital Root) of the integer, subtract that Mispar Katan from the original integer.... the remaining number will always have a Mispar Katan of '9' .
Example: '564' --> mispar katan is 15--> subtract 15 from 564 yields 549--> Mispar Katan of 549 is 18 which then becomes 9.

(Solution why it works: the initial calculating of Mispar Katan essentially parses out all of the "non-9" units i.e. the units of כלל and the units of Ones. Of course then when you subtract those units from the original number you're left with a number that is divisible by 9)

Now the Gemaras regarding תשיעי, עשירי, ואחת עשר by מעשר בהמה takes on a whole new depth of meaning. עיין שם Also, consider redemption of הקדש קרקע in פר' בחוקתי --  the 9 Yisraelim and one Kohen learnt out from the 10 mentions of Kohen.

Also see the mathematical concepts of Mispar Katan a.k.a. Digital Roots and also Modular Arithmetic for this.

[ועוד א ב ג ד ה ו ז ח ט , אדם. י-- ייחוד דיליה מלכות, עשיראה דאדם [זוהר פנחס רכד