Letter Notation Part III

Arrays with more than two variable, and PEdit
First, a short reminder of the supporting array notation given in part II:
 * (i) (0,1)|x = 10^x
 * (ii) (m,n+1)|x = (m,n)|(m,n)|...|(m,n)|10^frac(x) with int(x) (m,n)'s
 * (iii) (m,x)|10 = (m,int(x+1))|2*5^frac(x)
 * (iv) For x<2: (m+1,0)|x = (m,3)|x
 * (v) For x≥2: (m+1,0)|x = (m,x)|10

(where m,n ≥ 0 are integers and x≥0 is real)

This can be easily extended to a multivariable array notation, like so: The first 5 rules are a simple and direct extention of the 2-variable arrays notation, and Rule vii simply states that leading zeros can be ommitted.
 * (i) For x≤1: (anything)|x = 10^x
 * (ii) (a,b,c,...,n+1)|x = (a,b,c,...,n)|(a,b,c,...,n)|...|(a,b,c,...,n)|10^frac(x) with in(x) (a,b,c,...,n)'s
 * (iii) (a,b,c,...,x)|10 = (a,b,c,...,int(x+1))|2*5^frac(x)
 * (iv) For 1)|10^(x-1)
 * (v) For x≥2: (a,b,...,m+1, )|x = (a,b,...,m,x,)|10
 * (vi) (a,b,...,x, )|10 = (a,b,...,int(x),frac(x)*10,)|10
 * (vii) (0,...,0,a,b,...,m)|x = (a,b,...,m)|x

Rule vi is an interesting one, though. It basically tells us that if we have an array which ends with (...,x,0,0,...,0) then the digits of the fractional part of x are to be distributed among the zeros. For example:

(22,7,3.14159,0,0,0,0,0)|10 = (22,7,3,1,4,1,5,9)|10.

Now, all that is left to do is to define P:
 * For x<2: Px = (1,0,1)|x
 * For x≥2: Px =  (10^frac(x),0,...,0)|10 with int(x) zeros.

And that's it!

P-Canonical FormsEdit
Just like the previous letters, any number can be written as Px (for some real number x). Here, it is actually  the binary form of xPn = P(n+log x) which has the most intuitive meaning (for n≥2):

In terms of the array notation, n+1 tells us how many numbers are in the array and the digits of x tell us the what those numbers are. For example: 1.2358P4 = (1,2,3,5,8)|10

And in terms of FGH ordinals, n gives us the maximum power of ω and the digits of x give us the coefficents of the various powers of ω: 1.2358P4 ~ fω⁴+ω³2+ω²3+ω5+8(10)

(actually, these neat relations are also true for n=1 and x≥2, so 2.5P1 = (2,5)|10)

Of-course, for numbers between P2 and P10, the P-Canonical Form is also the Universal Canonical Form.

Examples of P-Canonical Forms Edit

 * 1 = 1P0 = P0
 * 10 = 1P1 = P1
 * 100 ≈ 1.01995P1 ≈ P1.00858
 * 1010 = F2 ≈ 1.046P1 ≈ P1.01995
 * Tritri = {3,3,3} ≈ J3.0897 ≈ 1.1732P1 ≈ P1.0694
 * Graham's Number ≈ K64.613 ≈ L2.04123≈ 1.116P1 ≈ P1.1206
 * {10,3,2,2} = L3 ≈ M1.4771 ≈ N1.477 = 1.169P1 ≈ P1.205
 * Conway's Tetratri = 3→3→3→3 ≈ L3.026 ≈ M1.4787 ≈ N1.170 ≈ L3.011 ≈ 1.477P1 ≈ P1.206
 * Conway's Tetratet = 4→4→4→4 ≈ M2.432 ≈ N1.386 = 1.386P1 ≈ P1.224
 * Grand Tridecal = {10,10,10,2} = N2 = 2P1 ≈ P1.30103
 * Biggol = {10,10,100,2} = M100 ≈ (2,1)|2.0037 ≈ N2.00011 = 2.00011P1 ≈ P1.30105
 * N2.1 = (2,1)|10 = 2.1P1 ≈ P1.322
 * Supertet = {4,4,4,4} ≈ (3,4)|3.55 ≈ N3.3356 ≈ P1.523
 * N5.7 = (5,7)|10 = 5.7P1 ≈ P1.756
 * General = {10,10,10,10} = N10 = P2
 * Troogol = {10,10,10,100} = N100 ≈ NN1.020 ≈ (1,0,1)|2.0016 ≈ 1.00070P2  ≈ P2.00030
 * Fish number 1 ≈ (1,0,1)|63 ≈ 1.01P2 ≈ P2.004
 * Triggol = {10,10,10,100,2} ≈ (2,0,0)|100 ≈ 2P2 ≈ P2.301
 * Pentatri = {3,3,3,3,3} ≈ (2,2,3)|2.38 ≈ 2.221P2 ≈ P2.346
 * 16th Goodstein number ≈ (2,2,3)|3.55 ≈ 2.226P2 ≈ P2.347
 * Superpent = {5,5,5,5,5} ≈(4,4,5)|5.76 ≈ 4.447P2 ≈ P2.648
 * Pentadecal = {10,10,10,10,10} = (1,0,0,0)|10 = 1P3 = P3
 * 17th Goodstein number ≈ (3,3,3,4)|4.67 ≈ 3.334P3 ≈ P3.523
 * Superhex = {6,6,6,6,6,6} = (5,5,5,6)|5.76 ≈5.556P3 = P3.745
 * Hexadecal (?) = {10,10,10,10,10,10} = (1,0,0,0,0)|10 = 1P4 = P4
 * 18th Goodstein number ≈ (5,5,5,5,5,6)|6.83 ≈ 5.556P5 ≈ P5.745
 * 19th Goodstein number ≈ (7,7,7,7,7,7,7,8)|8.95 ≈ 7.778P5 ≈ P7.891
 * Iteral = {10,10,10,10,10,10,10,10,10,10} = (1,0,0,0,0,0,0,0,0)|10 = 1P8 = P8
 * {10,12 (1) 2} = (1,0,0,0,0,0,0,0,0,0,0)|10 = 1P10 = P10 (=Q2)

Bonus: A Continuous Generalization of General Bowers Linear ArraysEdit
It turns out that for integer arguments, we get:
 * {10,a,b,c,.....,n} = (n-1,...,c-1,b)|a

And replacing the 'a' by any real number x gives us a continuous version of Linear Arrays of arbitrary length. Generalizing this further to any integer base b>2 is also possible, using the usual trick of replacing all 10's with b and all 5's with b/2.

At any rate, this "bonus benifit" does not extend beyond Linear Arrays, because Pn = {10,n+2 (1) 2} rather than {10,n (1) 2}.

The Binary Canonical FormsEdit
In Part I we've defined xAn as A(n+log x) for any 1<x<10 and A being either E or F or G or H.

Now we'll define Binary Forms for the other letters. Given a letter A and x<10:
 * (i) If A∈{E,F,G,H,K,L,P} and x≥1 then xAn = A(n+log x)
 * (ii) if A∈{J,M} and x≥2 then xAn = A(n+log5(x/2))
 * (iii) For the letter N: xNn=N(n+x/10)

And the Universal Binary Canonical Form of a given number x is the binary counterpart of its Universal Canonical Form.

Given these definitions, we get a nice intuitive interpertation for the binary forms:
 * xEn = x*10^n
 * xFn = a power tower of n tens with an x on top
 * mGn = 10↑↑10↑↑...10↑↑m (with n 10's)
 * mHn = 10↑↑↑10↑↑↑...10↑↑↑m (with n 10's)
 * mJn = 10↑↑...↑↑m with n arrows = {10,m,n}
 * mKn = JJ...JJm (with n J's)
 * mLn = KK...KKm (with n K's)
 * mMn = (1,n)|m
 * mNn = (n,m)|10
 * xPn = ()|10

What's Coming NextEdit
The system defined above is - I believe - quite intuitive. It's a nice self-contained ωω-level notation which lends itself to a simple interpertation. It also serves as a continuous extension of Bowers Linear Arrays.

So I find it very tempting to just keep it that way, and say that letter notation ends at P10. I will post one possible definition for Q (which is an ε₀-level function) later, but I doubt it will become part of the "official" notation.

I also plan to do two more things:

(1) Create a completely new (and much simpler) continuous notation with smoother interpolation rules. Of-course, to avoid confusion, I won't be using letters for that one.

(2) Show a variant of letter notation which allows us to: (a) order all numbers in a lexicographic order and (b) know the actual value of any given number without any calculation (for example, the number 10^10^(5.374*10^7415) will be encoded as F4-3-7415-5374)