My Letter Notation

Motivation
The idea here is to extend the concept of ordinary scientific notation to much larger numbers. The properties of scientific notations we wish to preserve here are:

(1) Any number has a unique standard representation in the system.

(2) Given the standard representation of two numbers, one can immediately tell which one is larger without any calculations.

The format of the proposed notation
Our final notation will look like this:

where can be one of the following: E,F,G,H,J,K,L,M,N,P

and can be any positive real number (nonintegers included).

The First Levels: A Continuous version of Knuth Arrows
We'll define:

Ex=10^x

Fx = EEEE...EEE(10^frac(x)) with int(x) E's

Gx = FFFF...FFF(10^frac(x)) with int(x) F's

Hx = GGGG...GGG(10^frac(x)) with int(x) G's

Note that according to these definitions we have:

(1) For x≤1: Ex = Fx = Gx = Hx = 10^x

(2) For any integer n:

En = 10↑n

Fn = 10↑↑n

Gn = 10↑↑↑n

Hn = 10↑↑↑↑n

So the above definitions are - indeed - an extention of Knuth arrows to nonintegers.

Letter-Canonical Forms
If x is a number greater than 1 and A is a one of the letters E,F,G,H then there is a unique number y such that:

x = Ay.

And we call "Ay" the A-Canonical Form of the number x.

For example, the E-Canonical form of 1000 is E3:
 * E3 = 10^3 = 1000.

And the F-Canonical form of 1000 is about F1.47712:
 * F1.47712 = E(10^0.47712) ≈ E3 = 10^3 = 1000.

Binary-Letter-Canonical Forms
To recreate ordinary scientific notation, we'll define a binary version of the letter functions like this:

Let A be one of the letters E,F,G,H. Let n be a nonnegative integer and x be a real number between 1 and 10. Then:

xAn = A(n+log(x))

For Example:
 * 7E3 = E(3+log(7)) = 10^(3+log(7)) = 7*10^3 = 7000
 * 7F3 = F(3+log(7)) = EEE(10^log(7)) = EEE7 = 10^10^10^7

So xEn is nothing more than ordinary scientific notation.

And xFn is a power tower of n 10's topped by an x (which is equivalent to Ex#n in Saibian's Hyper-E notation)

And again, given any specific letter (E,F,G OR H), ANY number greater than 1 has a unique representation as xAn (with 1≤x<n). So we can call this the Binary-A-Canonical Form of x.

Universal-Canonical Forms (up to H10)
A quick a look at the definitions of E,F,G and H tells us that:
 * F2=E10
 * G2=F10
 * H2=G10

Realizing this, we can define a "Universal-Canonical Form" of any number x<H10 like this:

(1) if x<100 (that's E2) then we write down the E-Canonical Form of x.

(2) Otherwise, we write x as Ay for some letter A and 2≤y<10.

It is easy to see that since 2 = 10, the letter A in (2) is uniquely defined.

Examples of Universal-Canonical Forms (up to H10)

 * 1 = 1E0 = E0
 * 2 = 2E0 ≈ E0.3010
 * 3= 3E0 ≈ E0.4771
 * 10 = 1E1 = E1
 * 11 = 1.1E1 ≈ E1.0414
 * 20 = 2E1 ≈ E1.3010
 * 100 = 1E2 = E2
 * 1000 = 1E3 = E3
 * 2016 = 2.016E3 ≈ E3.3045
 * 10000 = 1E4 = E4
 * 1 million = 1E6 = E6
 * 1 billion = 1E9 = E9
 * 10 billion = 1F2 = F2
 * 1 trillion ≈ 1.079F2 ≈ F2.033
 * 1 quadrillion ≈ 1.176F2 ≈ F2.070
 * Avogadro's Number ≈ 6*10^23 ≈ 1.376F2 ≈ F2.139
 * googol = 10^100 = 2F2 ≈ F2.301
 * googolchime = 10^1000 = 3F2 ≈ F2.477
 * googolgong = 10^100000 = 5F2 ≈ F2.699
 * trialogue = 10^10^10 = 1F3 = F3
 * googolplex = 10^10^100 = 2F3 ≈ F3.301
 * {4,4,3} = 4^4^4^4 ≈ 10^10^153,9 ≈ 2.187F3 ≈ F3.340
 * tetralogue = 10↑↑4 = 1F4 = F4
 * Skew's number ≈ 10^10^10^34 ≈ 1.531F4 ≈ F4.1851
 * Gag(4) = 2↑↑7-3 ≈ 10^10^10^19728 ≈4.295F5 ≈ F5.633
 * Decker = 10↑↑10 = 1G2 = G2 (and the F-Canonical form is F10)
 * Giggol = 10↑↑100 ≈ 1.301G2 ≈ G2.114 (and the F-Canonical form is F100)
 * Mega ≈ E619#256 ≈ 1.382G2 ≈ G2.141 (and the F-Canonical form is ≈F257.446)
 * 10↑↑(10↑10) = 2G2 ≈ G2.301
 * Tritri = {3,3,4} ≈ 2.045G2 ≈ G2.311
 * Googolstack = 10↑↑googol ≈ 2.301G2 ≈ G2.361
 * Triataxis = 10↑↑10↑↑10 = 1G3 ≈ G3
 * {4,4,4} ≈ 3.340G3  ≈ G3.524
 * 10↑↑↑9 = 1G9 ≈ G9
 * 10↑↑↑10 = 1H2 ≈H2
 * 10↑↑↑11 = 1.00758H2 ≈ H2.00328 (the G-Canonical form is of-course G11)
 * Megiston ≈ 1.00764H2 ≈ H2.00330
 * 10↑↑↑12 = 1.01414H2 ≈ H2.00610 (can also be written as G12)
 * 2 in a hexagon ≈ 10↑↑↑Mega ≈ 2.141H2 ≈H2.331
 * Grahal = {3,3,5} ≈ 2.311H2 ≈ H2.364
 * 10↑↑↑↑3 = 1H3 = H3
 * Tritet = {4,4,5} ≈ 3.524H3 ≈ H3.547
 * 10↑↑↑↑4 = 1H4 = H4
 * {5,5,5} ≈ 4.655H4 ≈ H4.668
 * 10↑↑↑↑5 = 1H5 = H5
 * {6,6,5} ≈ 5.753H5 ≈ H5.760
 * 10↑↑↑↑6 = 1H6 = H6
 * {7,7,5} ≈ 6.830H6 ≈ H6.834
 * 10↑↑↑↑7 = 1H7 = H7
 * {8,8,5} ≈ 7.895H7 ≈ H7.897
 * 10↑↑↑↑8 = 1H8 = H8
 * {9,9,5} ≈ 8.951H8 ≈ H8.952
 * 10↑↑↑↑9 = 1H9 = H9
 * 10↑↑↑↑10 = H10 (which is not a universal canonical form because 10 isn't less than 10. We'll later see that the universal canonical form of 10↑↑↑↑10 is J4)

Comparing Canonical Forms
Let A and B be one of the letters E,F,G,H and x,y positive real numbers.

Then:

(1) Ax > Ay if and only if x>y

(2) If Ax and By are both Universal Canonical Forms, then:
 * (i) If A and B are the same letter then Ax>By if and only if x>y
 * (ii) if A and B are different letters, Ax>By of and only if A comes after B in the alphabet.

Note that rule 2ii only holds if both numbers are written in Universal Canonical Form. For example E100 (a googol) is bigger than F2 (ten billion) even though F comes after E. This is okay, because E100 is not canonical (100 being greater than 10).

Bonus: A Continuous Generalization on of Bowers Linear Arrays
This wasn't my original intention, but it so happens that my letter notation (all the way up to P) is completely equivalent to Bowers Arrays for integer arguments. So one could use the above definitions to write any number as a Bowers array. So far, we have:

{10,x,2} = Ex

{10,x,3} = Fx

{10,x,4} = Gx

{10,x,5} = Hx

Note that the noninteger number can only be in the second place of the array, but that's not really a limitation because one can write any number in the form {10,x,n}. Also, it isn't difficult to create a "letter notation" in bases other than 10 (just replace all the 10's in the definition with whatever base you wish). So we can actually write any number of the form:

{b,x,n} where b>2 is an integer, x is real and n∈{1,2,3,4}

For example:

{3,2.5,3} = F2.5 (in base 3) = EE(3^0.5) in base 3 = 3^3^3^0.5 ≈ 1581.6 which is indeed between {3,3,2}=27 and {3,3,3}=7625597484987.

And that's all for now. Later I'll post the definitions of the letters J-P which will give us a "Universal Canonical Form" for any number under P10 = {10,12 (1) 2}.

Oh, and your feedback is most welcome :-)