User blog:B1mb0w/Alpha Function

Alpha Function

The Alpha Function is a relatively simple function with only one parameter:

Alpha(r)   where r is any real number

Alpha(r) is derived from the Strong D Function with a variable number of input parameters. See User Blog:B1mb0w/Strong D Function for the definition of D.

I have included Visual Basic code at the end of this blog to define the Alpha function exactly.

What is the Alpha Function

My motivation to create this function was to develop a finely grained number notation system for really big numbers. Alpha(1) for example can be used to reference the number 2. Therefore 1 is the Alpha Index for the number 2.

Obviously Alpha needs to reference big numbers very quickly to be useful, therefore it uses the Strong D Function for this purpose.

The Alpha Function should also be strictly hierarchical and for every number a > b: a must reference larger numbers so that Alpha(a) > Alpha(b) in all cases.

The function is finely grained. It accepts a real number input and offers some finesse to locate large big numbers. Here is an example:

Alpha(4.14) >> Graham's number >> Alpha(4.139)

Examples of Alpha Notation numbers

Alpha(0)  =  D      = 0

Alpha(1)  =  D(0)     = 1

Alpha(1.5) = D(1)     = 2

Alpha(2)  =  D(1,0)   = 3

Alpha(2.2) = D(1,2)   = 5

Alpha(3.3) = D(3,3)   = 1697 Alpha(3.5) = D(4,0)   >> Googol

Alpha(3.7) = D(4,2)   >> Googolplex

Alpha(4.001) = D(1,0,1) >> g2 where g64 is Graham's number

Alpha(4.139) = D(1,9,7) >> Graham's number

Alpha(5)    = D(5,0,0) = D(4, D(4,5,5), D(4,5,5)) >> to be investigated

Alpha(6)    = D(9,0,0) = D(8, D(8,9,9), D(8,9,9)) >> to be investigated

Alpha(10)   = D(17,0,0,0) = D(16, D(16,17,17,17), D(16,17,17,17), D(16,17,17,17)) >> to be investigated

Alpha(15)   = D(57,0,0,0) >> to be investigated

To Be Investigated

I hope to calculate the Alpha index for Tree(3) next. It will be interesting to see where this lands.

Comments and Questions

I am new to Googology but look forward to comments and questions and I hope to learn a lot by contributing.

B1mb0w.

Alpha Function Definition

The following is VBA visual basic code and will run as a macro in Microsoft Excel. This function creates a string literal a Strong D Function equal to the Alpha function with any Real number input. The function does not attempt to evaluate the Strong D Function and the run time is therefore very fast.

Function Alpha(r As Double) As String

Dim d As Double, e As Double, i As Integer, j As Integer, k As Integer

d = Int(r) i = 0 While d >= 1 d = Int(d) / 2 If d >= 1 Then i = i + 1 Wend

e = 2 For j = 1 To i   e = 2 ^ e    Next j

Alpha = "D(" If r >= 0.5 Then   d = (r - 2 ^ i) * 2 ^ i    j = Int(d)    Alpha = Alpha & (j + 1)    d = d - j

For j = 1 To i       d = d * (e + 1) Alpha = Alpha & "," & Int(d) d = d - Int(d) Next j   End If

Alpha = Alpha & ")" End Function