My legacy -illion system is a notation that catalogues higher tiers of -illions.
First, we need a definition of the notation. We define the primitive system first. The primitive system is similar to many “linear” notations in that it takes any amount of integers. It is written as follows:
x[a,b,c,d,...]
Here the numbers (b,c,d,e…) comprise the main array, and are called entries. "x" is called the multiplier. First entry must be nonnegative integers and all subsequent entries and must be integers greater than 0. "x" can be any real numbers since it is the multiplier of all -illions.
x[0] = x*1000
1[a] = 10^(3a+3)
x[a] = x*10^(3a+3)
x[a,b] = x[1[a,b-1]/1000]
x[#,1] = x[#]
x[#,b,c] = x[#,1[b,c-1]]
If the x parameter is omitted, [#] = 1[#]
If there are two or more distinct rules to apply to a single expression, the upper rule will be applied if it can be evaluated, otherwise the lower rule. Here # denotes a portion of the array. It can also be empty.
Examples:
1[2,1]
= 1[10^(3*2),1]
= 1[10^(3*2)]
= 1[10^6]
= 1[1000000]
= 10^(3*1,000,000+3)
= 10^3,000,003 (micrillion)
1[1,3]
= 1[10^3,2]
= 1[1000,2]
= 1[10^(3*1000),1]
= 1[10^3000,1]
= 10^(3*10^3,000+3) (killillion)
1[1,2,2]
= 1[1,1[1,2,1]]
= 1[1,1[1,2]]
= 1[1,1[1000]]
= 1[1,10^(3*1000+3)]
= 1[1,10^3003] = the first tier millillion -illion number
Expression: Ra#b#c#..., where a, b and c are natural numbers. @ stands for the latter of the expression.
Basic rules:
If the expression is merely expressed by Ra, Ra = 10^(3a+3).
If the last arguments are 1, R@#1 = R@.
If there are arguments after 1, R@#1#a = R@.
Else, Ra#b = R(10^3a)#b-1.
Advanced rules:
R@#a#b = R@#(R@#(10^3a)#b-1)
R4#3
= R(10^(4*3))#2
= R(10^12)#2
= R1,000,000,000,000#2
= R(10^3,000,000,000,000)
= 10^(3*10^3,000,000,000,000) a.k.a. terillion
R1#1#2
= R1#(R1#1,000#1)
= R1#(R1#1,000)
= R1#(R1,000#999)
R3#2#4
= R3#(R3#2#3)
= R3#(R3#(R3#2#2))
= R3#(R3#(R3#(R3#2#1)))
= R3#(R3#(R3#(R3#2)))
= R3#(R3#(R3#(R1,000,000,000#1)))
= R3#(R3#(R3#(R1,000,000,000)))
= R3#(R3#(R3#(10^(3*1,000,000,000+3))))
= R3#(R3#(R3#(10^3,000,000,003)))
We can have some new -illion system notation (R#) extensions using Extensible-E system structures, using the same rules as in Extensible-E system mentioned by Sbiis Saibian.
The one-entry arrays are equal to the basic -illions, such as 1[1] = 10^(3+3) = 10^6 = 1,000,000 (million), 1[2] = 10^9 = 1,000,000,000 (billion), 1[3] = 10^12 = 1,000,000,000,000 (trillion), and so on. FGH level f2.
For two-entry arrays, it is functionally equal to 1[a,b] = a-th of tier b -illion.
The original definition was ill-defined. I remade the system with the second definition above.
Millillion = 1[1000] = 1[1,2]
Micrillion = 1[1000000] = 1[2,2]
Nanillion = 1[10^9] = 1[3,2]
Picillion = 1[10^12] = 1[4,2]
Femtillion = 1[10^15] = 1[5,2]
Attillion = 1[10^18] = 1[6,2]
Zeptillion = 1[10^21] = 1[7,2]
Yoctillion = 1[10^24] = 1[8,2]
Xonillion = 1[9,2]
Vecillion = 1[10,2]
...
Multillion = 1[14,4]