I decided to define the notation based on Kiowa alphabet, which is ordered according to mostly phonetic principles. Note that I am using single capital letters to represent actual letters, and single small letters to represent variables.
The notation has the format using capital letters first. Digraphs use a capital letter followed by a small letter. The expressions in this notation are of this form:
(AAuEIOUBFPVDJTTh...)
Here "AAuEIOUBFPVDJTTh..." are a sequence of actual letters in the Latin alphabet. Wrapping braces are simply to distinguish the expressions from actual words. () is also a valid expression. An example of a valid expression is (AAuE).
Here is the set of the alphabet to represent the number:
(A, Au, E, I, O, U, B, F, P, V, D, J, T, Th, G, C, K, Q, Ch, X, S, Z, L, Y, W, H, M, N)
where:
A = 0
Au = 1
E = 2
I = 3
O = 4
U = 5
B = 6
F = 7
P = 8
V = 9
D = 10
J = 11
T = 12
Th = 13
G = 14
C = 15
K = 16
Q = 17
Ch = 18
X = 19
S = 20
Z = 21
L = 22
Y = 23
W = 24
H = 25
M = 26
N = 27
and the letters represent base-28 numeral system.
Every expression will output large numbers based on base-28 numeral system. To solve the expression, we need some rules as follows:
() = (A) = 0
For the expressions with two or more letters, do the same rules as base-28 numeral system.
(ME) = 26*28 + 2 = 730
(YOU) = 23*28^2 + 4*28 + 5 = 18,149
(GOOGOL) = 14*28^5 + 4*28^4 + 4*28^3 + 14*28^2 + 4*28 + 22 = 243,502,694