We define the following operators on the set of digits (any base is fine, we use 10 hereafter):
- Left t:
ty = -ywhereyis some number. This operator is not applied unless it is the leftmost character or preceeded by ana. - Binary a:
xay = x^(-y) - Right t:
xt = 10 * x - Unary a:
ay = -(y). These are in order of precedence: later the more predecent.
We pronounce a anti and t as tee (like ty in twenty).
The binary a is right associative.
A right t followed by a digit ytx is interpreted as y * 10 + x.
Consecutive digits imply summation of the two component numbers.
t is usually the right t. It is the left t it is either the leftmost
character or has an a to the right.
We belive the above is enough to define:
x aa y = x ^ y(we treat the rightaas unary and the left one as binary).x ta y = (10 x) ^ -yx taa y = (10 x) ^ y.
11 = 2 since the adjacent 1 digits add.
95a15a15a15a1 = 9.8 since the 4 5a1s add to give 4/5 = 8/10.
1tt2t1aa2a1= 11 (we mean eleven) since the rightmost a gives 1/2,
then the aa gives 1tt2t1 to the 1/2 (so square root of 1tt2t1),
and 1tt2t1 is 121 since all the ts are right ts.
2aa2a1 = sqrt(2) since 2a1 is a half so we have the square root of 2.
So now, we note that algebraic real numbers are probably all expressible in this.
By we've yet to use the left t...
t1a2a1 = i (yes, the imaginary unit), since the left t negates the 1 and
we take its square root.
So far, we probably get Q[i], the complex numbers with rational coefficients. But now:
2at1a2a1 = 2^i = cos(ln(2)) + i sin(ln(2)) which we're not so sure about...
Digit is a set of terminals that can vary by base (as long as a and t aren't rendered ambiguous).
Sum := NumberLT | NumberLT Sum
NumberLT := t NumberBA | NumberBA
NumberBA := Sum2 a NumberLT | Sum2 | a NumberBA
Sum2 := NumberRT | NumberRT NumberRT
NumberRT := NumberRT t | Digit