The question is just what is on the title, but I'll describe the context for completion:
Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function applied to an argument N times:
0 = (λ f x → x) 1 = (λ f x → f x) 2 = (λ f x → f (f x))) 3 = (λ f x → f (f (f x))))
That representation gives us a very simple formulas to addition, multiplication and exponentiation:
add = λ n f x → f (n f x) mul = λ m n f → m (n f) exp = λ m n → n m
The representation of their inverses is not as straightforward, though. Division implementations either require recursion through the Y-combinator, or are huge formulas. This makes me suspect that church numbers are not an ideal representation of fractionals, so I've been looking for a better alternative. I guess the problem is fundamental: there is something about the nature of numbers I am missing. This is why I ask the following question: is there any kind of system/encoding in which fractional numbers can be represented elegantly? One for which algorithms such as division, logarithm and sine are as simple as the
Put short, what is the most elegant representation of fractional numbers you know?