# How does one define transcendental numbers (such as Pi) in theory of general recursive functions

On a turing machine and in the lambda calculus one can define transcendental numbers such as Pi, the golden ratio, etc. These are computatible functions with 0-arity that never terminate. In the theory of general recursive functions (which also defines the same class of functions as turing machines and lambda calculus) the output is always a natural number N, or undefined. How does one define transcendental numbers in the theory of general recursive functions (mathematical functions defined by Gödel)

If you are only interested in transcendental (or at least only irrational) numbers, you can safely identify the number $$x$$ with the function $$d_x : \mathbb{N} \to \mathbb{N}$$ that on input $$n$$ outputs the $$n$$-th digit of the decimal expansion of $$x$$.
If you want all real numbers, that does not lead to a satisfactory theory of computation. A suitable way to handle reals there is to let a function $$f : \mathbb{N} \to \mathbb{N}$$ code a real number $$x$$ if $$\forall n \in \mathbb{N} \ |\nu(f(n)) - x| < 2^{-n}$$ where $$\nu : \mathbb{N} \to \mathbb{Q}$$ is some standard bijection. This means that there are many functions coding the same real, but that is unavoidable.