## 2 Materials and Methods
\begin{cases}m,&\text{if $n=1$},\\ \big{(}\langle m\rangle_{1},\langle m\rangle_{2}\big{)}&\text{if $n=2$},\\ \big{(}\langle m\rangle_{1},\langle\langle m\rangle_{2}\rangle_{1},\langle \langle m\rangle_{2}\rangle_{2}\big{)},&\text{if $n=3$},\\ \big{(}\langle m\rangle_{1},\langle\langle m\rangle_{2}\rangle_{1},\ldots, \langle\langle m\rangle_{2}^{n-2}\rangle_{1},\langle\langle m\rangle_{2}^{n- 2}\rangle_{2}\big{)},&\text{if $n\geq 4$},\end{cases} \] where \(\langle m\rangle_{2}^{n}\) denotes the \(n\)-fold successive application of \(\langle\cdot\rangle_{2}\) to \(m\);
* For a _non_-atomic machine-readable language \(\mathfrak{A}=\{(a,\mathfrak{B},[\cdot]_{\mathfrak{A}}):a\in\mathcal{A}\}\) and a general machine-readable language \(\mathfrak{B}\) with canonical numbering \(\varphi_{\mathfrak{B}}:\mathbb{N}\supseteq\rightarrow\mathfrak{B}\), we define \[\varphi_{\mathfrak{A}}(m):=\big{(}a,\mathfrak{B},[\cdot]_{\mathfrak{A}}\big{)} \quad:\Leftrightarrow\quad[\varphi_{\mathfrak{B}}(m)]_{\mathfrak{A}}=a,\] with \(\text{dom}(\varphi_{\mathfrak{A}}):=\{m\in\text{dom}(\varphi_{\mathfrak{B}}): \varphi_{\mathfrak{B}}(m)\in\text{dom}([\cdot]_{\mathfrak{A}})\}\) accordingly. Among other things, and together with the relevant pairing function \((\langle\cdot\rangle_{1},\langle\cdot\rangle_{2})\), the canonical numbering facilitates the definition of machine-readable languages for tuplesof the form \((a_{1},a_{2})\in\mathcal{A}_{1}\times\mathcal{A}_{2}\), provided we have already defined machine-readable languages for the abstract sets \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\). Referring to this article's fundamental problem statement, if we want to represent abstract information on a digital machine in a sufficient way, we need to specify a machine-readable language for the relevant abstract set, and then investigate the language's structural properties. Albeit rarely explicit, this principle is used throughout the literature of computable analysis. In the context of Banach spaces, it is strongly related to the definitions of computability structures (Chapter 2.1, p. 80 ff in [15]). Further, any canonical numbering \(\varphi_{\mathfrak{R}}\) as defined above is essentially a numbering in the sense of a concept that is fundamental in computability theory (Chapter 1.4, p. 12 in [16]). As indicated before, formal approaches of this form are necessary for a mathematically rigorous theory of computable analysis. Yet, they are somewhat cumbersome in use. In Definition 3, for example, we have implicitly introduced a machine-readable language for the set of computable real numbers by defining the relation \([(n)_{n\in\mathbb{N}},\xi]_{\mathfrak{R}}=x\). This convention is an abuse of notation regarding the just-established formalization of machine-readable languages. Strictly speaking, we first have to define machine-readable languages for the set of triples \((g,h_{1},h_{2}):g,h_{1},h_{2}\in\mathcal{U}(1)\). Then, we have to define a machine-readable language for the set of computable sequences of rational numbers. Finally, we have to define a machine-readable language for the set of pairs \(((r_{n})_{n\in\mathbb{N}},\xi)\) as above, based on which we can define the machine-readable language for the set of computable real numbers in the sense of Definition 3. Intuitively, on the other hand, it is evident from Definition 3 that we describe a computable real number by a suitable pair \(((r_{n})_{n\in\mathbb{N}},\xi)\), and we can implement computable mappings on computable real numbers by applying \(\mu\)-recursive functions to the \"programs\" (with respect to the universal \(\mu\)-recursive function \(U\)) of the underlying quadruple \((g,h_{1},h_{2},\xi)\). Keeping the formal definition in mind, we will, with some abuse of nomenclature and notation, employ the following conventions for mathematical ease:  
* A standard description of computable real number \(x\) consists of a pair \(((r_{n})_{n\in\mathbb{N}},\xi)\) that characterizes \(x\) in the sense of Definition 3, and we write \(x=[(r_{n})_{n\in\mathbb{N}},\xi]_{\mathfrak{R}}\).