  If we start the Turing machine \(M\) on a tape containing \(n\) and it eventually halts with its tape containing \(m\), then we say that \(M\) on input \(n\), computes \(m: M(n) = m\). If, when we start \(M\) on input \(n\), it either never halts, or when it halts, its tape does not contain a natural number, e.g., because it has leading 0’s, or digits interspersed with blank symbols, then we say that \(M(n)\) is undefined, in symbols: \(M(n) =\nearrow\).  We can thus associate with each Turing machine, \(M\), a partial function, \(M: \mathbf{N} \rightarrow \mathbf{N} \cup \{\nearrow\}\). We say that the function \(M\) is total if for all \(n\in \mathbf{N}\), \(M(n) \in \mathbf{N}\), i.e., M\((n)\) is always defined.