p 99
Computability of irrational numbers, etc.
For example, if we wish to consider the computability of an irrational number, we must seriously take the computational errors (the round-off errors; no other error is considered in the present context) into account. Roughly speaking, that a mathematical object is computable implies that there is an algorithm (a definite finite means) to give the object within prescribed errors.
M. B. Pour-El and J. I. Richards, Computability in Analysis and Physics (Springer, 1989)
is an excellent reference book.
There is a theory of computation that handles a real number as a single entity [an accessible introduction may be: J. F. Traub, ``A continuous model of computation' Physics Today May, 1999, p39]. This type of more generalized theory has been developed by Blum et al. [L. Blum, M. Shub and S. Smale, ``On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines,'' Bull. Amer. Math. Soc. 21 , 1-46 (1989); L. Blum, F, Cucker, M. Shub and S. Smale, Complexity and Real Computation (Springer, 1997).].
Is computability relevant to physics?
There is a comment (by the present author) [INtroduction to the book by Pouor-El and Richards quoted above] in Butsuri 45(6) 421 (1990)
but it is in Japanese [Butsuri is a counterpart of Physics Today of Physcial Society of Japan]. A translation will be prepared later.