Editing Computable number (section)

==Use in place of the reals==

The computable numbers include the specific real numbers which appear in practice, including all real [[algebraic number]]s, as well as ''e'', ''π'', and many other [[transcendental number]]s. Though the computable reals exhaust those reals we can calculate or approximate, the assumption that all reals are computable leads to substantially different conclusions about the real numbers. The question naturally arises of whether it is possible to dispose of the full set of reals and use computable numbers for all of mathematics. This idea is appealing from a [[constructivism (mathematics)|constructivist]] point of view, and has been pursued by the Russian school of constructive mathematics.<ref>{{cite journal
 | last = Kushner | first = Boris A.
 | doi = 10.2307/27641983
 | issue = 6
 | journal = [[The American Mathematical Monthly]]
 | mr = 2231143
 | pages = 559–566
 | title = The constructive mathematics of A. A. Markov
 | volume = 113
 | year = 2006| jstor = 27641983
 }}</ref>

To actually develop analysis over computable numbers, some care must be taken. For example, if one uses the classical definition of a sequence, the set of computable numbers is not closed under the basic operation of taking the [[supremum]] of a [[bounded sequence]] (for example, consider a [[Specker sequence]], see the section above). This difficulty is addressed by considering only sequences which have a computable [[modulus of convergence]]. The resulting mathematical theory is called [[computable analysis]].