Editing Computable number (section)

===Properties as a field===
The arithmetical operations on computable numbers are themselves computable in the sense that whenever real numbers ''a'' and ''b'' are computable then the following real numbers are also computable: ''a'' + ''b'', ''a'' - ''b'', ''ab'', and ''a''/''b'' if ''b'' is nonzero.
These operations are actually ''uniformly computable''; for example, there is a Turing machine which on input (''A'',''B'',<math>\epsilon</math>) produces output ''r'', where ''A'' is the description of a Turing machine approximating ''a'', ''B'' is the description of a Turing machine approximating ''b'', and ''r'' is an <math>\epsilon</math> approximation of ''a'' + ''b''.

The fact that computable real numbers form a [[field (mathematics)|field]] was first proved by [[Henry Gordon Rice]] in 1954.{{sfnp|Rice|1954}}

Computable reals however do not form a [[computable field]], because the definition of a computable field requires effective equality.