primitive recursive number
Definition. A non-negative real number is said to be primitive recursive if there is a primitive recursive function such that
A real number is primitive recursive if is, and a complex number is primitive recursive if both and are.
we can define so that
Here, we assume that is non-negative.
In addition, we can show that is primitive recursive for any non-negative integer .
Suppose . Write in its decimal representation
Then . Multiply by to get its decimal representation
Then , so that By induction, we see that
Define by . Then is primitive recursive. Next,
Remark. It can be shown that is primitive recursive. A proof of this can be found in the link below.
- 1 S. G. Simpson, http://www.math.psu.edu/simpson/courses/math558/fom.pdfFoundations of Mathematics. (2009).
|Title||primitive recursive number|
|Date of creation||2013-03-22 19:06:06|
|Last modified on||2013-03-22 19:06:06|
|Last modified by||CWoo (3771)|