Computable number - Printable Version +- MKLab (https://mklab.gr) +-- Forum: [INDEX] (https://mklab.gr/forumdisplay.php?fid=1) +--- Forum: MATHEMATICS (https://mklab.gr/forumdisplay.php?fid=3) +---- Forum: ARTICLES (https://mklab.gr/forumdisplay.php?fid=13) +---- Thread: Computable number (/showthread.php?tid=1121)

Computable number - mklabgr - 07-15-2026 Computable number Summary A computable number is a real number that can be calculated to any desired level of precision by a finite, well-defined procedure, such as an algorithm or a Turing machine. The concept, introduced in the 1930s through the work of mathematicians such as Alan Turing and Alonzo Church, provides a rigorous foundation for understanding the limits of computation. A number is considered computable if there exists a method that can generate its decimal expansion (or any other representation) step by step, with the ability to approximate it as accurately as needed. Examples include familiar numbers such as integers, rational numbers, and many famous irrational numbers like π and √2. The study of computable numbers reveals that computation has fundamental boundaries. Although there are infinitely many computable numbers, they form only a small subset of all real numbers; most real numbers are uncomputable, meaning no algorithm can ever fully generate their digits. This distinction between computable and uncomputable numbers has deep implications for mathematics, computer science, and the philosophy of information, showing that some aspects of reality may exist beyond the reach of any computational process. Understanding computable numbers helps define what computers can achieve and where the limits of algorithms ultimately lie. ARTICLE