Al-Kashi, Pi and Decimal Fractions: Precision in Samarkand
How Jamshid al-Kashi calculated pi to remarkable precision, explained decimal fractions and turned demanding arithmetic into teachable procedures.
Jamshid al-Kashi, who died around 1429, was a Persian mathematician and astronomer associated with Ulugh Beg’s scholarly circle in Samarkand. His writings combined exceptional numerical precision with clear algorithms for computation, including a systematic treatment of decimal fractions.
The cover image is an original AI-assisted scholarly reconstruction, not a portrait of al-Kashi or a reproduction of a specific manuscript.
Pi emerged from polygons and repeated calculation
In his Risala al-Muhitiyya, or Treatise on the Circumference, al-Kashi calculated the ratio of a circle’s circumference to its diameter to a precision equivalent to about sixteen decimal places. He used polygonal geometry and high-precision arithmetic rather than a modern infinite-series formula.
The achievement was not just a long string of digits. He controlled error through successive stages and expressed a practical goal: sufficient precision for a circle on an immense astronomical scale.
Decimal fractions became a working language
Fractions with powers of ten existed before al-Kashi, including in Chinese and Islamic mathematics. His Key to Arithmetic gave a sustained, teachable account of decimal computation and used it across multiplication, roots and measurement.
It is therefore better to say that al-Kashi developed and systematized decimal fractions than that he single-handedly invented every decimal idea. Mathematical notation has multiple lineages, and his written conventions were not identical to the modern decimal point.
Algorithms served astronomy and administration
Accurate tables require repeated arithmetic. Al-Kashi described methods for extracting roots and calculating trigonometric values, supporting the observational program at Samarkand. The same procedural clarity also made his arithmetic useful for education and practical calculation.
Manuscript copying introduces another layer of uncertainty. A misplaced digit can destroy a high-precision result, so historians compare copies and reconstruct notation before translating it into contemporary symbols.
How to present al-Kashi’s result
- Distinguish the value he obtained from modern notation used to display it.
- Explain the polygonal method rather than praising digits alone.
- Acknowledge earlier decimal traditions while identifying his systematic contribution.
- Connect numerical work to instruments, tables and teaching.
Al-Kashi’s mathematics shows precision as an intellectual practice. Reliable digits came from organized procedures, error control and an institution capable of sustaining difficult calculation.
Sources
- MacTutor: Jamshid al-Kashi — biography, pi and decimal fractions.
- MacTutor: Arabic Mathematics — broader mathematical lineages and methods.
- British Society for the History of Mathematics: Al-Kashi — detailed historical study.
