Slonimski's Theorem is an observation by

Hayyim Selig Slonimski Ḥayyim Selig ben Ya'akov Slonimski (; March 31, 1810 – May 15, 1904), also known by Hebrew abbreviations, his acronym ḤaZaS (), was a Hebrew publisher, mathematician, astronomer, inventor, science writer, and rabbi. He was among the first t ...

that the sequence of carry digits in a multiplication table is the

Farey sequence In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which have denominators less than or equal to ''n'', arranged in order of increasing size. Wi ...

. This observation allowed Slonimski to create very compact multiplication tables for use in hand calculations. He received several awards for different devices for presenting these tables. The most common format were Joffe Bars similar to Napier's Rods. Joffe Bars were popular in Eastern Europe in the late 19th and early 20th century.

References

*. Provides a derivation of Slonimski's theorem, and some details on the calculating machine. *. Provides a complete set of tables. *{{citation, mr=3111378 , last1=Monnier, first1= Valéry, last2=Szrek, first2= Walter, last3= Zalewski, first3= Janusz , title=Chaim Selig Slonimski and his adding devices , journal=IEEE Ann. Hist. Comput. , volume=35 , year=2013, issue= 3, pages= 42–53, doi=10.1109/MAHC.2013.13 Mechanical calculators Multiplication