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 his acronym ḤaZaS (), was a Hebrew publisher, mathematician, astronomer, inventor, science writer, and rabbi. He was among the first to write books on scie ...

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

Farey sequence In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

. 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 Napier's bones is a manually-operated calculating device created by John Napier of Merchiston, Scotland for the calculation of products and quotients of numbers. The method was based on lattice multiplication, and also called ''rabdology'', a word ...

. 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 , last=Monnier, first= 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 Mechanical calculators Multiplication