# Compendium of "pencil-and-paper numerical methods"

NB: I replaced "algorithms" in the title with "numerical methods", since my original wording seemed to be causing confusion. I made a similar change in various places throughout the post.

I'm looking for a compendium of numerical methods for pencil-and-paper calculations. Such methods not only describe the operations to be performed, but also how one lays out the calculation on the page. IOW, such a method includes details about arranging the figures appearing in the computation into columns, or rows, or tables, etc., on the page (or on the blackboard).

Here, I am interested only in the most efficient methods for the given tasks (as opposed to less useful methods that are only of historical interest; e.g. I have no interest in methods for multiplying Roman numerals, however clever they may be).

I am primarily interested in methods used in arithmetic (e.g. the extended Euclidean algorithm) and algebra (e.g. Ruffini's method for dividing polynomials).

Once more, I stress that I am looking for methods that include explicit instructions for how to lay the computations out on the page. IOW, the layout is part of the method.

• I have to confess that I haven't seen any algorithms specified with explicit layout instructions after learning long multiplication and long division in school. Do you have any examples of what you have in mind?
– user856
Apr 23, 2018 at 14:18
• simplex algorithm and gaussian elimination perhaps? I'm not sure though, I can't imagine anyone being so tyrannical about the layout of anything. Apr 23, 2018 at 14:27
• @Countingstuff: when I was taught multiplication of two multi-digit numbers in elementary school, the instructions included where digits were to be placed relative to each other. That is layout.
– kjo
Apr 23, 2018 at 17:00
• @rahul: if you understand Portuguese, here's an example of the extended Euclid algorithm, including how the intermediate results are to be arranged on the board: youtube.com/watch?v=V02GMdq964w&t=3562 . BTW, long multiplication and long division as taught in elementary school are excellent examples of the sort of algorithm I have in mind. I don't understand why a question about such algorithms merits downvotes.
– kjo
Apr 23, 2018 at 17:06
• math.stackexchange.com/questions/683774/… has descriptions of a few different methods for arithmetic division.
– MJD
Sep 25, 2021 at 17:39

I hope that after almost 6 years I may help with an answer. If you read at least some German, there are some classic textbooks where you should find what you are looking for.

• B. E. R. Schurig, Lehrbuch der Arithmetik zum Gebrauch an niedern und höheren Lehranstalten, Leipzig: Friedrich Brandstetter 1883–1885 (3 vols.).
• J. Lüroth, Vorlesungen über numerisches Rechnen, Leipzig: Teubner 1900.
• C. Runge, Die Praxis der Gleichungen, Berlin – Leipzig: Walter de Gruyter 1921.
• C. Runge – H. König, Vorlesungen über numerisches Rechnen, Die Grundlehren der mathematischen
Wissenschaften XI, Berlin: Julius Springer 1924.
• R. W. Doerfler, Dead Reckoning. Calculating without Instruments, Houston 1993.

You can find them all on archive.org. The books by Lüroth 1900, Runge 1921, and Runge – König 1924 are still available as reprints. The most advanced level is offered by Runge – König, including numerical differentiation/integration and solving differential equations using paper and pencil methods.

• You made my day. Thank you!
– kjo
Dec 8, 2023 at 14:12