Document integrity: Practical Application of Cryptographic Checksums, with an introduction to signing and encryption, contains all hyperlinks, framed and visible in the browser or in the separate PDF viewer. English: PDF DIN A4 • A4 two-column • US Letter • US Letter two-column, German: PDF DIN A4 • A4 two-column • US Letter • US Letter two-column.

Referenced in: “Künstliche Intelligenz - Ethik und Recht”, Hoeren/Pinelli (Hrsg.), 2022, footnote 49 on page 11, Verlag C.H. Beck, (book review by Roland Kemper) • M2M-Geschäfte, Vertragsschluss und vertragliche Struk­tu­ren im automatisierten E-Procurement”, Dajana Mohr, Tectum Verlag 2022, footnote 283 on page 72, PDF sample  •  German legal magazine JuristenZeitung, 18.09.2020, “Daten im Rechtsverkehr – Überlegungen für ein allgemeines Datenvertragsrecht” (footnote 14, page 881), Thomas Hoeren and Stefean Pinelli, Verlag Mohr Siebeck  •  Wikipedia article “Checksum” (EN) • Wikipedia article “Prüfsumme  • Top 10 (EN) • Top 10  • Top 10

Scientific typesetting: Free Typesetting Software for the Professional Document Preparation Top 10 • Top 10 (EN)

Introduction to the indirect proof: “Proof of the Irrationality of the Square Root of 2 Top 10 • Top 10 (EN)

Mathematical proofs are used to prove the cor­rect­ness of formulas and assumed facts. Sev­er­al basic procedures and ways of veri­fy­ing mathematical correctness are in use, all of which follow a fixed scheme.

For the proof of the irrationality of the square root of 2 dif­fer­ent proofs exist. This article explains the indirect proof vari­ant in detail and introduces one of the possible short spell­ings in mathematical notation. Exercises with solutions con­soli­date what has been learned. 12 pages::

      A4  •  A4 duplex  •  US-Letter  •  US-Letter duplex

School mathematics: Detailed so­lu­tions to the first 27 text problems of A first book in Al­­ge­­bra, author Wallace C. Boyden, added, a textbook for be­gin­ners in algebra: Text problems with solutions, 10 pages (PDF, A4 format, in English)..

Maps on the German language and Germany. Directory overview at, sample maps: Deutsches Reich physikalisch, Deutsches Reich Staatenkarte 1913. Prefer the PNG version to the preselected JPEG version, the PNG file has the better quality and can also be directly linked (hotlinked), listed on the right in the menu “DOWNLOAD OPTIONS” or under “SHOW ALL”.

Youtube playlist „Deutsche Sprache: “Why Germans can say things no one else can.”  • “Die Lebendigkeit der deutschen Sprache (J.G.Fichte)”  • “The German Language: Its Fascinating History, Diverse Dialects & More!”  •  Recommended reading on spelling: • Youtube playlist “Deutsches Reich •  Overview of all YouTube playlists

Maps on the German Language

“Völker- und Sprachenkarte von Europa”, “Map of Peoples and Languages of Europe”, scale 1:30 000 000.
Bibliographisches Institut A.G., Leipzig; “Meyers Geographischer Handatlas”, “[...] siebente neubearbeitete
und vermehrte Auflage [...], 1928” Raw file, WP-article “Meyers Handatlas”. Read also the Metapedia
article “Deutsche Sprache”:

Distribution [resp. dissemination] of the German language around 1913, original maps, freely and directly
available at the Internet Archive (title page map 1, title page map 2). The best resolution quality is the
respective TIFF or PNG file versions on the right under “DOWNLOAD OPTIONS” or under “SHOW ALL”. They
can also be linked directly, such as the two following thumbnails. Click on them again during
or after loading for a detailed view:

“Verbreitung der Deutschen Mundarten”, “Distribution of the German dialects”, according to Prof. Dr. W. Stammler,
scale 1:8000000. Bibliographisches Institut A.G., Leipzig; “Meyers Geographischer Handatlas”, “[...] siebente
neubearbeitete und vermehrte Auflage [...], 1928” Raw file, WP article “Meyers Handatlas”. Also read the
Metapedia article „Deutsche Mundarten“, “German dialects”, at

As of: February 24, 2024

Imprint • Privacy Statement • OpenPGP key • Biographical Notes • German