Bi-quinary coded decimal

{{short description|Numeral encoding scheme}} {{Use dmy dates|date=May 2019|cs1-dates=y}} {{anchor|2-out-of-7|quibinary}} {{main article|Decimal computer}}

{{Multiple image | image1 = Code_Biquinaer.svg | caption1 = Biquinary code example<ref name="Ledley_1960"/> | image2 = Code Biquinaer reflektiert.svg | caption2 = Reflected biquinary code | total_width = 200 }}

[[Image:Soroban.JPG|349x349px|thumb|Japanese abacus. The right side represents {{formatnum:1234567890}} in bi-quinary: each column is one digit, with the lower beads representing "ones" and the upper beads "fives".]]

'''Bi-quinary coded decimal''' is a numeral encoding scheme used in many abacuses and in some early computers, notably the Colossus.<ref>{{cite web|url=https://www.youtube.com/watch?v=thrx3SBEpL8&list=WL&index=17&t=0s |archive-url=https://ghostarchive.org/varchive/youtube/20211212/thrx3SBEpL8| archive-date=2021-12-12 |url-status=live|title=Why Use Binary? - Computerphile |publisher=YouTube |date=2015-12-04 |access-date=2020-12-10}}{{cbignore}}</ref> The term '''''bi-quinary''''' indicates that the code comprises both a two-state (''bi'') and a five-state (''quin''ary) component. The encoding resembles that used by many abacuses, with four beads indicating the five values either from 0 through 4 or from 5 through 9 and another bead indicating which of those ranges (which can alternatively be thought of as +5).

Several human languages, most notably Fula and Wolof also use biquinary systems. For example, the Fula word for 6, ''jowi e go'o'', literally means ''five [plus] one''. Roman numerals use a symbolic, rather than positional, bi-quinary base, even though Latin is completely decimal.

The Korean finger counting system Chisanbop uses a bi-quinary system, where each finger represents a one and a thumb represents a five, allowing one to count from 0 to 99 with two hands.

One advantage of one bi-quinary encoding scheme on digital computers is that it must have two bits set (one in the binary field and one in the quinary field), providing a built-in checksum to verify if the number is valid or not. (Stuck bits happened frequently with computers using mechanical relays.)

==Examples==

Several different representations of bi-quinary coded decimal have been used by different machines. The two-state component is encoded as one or two bits, and the five-state component is encoded using three to five bits. Some examples are: * Roman and Chinese abacuses * Stibitz<ref name="Stibitz_1957"/> relay calculators at Bell Labs from Model II onwards * FACOM 128 relay calculators at Fujitsu ===IBM 650=== {{anchor|IBM650code}} The IBM 650 uses seven bits: two ''bi'' bits (0 and 5) and five ''quinary'' bits (0, 1, 2, 3, 4), with error checking.

Exactly one ''bi'' bit and one ''quinary'' bit is set in a valid digit. The bi-quinary encoding of the internal workings of the machine are evident in the arrangement of its lights – the ''bi'' bits form the top of a T for each digit, and the ''quinary'' bits form the vertical stem.

{| class="wikitable" |- ! Value || 05-01234 bits<ref name="Ledley_1960"/> | rowspan="11" | thumb|center|IBM 650 front panel while running, with active bits just discernible thumb|center|Close-up of IBM 650 indicators while running, with active bits visible |- | 0 || 10-10000 |- | 1 || 10-01000 |- | 2 || 10-00100 |- | 3 || 10-00010 |- | 4 || 10-00001 |- | 5 || 01-10000 |- | 6 || 01-01000 |- | 7 || 01-00100 |- | 8 || 01-00010 |- | 9 || 01-00001 |}

===Remington Rand 409=== The Remington Rand 409 has five bits: one ''quinary'' bit (tube) for each of 1, 3, 5, and 7 - only one of these would be on at the time. The fifth ''bi'' bit represented 9 if none of the others were on; otherwise it added 1 to the value represented by the other ''quinary'' bit. The machine was sold in the two models UNIVAC 60 and UNIVAC 120.

{| class="wikitable" |- ! Value || 1357-9 bits |- | 0 || 0000-0 |- | 1 || 1000-0 |- | 2 || 1000-1 |- | 3 || 0100-0 |- | 4 || 0100-1 |- | 5 || 0010-0 |- | 6 || 0010-1 |- | 7 || 0001-0 |- | 8 || 0001-1 |- | 9 || 0000-1 |} ===UNIVAC Solid State=== The UNIVAC Solid State uses four bits: one ''bi'' bit (5), three binary coded ''quinary'' bits (4 2 1)<ref name="Steinbuch_1962"/><ref name="Steinbuch-Wagner_1967"/><ref name="Steinbuch-Weber-Heinemann_1974"/><ref name="Dokter_1973"/><ref name="Dokter_1975"/><ref name="Savard_2018_Decimal"/> and one parity check bit

{| class="wikitable" |- ! Value || p-5-421 bits |- | 0 || 1-0-000 |- | 1 || 0-0-001 |- | 2 || 0-0-010 |- | 3 || 1-0-011 |- | 4 || 0-0-100 |- | 5 || 0-1-000 |- | 6 || 1-1-001 |- | 7 || 1-1-010 |- | 8 || 0-1-011 |- | 9 || 1-1-100 |} ===UNIVAC LARC=== The UNIVAC LARC has four bits:<ref name="Savard_2018_Decimal"/> one ''bi'' bit (5), three Johnson counter-coded ''quinary'' bits and one parity check bit.

{| class="wikitable" |- ! Value || p-5-qqq bits |- | 0 || 1-0-000 |- | 1 || 0-0-001 |- | 2 || 1-0-011 |- | 3 || 0-0-111 |- | 4 || 1-0-110 |- | 5 || 0-1-000 |- | 6 || 1-1-001 |- | 7 || 0-1-011 |- | 8 || 1-1-111 |- | 9 || 0-1-110 |}

==See also== * {{annotated link|Binary-coded decimal}} * Binary number * Chisanbop * {{annotated link|Decimal computer}} * FACOM 128 * Finger binary * Quinary * Two-out-of-five code

==References== {{Reflist|refs= <ref name="Steinbuch_1962">{{cite book |title=Taschenbuch der Nachrichtenverarbeitung |language=de |editor-first=Karl W. |editor-last=Steinbuch |editor-link=Karl W. Steinbuch |author-first=Erich R. |author-last=Berger |chapter=1.3.3. Die Codierung von Zahlen |date=1962 |edition=1 |publisher=Springer-Verlag OHG |location=Karlsruhe, Germany |publication-place=Berlin / Göttingen / New York |lccn=62-14511 |pages=68–75}}</ref> <ref name="Steinbuch-Wagner_1967">{{cite book |title=Taschenbuch der Nachrichtenverarbeitung |language=de |editor-first1=Karl W. |editor-last1=Steinbuch |editor-link1=Karl W. Steinbuch |editor-first2=Siegfried W. |editor-last2=Wagner |author-first1=Erich R. |author-last1=Berger |author-first2=Wolfgang |author-last2=Händler |author-link2=Wolfgang Händler |date=1967 |orig-year=1962 |edition=2 |publisher=Springer-Verlag OHG |location=Berlin, Germany |id=Title No. 1036 |lccn=67-21079}}</ref> <ref name="Steinbuch-Weber-Heinemann_1974">{{cite book |title=Taschenbuch der Informatik - Band II - Struktur und Programmierung von EDV-Systemen |language=de |editor-first1=Karl W. |editor-last1=Steinbuch |editor-link1=Karl W. Steinbuch |editor-first2=Wolfgang |editor-last2=Weber  |editor-first3=Traute |editor-last3=Heinemann |date=1974 |orig-year=1967 |edition=3 |volume=2 |work=Taschenbuch der Nachrichtenverarbeitung |publisher=Springer-Verlag |location=Berlin, Germany |isbn=3-540-06241-6 |lccn=73-80607}}</ref> <ref name="Dokter_1973">{{cite book |title=Digital Electronics |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |date=1973-06-18 |series=Philips Technical Library (PTL) / Macmillan Education |publisher=The Macmillan Press Ltd. / N. V. Philips' Gloeilampenfabrieken |edition=Reprint of 1st English |location=Eindhoven, Netherlands |sbn=333-13360-9 |isbn=978-1-349-01419-4 |doi=10.1007/978-1-349-01417-0 |url=https://books.google.com/books?id=hlRdDwAAQBAJ |access-date=2020-05-11 }}{{Dead link|date=October 2023 |bot=InternetArchiveBot |fix-attempted=yes }} (270 pages) (NB. This is based on a translation of volume I of the two-volume German edition.)</ref> <ref name="Dokter_1975">{{cite book |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |title=Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik |language=de |series=Philips Fachbücher |publisher=Deutsche Philips GmbH |location=Hamburg, Germany |volume=I |date=1975 |orig-year=1969 |edition=improved and extended 5th |isbn=3-87145-272-6 |page=50}} (xii+327+3 pages) (NB. The German edition of volume I was published in 1969, 1971, two editions in 1972, and 1975. Volume II was published in 1970, 1972, 1973, and 1975.)</ref> <ref name="Stibitz_1957">{{cite book |title=Mathematics and Computers |author-first1=George Robert |author-last1=Stibitz |author-link1=George Robert Stibitz |author-first2=Jules A. |author-last2=Larrivee |date=1957 |edition=1 |publisher=McGraw-Hill Book Company, Inc. |publication-place=New York, US / Toronto, Canada / London, UK |location=Underhill, Vermont, US |lccn=56-10331 |page=105}} (10+228 pages)</ref> <ref name="Savard_2018_Decimal">{{cite web |title=Decimal Representations |author-first=John J. G. |author-last=Savard |date=2018 |orig-year=2006 |work=quadibloc |url=http://www.quadibloc.com/comp/cp0203.htm |access-date=2018-07-16 |url-status=live |archive-url=https://web.archive.org/web/20180716101321/http://www.quadibloc.com/comp/cp0203.htm |archive-date=2018-07-16}}</ref> <ref name="Ledley_1960">{{cite book |title=Digital Computer and Control Engineering |chapter=Part 4. Logical Design of Digital-Computer Circuitry; Chapter 15. Serial Arithmetic Operations; Chapter 15-7. Additional Topics |author-first1=Robert Steven |author-last1=Ledley |author-link1=Robert Steven Ledley |author-first2=Louis S. |author-last2=Rotolo |author-first3=James Bruce |author-last3=Wilson |publisher=McGraw-Hill Book Company, Inc. (printer: The Maple Press Company, York, Pennsylvania, US) |publication-place=New York, US |series=McGraw-Hill Electrical and Electronic Engineering Series |edition=1 |date=1960 |sbn=07036981-X |isbn=0-07036981-X |id={{ISBN|978-0-07036981-8}}. ark:/13960/t72v3b312 |issn=2574-7916 |ol=OL5776493M |lccn=59015055 |oclc=1033638267 |pages=517–518 |url=http://bitsavers.informatik.uni-stuttgart.de/pdf/columbiaUniv/Ledley_Digital_Computer_and_Control_Engineering_1960.pdf |access-date=2021-02-19 |url-status=live |archive-url=https://web.archive.org/web/20210219203314/http://bitsavers.informatik.uni-stuttgart.de/pdf/columbiaUniv/Ledley_Digital_Computer_and_Control_Engineering_1960.pdf |archive-date=2021-02-19 |quote-page=518 |quote=[…] The use of the biquinary code in this respect is typical. The binary part (i.e., the most significant bit) and the quinary part (the other 4 bits) are first added separately; then the quinary carry is added to the binary part. If a binary carry is generated, this is propagated to the quinary part of the next decimal digit to the left. […]}} [https://archive.org/details/digitalcomputerc00ledl] (xxiv+835+1 pages)</ref> }}

==Further reading== * {{cite book |title=Military Handbook: Encoders - Shaft Angle To Digital |publisher=United States Department of Defense |id=MIL-HDBK-231A |date=1991-09-30 |url=http://everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf |access-date=2020-07-25 |url-status=live |archive-url=https://web.archive.org/web/20200725051128/http://everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf |archive-date=2020-07-25}} (NB. Supersedes MIL-HDBK-231(AS) (1970-07-01).)

{{DEFAULTSORT:Bi-Quinary Coded Decimal}} Category:Computer arithmetic Category:Numeral systems