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<title>Bi-quinary coded decimal</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Bi-quinary coded decimal</span></span>

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:192px;max-width:192px"><div class="trow"><div class="tsingle" style="width:94px;max-width:94px"><div class="thumbimage" style="height:119px;overflow:hidden"><span typeof="mw:File"></span></div><div class="thumbcaption">Biquinary code example<sup id="cite_ref-Ledley_1960_1-0" class="reference"><a href="#cite_note-Ledley_1960-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></div></div><div class="tsingle" style="width:94px;max-width:94px"><div class="thumbimage" style="height:119px;overflow:hidden"><span typeof="mw:File"></span></div><div class="thumbcaption">Reflected biquinary code</div></div></div></div></div>

<p><b>Bi-quinary coded decimal</b> is a <a href="Numeral_system" title="Numeral system">numeral encoding scheme</a> used in many <a href="Abacus" title="Abacus">abacuses</a> and in some <a href="Early_computer" class="mw-redirect" title="Early computer">early computers</a>, notably the <a href="Colossus_computer" title="Colossus computer">Colossus</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The term <i><b>bi-quinary</b></i> indicates that the code comprises both a two-state (<i>bi</i>) and a five-state (<i>quin</i>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).

</p><p>Several human languages, most notably <a href="Fula_language" title="Fula language">Fula</a> and <a href="Wolof_language" title="Wolof language">Wolof</a> also use biquinary systems. For example, the Fula word for 6, <i>jowi e go'o</i>, literally means <i>five [plus] one</i>. <a href="Roman_numerals" title="Roman numerals">Roman numerals</a> use a symbolic, rather than positional, bi-quinary base, even though <a href="Latin" title="Latin">Latin</a> is completely decimal.

</p><p>The Korean finger counting system <a href="Chisanbop" title="Chisanbop">Chisanbop</a> 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.

</p><p>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 <a href="Checksum" title="Checksum">checksum</a> to verify if the number is valid or not. (Stuck bits happened frequently with computers using <a href="Relay" title="Relay">mechanical relays</a>.)

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>Several different representations of bi-quinary coded decimal have been used by different machines. The two-state component is encoded as one or two <a href="Bit" title="Bit">bits</a>, and the five-state component is encoded using three to five bits. Some examples are:

</p>

<ul><li>Roman and Chinese <a href="Abacus" title="Abacus">abacuses</a></li>

<li><a href="George_Stibitz" title="George Stibitz">Stibitz</a><sup id="cite_ref-Stibitz_1957_3-0" class="reference"><a href="#cite_note-Stibitz_1957-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> relay calculators at Bell Labs from <a href="Bell_Labs#Calculators" title="Bell Labs">Model II</a> onwards</li>

<li><a href="FACOM_128" title="FACOM 128">FACOM 128</a> relay calculators at <a href="Fujitsu" title="Fujitsu">Fujitsu</a></li></ul>

<p>

The <a href="IBM_650" title="IBM 650">IBM 650</a> uses seven bits: two <i>bi</i> bits (0 and 5) and five <i>quinary</i> bits (0, 1, 2, 3, 4), with error checking.

</p><p>Exactly one <i>bi</i> bit and one <i>quinary</i> 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 <i>bi</i> bits form the top of a T for each digit, and the <i>quinary</i> bits form the vertical stem.

</p>

<th>Value</th>

</th>

</td></tr>

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<td>10-10000

</td></tr>

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<td>10-01000

</td></tr>

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<td>10-00100

</td></tr>

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<td>10-00010

</td></tr>

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<td>10-00001

</td></tr>

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<td>01-10000

</td></tr>

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<td>01-01000

</td></tr>

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<td>01-00100

</td></tr>

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<td>01-00010

</td></tr>

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<td>01-00001

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Remington_Rand_409">Remington Rand 409</h3></div>

<p>The <a href="Remington_Rand_409" title="Remington Rand 409">Remington Rand 409</a> has five bits: one <i>quinary</i> bit (tube) for each of 1, 3, 5, and 7 - only one of these would be on at the time. The fifth <i>bi</i> bit represented 9 if none of the others were on; otherwise it added 1 to the value represented by the other <i>quinary</i> bit. The machine was sold in the two models <a href="UNIVAC_60" class="mw-redirect" title="UNIVAC 60">UNIVAC 60</a> and <a href="UNIVAC_120" class="mw-redirect" title="UNIVAC 120">UNIVAC 120</a>.

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<th>Value</th>

<th>1357-9 bits

</th></tr>

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<td>0000-0

</td></tr>

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<td>1000-0

</td></tr>

<tr>

<td>1000-1

</td></tr>

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<td>0100-0

</td></tr>

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<td>0100-1

</td></tr>

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<td>0010-0

</td></tr>

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<td>0010-1

</td></tr>

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<td>0001-0

</td></tr>

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<td>0001-1

</td></tr>

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<td>0000-1

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="UNIVAC_Solid_State">UNIVAC Solid State</h3></div>

<p>The <a href="UNIVAC_Solid_State" title="UNIVAC Solid State">UNIVAC Solid State</a> uses four bits: one <i>bi</i> bit (5), three binary coded <i>quinary</i> bits (4 2 1)<sup id="cite_ref-Steinbuch_1962_4-0" class="reference"><a href="#cite_note-Steinbuch_1962-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Steinbuch-Wagner_1967_5-0" class="reference"><a href="#cite_note-Steinbuch-Wagner_1967-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Steinbuch-Weber-Heinemann_1974_6-0" class="reference"><a href="#cite_note-Steinbuch-Weber-Heinemann_1974-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Dokter_1973_7-0" class="reference"><a href="#cite_note-Dokter_1973-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Dokter_1975_8-0" class="reference"><a href="#cite_note-Dokter_1975-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Savard_2018_Decimal_9-0" class="reference"><a href="#cite_note-Savard_2018_Decimal-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and one <a href="Parity_bit" title="Parity bit">parity check bit</a>

</p>

<th>Value</th>

<th>p-5-421 bits

</th></tr>

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<td>1-0-000

</td></tr>

<tr>

<td>0-0-001

</td></tr>

<tr>

<td>0-0-010

</td></tr>

<tr>

<td>1-0-011

</td></tr>

<tr>

<td>0-0-100

</td></tr>

<tr>

<td>0-1-000

</td></tr>

<tr>

<td>1-1-001

</td></tr>

<tr>

<td>1-1-010

</td></tr>

<tr>

<td>0-1-011

</td></tr>

<tr>

<td>1-1-100

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="UNIVAC_LARC">UNIVAC LARC</h3></div>

<p>The <a href="UNIVAC_LARC" title="UNIVAC LARC">UNIVAC LARC</a> has four bits:<sup id="cite_ref-Savard_2018_Decimal_9-1" class="reference"><a href="#cite_note-Savard_2018_Decimal-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> one <i>bi</i> bit (5), three <a href="Johnson_counter" class="mw-redirect" title="Johnson counter">Johnson counter</a>-coded <i>quinary</i> bits and one parity check bit.

</p>

<th>Value</th>

<th>p-5-qqq bits

</th></tr>

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<td>1-0-000

</td></tr>

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<td>0-0-001

</td></tr>

<tr>

<td>1-0-011

</td></tr>

<tr>

<td>0-0-111

</td></tr>

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<td>1-0-110

</td></tr>

<tr>

<td>0-1-000

</td></tr>

<tr>

<td>1-1-001

</td></tr>

<tr>

<td>0-1-011

</td></tr>

<tr>

<td>1-1-111

</td></tr>

<tr>

<td>0-1-110

</td></tr></tbody></table>

<ul><li><a href="Binary-coded_decimal" title="Binary-coded decimal">Binary-coded decimal</a></li>

<li><a href="Binary_number" title="Binary number">Binary number</a></li>

<li><a href="Chisanbop" title="Chisanbop">Chisanbop</a></li>

<li><a href="Finger_binary" title="Finger binary">Finger binary</a></li>

<li><a href="Quinary" title="Quinary">Quinary</a></li>

<li><a href="FACOM_128" title="FACOM 128">FACOM 128</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFLedleyRotoloWilson1960" class="citation book cs1"><a href="Robert_Steven_Ledley" class="mw-redirect" title="Robert Steven Ledley">Ledley, Robert Steven</a>; Rotolo, Louis S.; Wilson, James Bruce (1960). "Part 4. Logical Design of Digital-Computer Circuitry; Chapter 15. Serial Arithmetic Operations; Chapter 15-7. Additional Topics". <a rel="nofollow" class="external text" href="http://bitsavers.informatik.uni-stuttgart.de/pdf/columbiaUniv/Ledley_Digital_Computer_and_Control_Engineering_1960.pdf"><i>Digital Computer and Control Engineering</i></a> <span class="cs1-format">(PDF)</span>. McGraw-Hill Electrical and Electronic Engineering Series (1 ed.). New York, US: <a href="McGraw-Hill_Book_Company%2C_Inc." class="mw-redirect" title="McGraw-Hill Book Company, Inc.">McGraw-Hill Book Company, Inc.</a> (printer: The Maple Press Company, York, Pennsylvania, US). pp. <span class="nowrap">517–</span>518. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07036981-X</bdi>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2574-7916">2574-7916</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/59015055">59015055</a>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1033638267">1033638267</a>. <a href="OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL5776493M">5776493M</a>. <a href="SBN_(identifier)" class="mw-redirect" title="SBN (identifier)">SBN</a> <bdi>07036981-X</bdi>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07036981-8</bdi>. ark:/13960/t72v3b312. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210219203314/http://bitsavers.informatik.uni-stuttgart.de/pdf/columbiaUniv/Ledley_Digital_Computer_and_Control_Engineering_1960.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2021-02-19<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-02-19</span></span>. p. 518: <q>[…] 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. […]</q></cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span> <a rel="nofollow" class="external autonumber" href="https://archive.org/details/digitalcomputerc00ledl">[1]</a> (xxiv+835+1 pages)</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=thrx3SBEpL8&list=WL&index=17&t=0s">"Why Use Binary? - Computerphile"</a>. YouTube. 2015-12-04. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211212/thrx3SBEpL8">Archived</a> from the original on 2021-12-12<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-12-10</span></span>.</cite></span>

</li>

<li id="cite_note-Stibitz_1957-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stibitz_1957_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFStibitzLarrivee1957" class="citation book cs1 cs1-prop-location-test"><a href="George_Robert_Stibitz" class="mw-redirect" title="George Robert Stibitz">Stibitz, George Robert</a>; Larrivee, Jules A. (1957). Written at Underhill, Vermont, US. <i>Mathematics and Computers</i> (1 ed.). New York, US / Toronto, Canada / London, UK: <a href="McGraw-Hill_Book_Company%2C_Inc." class="mw-redirect" title="McGraw-Hill Book Company, Inc.">McGraw-Hill Book Company, Inc.</a> p. 105. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/56-10331">56-10331</a>.</cite> (10+228 pages)</span>

</li>

<li id="cite_note-Steinbuch_1962-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Steinbuch_1962_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBerger1962" class="citation book cs1 cs1-prop-location-test cs1-prop-foreign-lang-source">Berger, Erich R. (1962). "1.3.3. Die Codierung von Zahlen". Written at Karlsruhe, Germany. In <a href="Karl_W._Steinbuch" class="mw-redirect" title="Karl W. Steinbuch">Steinbuch, Karl W.</a> (ed.). <i>Taschenbuch der Nachrichtenverarbeitung</i> (in German) (1 ed.). Berlin / Göttingen / New York: <a href="Springer-Verlag_OHG" class="mw-redirect" title="Springer-Verlag OHG">Springer-Verlag OHG</a>. pp. <span class="nowrap">68–</span>75. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/62-14511">62-14511</a>.</cite></span>

</li>

<li id="cite_note-Steinbuch-Wagner_1967-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Steinbuch-Wagner_1967_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBergerHändler1967" class="citation book cs1 cs1-prop-foreign-lang-source">Berger, Erich R.; <a href="Wolfgang_H%C3%A4ndler" title="Wolfgang Händler">Händler, Wolfgang</a> (1967) [1962]. <a href="Karl_W._Steinbuch" class="mw-redirect" title="Karl W. Steinbuch">Steinbuch, Karl W.</a>; Wagner, Siegfried W. (eds.). <i>Taschenbuch der Nachrichtenverarbeitung</i> (in German) (2 ed.). Berlin, Germany: <a href="Springer-Verlag_OHG" class="mw-redirect" title="Springer-Verlag OHG">Springer-Verlag OHG</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/67-21079">67-21079</a>. Title No. 1036.</cite></span>

</li>

<li id="cite_note-Steinbuch-Weber-Heinemann_1974-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Steinbuch-Weber-Heinemann_1974_6-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFSteinbuchWeberHeinemann1974" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Karl_W._Steinbuch" class="mw-redirect" title="Karl W. Steinbuch">Steinbuch, Karl W.</a>; Weber, Wolfgang; Heinemann, Traute, eds. (1974) [1967]. <i>Taschenbuch der Informatik - Band II - Struktur und Programmierung von EDV-Systemen</i> (in German). Vol. 2 (3 ed.). Berlin, Germany: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-06241-6</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/73-80607">73-80607</a>.</cite> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|work=</code> ignored (help)</span></span>

</li>

<li id="cite_note-Dokter_1973-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dokter_1973_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFDokterSteinhauer1973" class="citation book cs1">Dokter, Folkert; Steinhauer, Jürgen (1973-06-18). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hlRdDwAAQBAJ"><i>Digital Electronics</i></a>. Philips Technical Library (PTL) / Macmillan Education (Reprint of 1st English ed.). Eindhoven, Netherlands: <a href="The_Macmillan_Press_Ltd." class="mw-redirect" title="The Macmillan Press Ltd.">The Macmillan Press Ltd.</a> / <a href="N._V._Philips'_Gloeilampenfabrieken" class="mw-redirect" title="N. V. Philips' Gloeilampenfabrieken">N. V. Philips' Gloeilampenfabrieken</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-349-01417-0">10.1007/978-1-349-01417-0</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-349-01419-4</bdi>. <a href="SBN_(identifier)" class="mw-redirect" title="SBN (identifier)">SBN</a> <bdi>333-13360-9</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-11</span></span>.</cite> (270 pages) (NB. This is based on a translation of volume I of the two-volume German edition.)</span>

</li>

<li id="cite_note-Dokter_1975-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dokter_1975_8-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFDokterSteinhauer1975" class="citation book cs1 cs1-prop-foreign-lang-source">Dokter, Folkert; Steinhauer, Jürgen (1975) [1969]. <i>Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik</i>. Philips Fachbücher (in German). Vol. I (improved and extended 5th ed.). Hamburg, Germany: <a href="Deutsche_Philips_GmbH" class="mw-redirect" title="Deutsche Philips GmbH">Deutsche Philips GmbH</a>. p. 50. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-87145-272-6</bdi>.</cite> (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.)</span>

</li>

<li id="cite_note-Savard_2018_Decimal-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Savard_2018_Decimal_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Savard_2018_Decimal_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFSavard2018" class="citation web cs1">Savard, John J. G. (2018) [2006]. <a rel="nofollow" class="external text" href="http://www.quadibloc.com/comp/cp0203.htm">"Decimal Representations"</a>. <i>quadibloc</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180716101321/http://www.quadibloc.com/comp/cp0203.htm">Archived</a> from the original on 2018-07-16<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-07-16</span></span>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf"><i>Military Handbook: Encoders - Shaft Angle To Digital</i></a> <span class="cs1-format">(PDF)</span>. <a href="United_States_Department_of_Defense" title="United States Department of Defense">United States Department of Defense</a>. 1991-09-30. MIL-HDBK-231A. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200725051128/http://everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2020-07-25<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-07-25</span></span>.</cite> (NB. Supersedes MIL-HDBK-231(AS) (1970-07-01).)</li></ul></div><div><div class="zim-footer">

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