Editing Significand

{{short description|Part of a number in scientific notation}}
{{use dmy dates|date=July 2020|cs1-dates=y}}
The '''significand'''<ref name="Kahan_2002"/> (also '''coefficient''',<ref name="Kahan_2002"/> sometimes '''argument''',<ref>{{Cite book |last=Clements |first=Alan |url=https://books.google.com/books?id=wUecAQAAQBAJ |title=Principles of Computer Hardware |date=2006-02-09 |publisher=OUP Oxford |isbn=978-0-19-927313-3 |language=en}}</ref> or more ambiguously '''mantissa''',<ref name="Gosling_1980"/> '''fraction''',<ref name="EE_1961"/><ref name="Savard_2005"/><ref group="nb" name="NB_Fraction"/> or '''characteristic'''<ref name="Knuth_ACP"/><ref name="Gosling_1980"/>) is the first (left) part of a number in [[scientific notation]] or related concepts in [[Floating-point arithmetic|floating-point]] representation, consisting of its [[significant digit]]s. For negative numbers, it does not include the initial minus sign.

Depending on the interpretation of the [[exponent]], the significand may represent an [[integer]] or a [[fractional number]], which may cause the term "mantissa" to be misleading, since [[Mantissa (logarithm)|the ''mantissa'' of a logarithm]] is always its fractional part.<ref>{{Cite book |last=Magazines |first=Hearst |url=https://books.google.com/books?id=Ad4DAAAAMBAJ&pg=PA291 |title=Popular Mechanics |date=February 1913 |publisher=Hearst Magazines |pages=291 |language=en}}</ref><ref>{{Cite book |last=Gupta |first=Dr Alok |url=https://books.google.com/books?id=IVTvDwAAQBAJ |title=Business Mathematics by Alok Gupta: SBPD Publications |date=2020-07-04 |publisher=SBPD publications |isbn=978-93-86908-16-2 |pages=140 |language=en}}</ref> Although the other names mentioned are common, ''significand'' is the word used by [[IEEE 754]], an important technical standard for floating-point arithmetic.<ref>{{Cite book |publisher=IEEE |title=754-1985 - IEEE Standard for Binary Floating-Point Arithmetic|doi=10.1109/IEEESTD.1985.82928 |isbn=0-7381-1165-1 }}</ref> In [[mathematics]], the term "argument" may also be ambiguous, since "the argument of a number" sometimes refers to the length of a circular arc from [[1]] to a number on the [[unit circle]] in the [[complex plane]].<ref>{{Cite book |last1=Gowers |first1=Timothy |url=https://books.google.com/books?id=ZOfUsvemJDMC |title=The Princeton Companion to Mathematics |last2=Barrow-Green |first2=June |last3=Leader |first3=Imre |date=2010-07-18 |publisher=Princeton University Press |isbn=978-1-4008-3039-8 |pages=201 |language=en}}</ref>

==Example==

The number 123.45 can be represented as a [[decimal]] floating-point number with the integer 12345 as the significand and a 10<sup>−2</sup> power term, also called [[characteristic (exponent notation)|characteristic]]s,<ref name="Burks_1946"/><ref name="Schmid_1974"/><ref name="Schmid_1983"/> where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic:

: 123.45 = 12345 × 10<sup>−2</sup>.

This same value can also be represented in [[scientific notation]] with the significand 1.2345 as a fractional coefficient, and +2 as the exponent (and 10 as the base):

: 123.45 = 1'''.'''2345 × 10<sup>+2</sup>.

Schmid, however, called this representation with a significand ranging between 1.0 and 10 a ''modified normalized form''.<ref name="Schmid_1974"/><ref name="Schmid_1983"/>

For base 2, this 1.xxxx form is also called a ''normalized significand''.

Finally, the value can be represented in the format given by the [[Language Independent Arithmetic]] standard and several programming language standards, including [[Ada (programming language)|Ada]], [[C (programming language)|C]], [[Fortran]] and [[Modula-2]], as

: 123.45 = 0'''.'''12345 × 10<sup>+3</sup>.

Schmid called this representation with a significand ranging between 0.1 and 1.0 the ''true normalized form''.<ref name="Schmid_1974"/><ref name="Schmid_1983"/>

==The hidden bit in floating point==

For a [[normalized number]], the most significant digit is always non-zero.  When working in [[binary number|binary]], this constraint uniquely determines this digit to always be 1. As such, it is not explicitly stored, being called the ''[[hidden bit]]''.

The significand is characterized by its width in [[Bit|(binary) digits]], and depending on the context, the hidden bit may or may not be counted toward the width. For example, the same [[IEEE 754]] [[Double-precision floating-point format|double-precision format]] is commonly described as having either a 53-bit significand, including the hidden bit, or a 52-bit significand,{{Citation needed|date=April 2024}} excluding the hidden bit. IEEE 754 defines the precision ''p'' to be the number of digits in the significand, including any implicit leading bit (e.g., ''p'' = 53 for the double-precision format), thus in a way independent from the encoding, and the term to express what is encoded (that is, the significand without its leading bit) is ''trailing significand field''.

==Floating-point mantissa==

In 1914, [[Leonardo Torres Quevedo]] introduced [[floating-point arithmetic]] in his ''Essays on Automatics'',<ref>Torres Quevedo, Leonardo. [https://quickclick.es/rop/pdf/publico/1914/1914_tomoI_2043_01.pdf Automática: Complemento de la Teoría de las Máquinas, (pdf)], pp. 575–583, Revista de Obras Públicas, 19 November 1914.</ref> where he proposed the format ''n''; ''m'', showing the need for a fixed-sized significand as currently used for floating-point data.<ref>Ronald T. Kneusel. ''[https://books.google.com/books?id=eq4ZDgAAQBAJ&dq=leonardo+torres+quevedo++electromechanical+machine+essays&pg=PA84 Numbers and Computers],'' Springer, pp. 84–85, 2017. {{ISBN|978-3319505084}}</ref>

In 1946, [[Arthur Burks]] used the terms ''mantissa'' and ''characteristic'' to describe the two parts of a floating-point number ([[Arthur Burks|Burks]]<ref name="Burks_1946"/> ''et al.'') by analogy with the then-prevalent [[common logarithm]] tables: the ''characteristic'' is the integer part of the logarithm (i.e. the exponent), and the ''mantissa'' is the fractional part. The usage remains common among [[computer scientist]]s today.

The term ''significand'' was introduced by [[George Elmer Forsythe|George Forsythe]] and [[Cleve Barry Moler|Cleve Moler]] in 1967<ref name="Forsythe_Moler_1967"/><ref name="Sterbenz_1974"/><ref name="Goldberg_1991"/><ref name="Savard_2005"/> and is the word used in the IEEE standard<ref name="IEEE-754-2019"/> as the coefficient in front of a scientific notation number discussed above. The fractional part is called the ''fraction''.

To understand both terms, notice that in binary, 1&nbsp;+&nbsp;mantissa&nbsp;≈&nbsp;significand, and the correspondence is exact when storing a power of two. This fact allows for a fast approximation of the base-2 logarithm, leading to algorithms e.g. for computing the [[Methods of computing square roots#Approximations_that_depend_on_the_floating_point_representation|fast square-root]] and [[Fast inverse square root|fast inverse-square-root]]. The implicit leading 1 is nothing but the hidden bit in IEEE 754 floating point, and the bitfield storing the remainder is thus the ''mantissa''.

However, whether or not the implicit 1 is included is a major point of confusion with both terms—and especially so with ''mantissa''. In keeping with the original usage in the context of log tables, it should not be present.

For those contexts where 1 is considered included, [[William Kahan]],<ref name="Kahan_2002"/> lead creator of IEEE 754, and [[Donald E. Knuth]], prominent computer programmer and author of ''[[The Art of Computer Programming]]'',<ref name="Knuth_ACP"/> condemn the use of ''mantissa''. This has led to declining use of the term ''mantissa'' in ''all'' contexts. In particular, the current IEEE 754 standard does not mention it.

==See also==
* [[Mantissa (logarithm)]]

==Notes==
{{Reflist|group="nb"|refs=
<ref group="nb" name="NB_Fraction">The term ''fraction'' is used in [[IEEE 754-1985]] with a different meaning: it is the fractional part of the significand, i.e. the significand without its explicit or implicit leading bit.</ref>
}}

==References==
{{Reflist|refs=
<ref name="Burks_1946">{{cite book |author-last1=Burks |author-first1=Arthur Walter |author-link1=Arthur Walter Burks |author-last2=Goldstine |author-first2=Herman H. |author-link2=Herman Goldstine |author-last3=von Neumann |author-first3=John |author-link3=John von Neumann |orig-date=1946 |title=Preliminary discussion of the logical design of an electronic computing instrument |type=Technical report, Institute for Advanced Study, Princeton, New Jersey, USA<!-- to U. S. Army Ordnance Department --> |chapter=5.3. |series=Collected Works of John von Neumann |volume=5 |editor-first=A. H. |editor-last=Taub |publisher=[[The Macmillan Company]] |publication-place=New York, USA |date=1963 |page=42<!-- 34-79 --> |url=https://www.cs.princeton.edu/courses/archive/fall10/cos375/Burks.pdf |access-date=2016-02-07 |quote=[…] Several of the digital computers being built or planned in this country and England are to contain a so-called "[[floating decimal point]]". This is a mechanism for expressing each word as a [[characteristic (exponent notation)|characteristic]] and a [[mantissa (floating point number)|mantissa]]—e.g. 123.45 would be carried in the machine as (0.12345,03), where the 3 is the exponent of 10 associated with the number. […]}}</ref>
<ref name="Kahan_2002">{{cite web |title=Names for Standardized Floating-Point Formats |author-first=William Morton |author-last=Kahan |author-link=William Morton Kahan |date=2002-04-19 |url=http://www.eecs.berkeley.edu/~wkahan/ieee754status/Names.pdf |access-date=2023-12-27 |url-status=live |archive-url=https://web.archive.org/web/20231227155514/https://people.eecs.berkeley.edu/~wkahan/ieee754status/Names.pdf |archive-date=2023-12-27 |quote=[…] ''m'' is the significand or coefficient or (wrongly) mantissa […]}} (8 pages)</ref>
<ref name="Schmid_1974">{{cite book |title=Decimal Computation |author-first=Hermann |author-last=Schmid<!-- General Electric Company, Binghamton, New York, USA --> |author-link=Hermann Schmid (computer scientist) |date=1974 |edition=1 |publisher=[[John Wiley & Sons, Inc.]] |location=Binghamton, New York, USA |isbn=0-471-76180-X |page=[https://archive.org/details/decimalcomputati0000schm/page/204 204]-205 |url=https://archive.org/details/decimalcomputati0000schm |url-access=registration |access-date=2016-01-03}}</ref>
<ref name="Schmid_1983">{{cite book |title=Decimal Computation |author-first=Hermann |author-last=Schmid<!-- General Electric Company, Binghamton, New York, USA --> |author-link=Hermann Schmid (computer scientist) |orig-date=1974 |date=1983 |edition=1 (reprint) |publisher=Robert E. Krieger Publishing Company |location=Malabar, Florida, USA |isbn=0-89874-318-4 |pages=204–205 |url=https://books.google.com/books?id=uEYZAQAAIAAJ |access-date=2016-01-03}} (NB. At least some batches of this reprint edition were [[misprint]]s with defective pages 115–146.<!-- they contain the contents of another book -->)</ref>
<ref name="Forsythe_Moler_1967">{{cite book |author-first1=George Elmer |author-last1=Forsythe |author-link1=George Elmer Forsythe |author-first2=Cleve Barry |author-last2=Moler |author-link2=Cleve Barry Moler |title=Computer Solution of Linear Algebraic Systems |date=September 1967 |publisher=[[Prentice-Hall]], [[Englewood Cliffs]] |location=New Jersey, USA |edition=1st |series=Automatic Computation |isbn=0-13-165779-8}}</ref>
<ref name="Sterbenz_1974">{{cite book |author-first=Pat H. |author-last=Sterbenz |title=Floating-Point Computation |date=1974-05-01 |edition=1 |series=Prentice-Hall Series in Automatic Computation |publisher=[[Prentice Hall]] |location=Englewood Cliffs, New Jersey, USA |isbn=0-13-322495-3<!-- 978-0-13-322495-5 -->}}</ref>
<ref name="Goldberg_1991">{{cite journal |author-first=David |author-last=Goldberg |author-link=David Goldberg (PARC) |title=What Every Computer Scientist Should Know About Floating-Point Arithmetic |location=[[Xerox Palo Alto Research Center]] (PARC), Palo Alto, California, USA |journal=[[Computing Surveys]] |date=March 1991 |volume=23 |number=1 |page=7 |publisher=[[Association for Computing Machinery, Inc.]] |url=http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf |access-date=2016-07-13 |url-status=live |archive-url=https://web.archive.org/web/20160713044143/http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf |archive-date=2016-07-13 |quote=[…] This term was introduced by [[George Elmer Forsythe|Forsythe]] and [[Cleve Barry Moler|Moler]] [1967], and has generally replaced the older term ''mantissa''. […]}} (NB. A newer edited version can be found here: [https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html])</ref>
<ref name="Savard_2005">{{cite web |title=Floating-Point Formats |at=A Note on Field Designations |author-first=John J. G. |author-last=Savard |date=2018 |orig-date=2005 |work=quadibloc |url=http://www.quadibloc.com/comp/cp0201.htm |access-date=2018-07-16 |url-status=live |archive-url=https://web.archive.org/web/20180703001709/http://www.quadibloc.com/comp/cp0201.htm |archive-date=2018-07-03}}</ref>
<ref name="Gosling_1980">{{cite book |title=Design of Arithmetic Units for Digital Computers |author-first=John B. |author-last=Gosling |editor-first=Frank H. |editor-last=Sumner |date=1980 |edition=1 |publisher=[[The Macmillan Press Ltd]] |location=Department of Computer Science, [[University of Manchester]], Manchester, UK |isbn=0-333-26397-9 |chapter=6.1 Floating-Point Notation / 6.8.5 Exponent Representation |series=Macmillan Computer Science Series |pages=74, 91, 137–138 |quote=[…] In [[floating-point representation]], a number ''x'' is represented by two signed numbers ''m'' and ''e'' such that ''x''&nbsp;= ''m'' &middot; ''b''<sup>''e''</sup> where ''m'' is the [[significand<!-- deliberate self-link! -->|mantissa]], ''e'' the [[exponent]] and ''b'' the [[base (exponentiation)|base]]. […] The mantissa is sometimes termed the characteristic and a version of the exponent also has this title from some authors. It is hoped that the terms here will be unambiguous. […] [w]e use a[n exponent] value which is shifted by half the binary range of the number. […] This special form is sometimes referred to as a [[biased exponent]], since it is the conventional value plus a constant. Some authors have called it a characteristic, but this term should not be used, since [[Control Data Corporation|CDC]] and others use this term for the mantissa. It is also referred to as an '[[Excess-K|excess -]]' representation, where, for example, - is 64 for a 7-bit exponent (2<sup>7&minus;1</sup>&nbsp;= 64). […]}} (NB. Gosling does not mention the term significand at all.)</ref>
<ref name="IEEE-754-2019">{{cite book |title=754-2019 - IEEE Standard for Floating-Point Arithmetic |publisher=[[IEEE]] |isbn=978-1-5044-5924-2 |doi=10.1109/IEEESTD.2019.8766229 |date=2019}}</ref>
<ref name="Knuth_ACP">{{cite book |title=The Art of Computer Programming |title-link=The Art of Computer Programming |author-last=Knuth |author-first=Donald E. |date=1997 |author-link=Donald Ervin Knuth |page=214 |volume=2 |publisher=Addison-Wesley |isbn=0-201-89684-2 |quote=[…] Other names are occasionally used for this purpose, notably 'characteristic' and 'mantissa'; but it is an abuse of terminology to call the fraction part a mantissa, since that term has quite a different meaning in connection with logarithms. Furthermore the English word mantissa means 'a worthless addition.' […]}}</ref>
<ref name="EE_1961">{{cite book |title=English Electric KDF9: Very high speed data processing system for Commerce, Industry, Science |type=Product flyer |date=c. 1961 |publisher=[[English Electric]] |id=Publication No. DP/103. 096320WP/RP0961 |url=http://www.ourcomputerheritage.org/KDF9_Flier.pdf |access-date=2020-07-27 |url-status=live |archive-url=https://web.archive.org/web/20200727143037/http://www.ourcomputerheritage.org/KDF9_Flier.pdf |archive-date=2020-07-27}}</ref>
}}

[[Category:Floating point]]
[[Category:Computer arithmetic]]
[[Category:Mathematical terminology]]