Significand

The number 123.45 can be represented as a decimal floating-point number with the integer 12345 as the significand and a 10⁻² power term, also called characteristics, where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic:

: 123.45 = 12345 × 10⁻².

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⁺².

Schmid, however, called this representation with a significand ranging between 1.0 and 10 a modified normalized form.

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, C, Fortran and Modula-2, as

: 123.45 = 0.12345 × 10⁺³.

Schmid called this representation with a significand ranging between 0.1 and 1.0 the true normalized form.

For a normalized number, the most significant digit is always non-zero. When working in 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 (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 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.

In 1914, Leonardo Torres Quevedo introduced floating-point arithmetic in his Essays on Automatics,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. where he proposed the format n; m, showing the need for a fixed-sized significand as currently used for floating-point data.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}}

In 1946, Arthur Burks used the terms mantissa and characteristic to describe the two parts of a floating-point number (Burks 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 scientists today.

The term significand was introduced by George Forsythe and Cleve Moler in 1967 and is the word used in the IEEE standard 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 + mantissa ≈ 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 fast square-root and 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, lead creator of IEEE 754, and Donald E. Knuth, prominent computer programmer and author of The Art of Computer Programming, 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.

• Mantissa (logarithm)

{{Reflist|group="nb"|refs=

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.

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{{Reflist|refs=

{{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 |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 |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 and a 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. […]}}

{{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)

{{cite book |title=Decimal Computation |author-first=Hermann |author-last=Schmid |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}}

{{cite book |title=Decimal Computation |author-first=Hermann |author-last=Schmid |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 misprints with defective pages 115–146.)

{{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}}

{{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}}

{{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 Forsythe and 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])

{{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}}

{{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 = m · b^e where m is the mantissa, e the exponent and b the 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 CDC and others use this term for the mantissa. It is also referred to as an 'excess -' representation, where, for example, - is 64 for a 7-bit exponent (2⁷⁻¹ = 64). […]}} (NB. Gosling does not mention the term significand at all.)

{{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}}

{{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.' […]}}

{{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}}

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Category:Floating point

Category:Computer arithmetic

Category:Mathematical terminology