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Significand

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The significand[1] (also coefficient,[1] sometimes argument,[2] or more ambiguously mantissa,[3] fraction,[4][5][nb 1] or characteristic[6][3]) is the first (left) part of a number in scientific notation or related concepts in floating-point representation, consisting of its significant digits.

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 the mantissa of a logarithm is always its fractional part.[7][8] Although the other names mentioned are common, significand is the word used by IEEE 754, an important technical standard for floating-point arithmetic.[9] 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.[10]

Example

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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,[11][12][13] where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic:

123.45 = 12345 × 10−2.

The 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.[12][13]

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.[12][13]

The hidden bit in floating point

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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] 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

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In 1914, Leonardo Torres Quevedo introduced floating-point arithmetic in his Essays on Automatics,[14] where he proposed the format n; m, showing the need for a fixed-sized significand as currently used for floating-point data.[15]

In 1946, Arthur Burks used the terms mantissa and characteristic to describe the two parts of a floating-point number (Burks[11] 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[16][17][18][5] and is the word used in the IEEE standard[19] 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,[1] lead creator of IEEE 754, and Donald E. Knuth, prominent computer programmer and author of The Art of Computer Programming,[6] 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

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Notes

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  1. ^ 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.

References

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  1. ^ a b c Kahan, William Morton (2002-04-19). "Names for Standardized Floating-Point Formats" (PDF). Archived (PDF) from the original on 2023-12-27. Retrieved 2023-12-27. […] m is the significand or coefficient or (wrongly) mantissa […] (8 pages)
  2. ^ Clements, Alan (2006-02-09). Principles of Computer Hardware. OUP Oxford. ISBN 978-0-19-927313-3.
  3. ^ a b Gosling, John B. (1980). "6.1 Floating-Point Notation / 6.8.5 Exponent Representation". In Sumner, Frank H. (ed.). Design of Arithmetic Units for Digital Computers. Macmillan Computer Science Series (1 ed.). Department of Computer Science, University of Manchester, Manchester, UK: The Macmillan Press Ltd. pp. 74, 91, 137–138. ISBN 0-333-26397-9. […] In floating-point representation, a number x is represented by two signed numbers m and e such that x = m · be 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.)
  4. ^ English Electric KDF9: Very high speed data processing system for Commerce, Industry, Science (PDF) (Product flyer). English Electric. c. 1961. Publication No. DP/103. 096320WP/RP0961. Archived (PDF) from the original on 2020-07-27. Retrieved 2020-07-27.
  5. ^ a b Savard, John J. G. (2018) [2005]. "Floating-Point Formats". quadibloc. A Note on Field Designations. Archived from the original on 2018-07-03. Retrieved 2018-07-16.
  6. ^ a b Knuth, Donald E. (1997). The Art of Computer Programming. Vol. 2. Addison-Wesley. p. 214. ISBN 0-201-89684-2. […] 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.' […]
  7. ^ Magazines, Hearst (February 1913). Popular Mechanics. Hearst Magazines. p. 291.
  8. ^ Gupta, Dr Alok (2020-07-04). Business Mathematics by Alok Gupta: SBPD Publications. SBPD publications. p. 140. ISBN 978-93-86908-16-2.
  9. ^ IEEE. IEEE 754-1985. doi:10.1109/IEEESTD.1985.82928. ISBN 0-7381-1165-1. Retrieved 2024-11-01. {{cite book}}: |website= ignored (help)
  10. ^ Gowers, Timothy; Barrow-Green, June; Leader, Imre (2010-07-18). The Princeton Companion to Mathematics. Princeton University Press. p. 201. ISBN 978-1-4008-3039-8.
  11. ^ a b Burks, Arthur Walter; Goldstine, Herman H.; von Neumann, John (1963) [1946]. "5.3.". In Taub, A. H. (ed.). Preliminary discussion of the logical design of an electronic computing instrument (PDF) (Technical report, Institute for Advanced Study, Princeton, New Jersey, USA). Collected Works of John von Neumann. Vol. 5. New York, USA: The Macmillan Company. p. 42. Retrieved 2016-02-07. […] 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. […]
  12. ^ a b c Schmid, Hermann (1974). Decimal Computation (1 ed.). Binghamton, New York, USA: John Wiley & Sons, Inc. p. 204-205. ISBN 0-471-76180-X. Retrieved 2016-01-03.
  13. ^ a b c Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. pp. 204–205. ISBN 0-89874-318-4. Retrieved 2016-01-03. (NB. At least some batches of this reprint edition were misprints with defective pages 115–146.)
  14. ^ Torres Quevedo, Leonardo. Automática: Complemento de la Teoría de las Máquinas, (pdf), pp. 575–583, Revista de Obras Públicas, 19 November 1914.
  15. ^ Ronald T. Kneusel. Numbers and Computers, Springer, pp. 84–85, 2017. ISBN 978-3319505084
  16. ^ Forsythe, George Elmer; Moler, Cleve Barry (September 1967). Computer Solution of Linear Algebraic Systems. Automatic Computation (1st ed.). New Jersey, USA: Prentice-Hall, Englewood Cliffs. ISBN 0-13-165779-8.
  17. ^ Sterbenz, Pat H. (1974-05-01). Floating-Point Computation. Prentice-Hall Series in Automatic Computation (1 ed.). Englewood Cliffs, New Jersey, USA: Prentice Hall. ISBN 0-13-322495-3.
  18. ^ Goldberg, David (March 1991). "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (PDF). Computing Surveys. 23 (1). Xerox Palo Alto Research Center (PARC), Palo Alto, California, USA: Association for Computing Machinery, Inc.: 7. Archived (PDF) from the original on 2016-07-13. Retrieved 2016-07-13. […] 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: [1])
  19. ^ 754-2019 - IEEE Standard for Floating-Point Arithmetic. IEEE. 2019. doi:10.1109/IEEESTD.2019.8766229. ISBN 978-1-5044-5924-2.
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