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The significand (also coefficient or mantissa) is the part of a floating-point number that contains its significant digits. Depending on the interpretation of the exponent, the significand may be considered to be an integer or a fraction.
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Examples
For example, the number 123.45 can be represented as a decimal floating-point number with integer significand 12345 and exponent −2. Its value is given by the arithmetic:
- 12345 × 10−2
This same value could also be represented in normalized form with the fractional (non-integer) coefficient 1.2345 and exponent +2:
- 1.2345 × 10+2
or in the format given by the Language Independent Arithmetic standard and several programming language standards (including Ada, C, Fortran and Modula-2) as:
- .12345 × 10+3
Significands and the hidden bit
In describing binary floating-point types, the significand is characterised by a certain width in bits (binary digits). Because the most significant bit is always 1 for normal numbers, this bit is typically not stored and is called the "hidden bit". Depending on the context, the hidden bit may or may not be counted in describing the width of the significand. For example, the same IEEE 754 double precision format is commonly described either as having a 53-bit significand/mantissa (including the hidden bit) or as having a 52-bit significand/mantissa (not including the hidden bit).
Use of "mantissa"
The original word used in American English to describe the coefficient of floating-point numbers in computer hardware, later called the significand, seems to have been mantissa (see Burks et al., below), and as of 2005 this usage remains common in computing and among computer scientists. However, this use of mantissa is discouraged by the IEEE floating-point standard committee and by some professionals such as William Kahan and Donald Knuth, because it conflicts with the pre-existing usage of mantissa for the fractional part of a logarithm (see also common logarithm).
The older meaning of mantissa is related to the IEEE's significand in that the fractional part of a logarithm is the logarithm of the significand for the same base, plus a constant depending on the normalization. (The integer part of the logarithm requires no such manipulation to relate to the floating-point exponent.)
The logarithmic meaning of mantissa dates to the 18th century (according to the OED), from its general English meaning (now archaic) of "minor addition", which stemmed from the Latin word for "makeweight" (which in turn may have come from Etruscan). Significand is a 20th century neologism.
Unfortunately, while significand was apparently coined on the model of earlier words like multiplicand, it is not etymologically correct since it means "that which is to be signified" rather than that which is significant.
References
- Burks, Arthur W.; Goldstine, Herman H.; Von Neumann, John (1946). Preliminary discussion of the logical design of an electronic computing instrument. Technical Report, Institute for Advanced Study, Princeton, NJ. In Von Neumann, Collected Works, Vol. 5, A. H. Taub, ed., MacMillan, New York, 1963, p. 42:
- 5.3. '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.'
Wikipedia content modification information:
- This page was last modified on 19 November 2008, at 21:04.
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