Significand

Significand

The significand (also coefficient or mantissa) is part of a floating-point number, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer or a fraction.

Contents

Examples

The number 123.45 can be represented as a decimal floating-point number with an integer significand of 12345 and an exponent of −2. Its value is given by the following arithmetic:

12345 × 10−2

This same value can also be represented in normalized form with a fractional coefficient of 1.2345 and an exponent of +2:

1.2345 × 10+2

Finally, this 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:

0.12345 × 10+3

Significands and the hidden bit

When working in binary, the significand is characterized by its width in binary digits (bits). Because the most significant bit is always 1 for a normalized number, this bit is not typically stored and is called the "hidden bit". Depending on the context, the hidden bit may or may not be counted towards the width of the significand. 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, not including the hidden bit. The notion of a hidden bit only applies to binary representations.

Use of "mantissa"

In American English, the original word for this seems to have been mantissa (Burks et al.), 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,[citation needed] because it conflicts with the pre-existing use of mantissa for the fractional part of a logarithm (see also common logarithm).

The fractional part of a logarithm, the original meaning of mantissa, is equal to the logarithm of the significand (for the same base) plus a constant depending on the normalization. By contrast, the relationship between the floating-point exponent and the integer part of the logarithm is not affected by normalization.

Etymology

The logarithmic meaning of mantissa dates to the 18th century (OED[Full citation needed]), from its general English meaning, now archaic, of "minor addition". This meaning stemmed from the Latin word for "makeweight", which in turn may have come from Etruscan. Significand is a 20th century neologism.

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.'

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