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. In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... For the traditional use of the word mantissa in mathematics, see common logarithm. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... The integers are commonly denoted by the above symbol. ... A cake divided into four equal quarters. ...

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 formula: The decimal (base ten or occasionally denary) numeral system has ten as its base. ...

12345 × 10⁻²

This same value could also be represented in normalized form with the fractional (non-integer) coefficient 1.2345 and exponent +2: Scientific notation is a scheme for writing numbers that is often used by scientists and mathematicians to easily write large and small numbers. ...

1.2345 × 10⁺²

Significands and the hidden bit

In describing binary floating-point types, the significand is characterized 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). The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... This article is about the unit of information. ... In computing, a normal number is a non-zero number in a floating-point representation which is within the balanced range supported by a given floating-point format. ... This article or section is in need of attention from an expert on the subject. ... The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. ... In computing, double precision is a computer numbering format that occupies two storage locations in computer memory at address and address+1. ...

Use of "mantissa"

main article: 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). For the traditional use of the word mantissa in mathematics, see common logarithm. ... 2005 (MMV) was a common year starting on Saturday of the Gregorian calendar. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. ... William Velvel Kahan (born June 5, 1933, in Toronto, Ontario, Canada) is an eminent mathematician and computer scientist. ... Donald Ervin Knuth ( or Ka-NOOTH[1], Chinese: [2]) (b. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, the common logarithm is the logarithm with base 10. ...

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. The Oxford English Dictionary print set The Oxford English Dictionary (OED) is a dictionary published by the Oxford University Press (OUP), and is generally regarded as the most comprehensive and scholarly dictionary of the English language. ... Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome. ... Languages in Iron Age Italy, 6th century BC Etruscan was a language spoken and written in the ancient region of Etruria (current Tuscany plus western Umbria and northern Latium) and in parts of what are now Lombardy, Veneto, and Emilia-Romagna (where the Etruscans were displaced by Gauls), in Italy. ... A neologism (Greek νεολογισμός [neologismos], from νέος [neos] new + λόγος [logos] word, speech, discourse + suffix -ισμός [-ismos] -ism) is a word, term, or phrase which has been recently created (coined) — often to apply to new concepts, to synthesize pre-existing concepts, or to make older terminology sound more contemporary. ...

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.'