Significand
Encyclopedia
The significand 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
.
floating-point number with an integer significand of 12345 and an exponent of −2. Its value is given by the following arithmetic:
This same value can also be represented in normalized form
with a fractional coefficient of 1.2345 and an exponent of +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:
(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.
. 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 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.
), 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.
Floating point
In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
or a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
.
Examples
The number 123.45 can be represented as a decimalDecimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
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
Scientific notation
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians, doctors, and engineers.In scientific...
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
Ada (programming language)
Ada is a structured, statically typed, imperative, wide-spectrum, and object-oriented high-level computer programming language, extended from Pascal and other languages...
, C
C (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
, Fortran
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
and Modula-2
Modula-2
Modula-2 is a computer programming language designed and developed between 1977 and 1980 by Niklaus Wirth at ETH Zurich as a revision of Pascal to serve as the sole programming language for the operating system and application software for the personal workstation Lilith...
, as:
- 0.12345 × 10+3
Significands and the hidden bit
When working in binary, the significand is characterized by its width in binary digitsBit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
(bits). Because the most significant bit is always 1 for a normalized number
Normalized number
A real number is called normalized, if it is in the form:\pm d_0.d_1d_2d_3\dots\times 10^nwhere n is an integer, d_0, d_1, d_2, d_3... are the digits of the number in base 10, and d_0 is not zero....
, 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
Double precision
In computing, double precision is a computer number format that occupies two adjacent storage locations in computer memory. A double-precision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point .Modern computers with 32-bit storage locations...
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 this usage remains common in computing and among computer scientistsComputer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
. However, this use of mantissa is discouraged by the IEEE floating-point standard
IEEE floating-point standard
IEEE 754–1985 was an industry standard for representingfloating-pointnumbers in computers, officially adopted in 1985 and superseded in 2008 byIEEE 754-2008. During its 23 years, it was the most widely used format for...
committee and by some professionals such as William Kahan
William Kahan
William Morton Kahan is a mathematician and computer scientist who received the Turing Award in 1989 for "his fundamental contributions to numerical analysis", and was named an ACM Fellow in 1994....
and Donald Knuth
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
, because it conflicts with the pre-existing use of mantissa for the fractional part
Fractional part
All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...
of a logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
(see also common logarithm
Common logarithm
The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L...
).
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 (OEDOxford English Dictionary
The Oxford English Dictionary , published by the Oxford University Press, is the self-styled premier dictionary of the English language. Two fully bound print editions of the OED have been published under its current name, in 1928 and 1989. The first edition was published in twelve volumes , and...
), from its general English meaning, now archaic, of "minor addition". This meaning stemmed from the Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
word for "makeweight", which in turn may have come from Etruscan
Etruscan language
The Etruscan language was spoken and written by the Etruscan civilization, in what is present-day Italy, in the ancient region of Etruria and in parts of Lombardy, Veneto, and Emilia-Romagna...
. Significand is a 20th century neologism.