Significand

Example

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

: 123.45 = 12345 × 10−2.

This 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.

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.

The hidden bit in floating point

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, 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

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

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

Notes

References

References

1. Clements, Alan. (2006-02-09). "Principles of Computer Hardware". OUP Oxford.
2. Magazines, Hearst. (February 1913). "Popular Mechanics". Hearst Magazines.
3. Gupta, Dr Alok. (2020-07-04). "Business Mathematics by Alok Gupta: SBPD Publications". SBPD publications.
4. "754-1985 - IEEE Standard for Binary Floating-Point Arithmetic". IEEE.
5. (2010-07-18). "The Princeton Companion to Mathematics". Princeton University Press.
6. Torres Quevedo, Leonardo. [https://quickclick.es/rop/pdf/publico/1914/1914_tomoI_2043_01.pdf Automática: Complemento de la Teoría de las Máquinas, (pdf)], pp. 575–583, Revista de Obras Públicas, 19 November 1914.
7. 978-3319505084
8. (1963). "Preliminary discussion of the logical design of an electronic computing instrument". [[The Macmillan Company]].
9. (2002-04-19). "Names for Standardized Floating-Point Formats".
10. (1974). "Decimal Computation". [[John Wiley & Sons, Inc.]].
11. (1983). "Decimal Computation". Robert E. Krieger Publishing Company.
12. (September 1967). "Computer Solution of Linear Algebraic Systems". [[Prentice-Hall]], [[Englewood Cliffs]].
13. (1974-05-01). "Floating-Point Computation". [[Prentice Hall]].
14. (March 1991). "What Every Computer Scientist Should Know About Floating-Point Arithmetic". [[Association for Computing Machinery, Inc.]].
15. (2018). "Floating-Point Formats". quadibloc.
16. (1980). "Design of Arithmetic Units for Digital Computers". [[The Macmillan Press Ltd]].
17. (2019). "754-2019 - IEEE Standard for Floating-Point Arithmetic". [[IEEE]].
18. (1997). "The Art of Computer Programming". Addison-Wesley.
19. (c. 1961). "English Electric KDF9: Very high speed data processing system for Commerce, Industry, Science". [[English Electric]].