Units of information

In computing and telecommunications, a unit of information is the capacity of some standard data storage system or communication channel, used to measure the capacities of other systems and channels. In information theory, units of information are also used to measure the information contents or entropy of random variables.

The most common units are the bit, the capacity of a system which can exist in only two states, and the byte (or octet), which is equivalent to eight bits. Multiples of these units can be formed from these with the SI prefixes (power-of-ten prefixes) or the newer IEC binary prefixes (binary power prefixes). Information capacity is a dimensionless quantity.

Primary units

Comparison of units of information: bit, trit, nat, ban. Quantity of information is the height of bars. Dark green level is the "Nat" unit.

In 1928, Ralph Hartley observed a fundamental storage principle,^[1] which was further formalized by Claude Shannon in 1945: the information that can be stored in a system is proportional to the logarithm log_b N of the number N of possible states of that system. Changing the basis of the logarithm from b to a different number c has the effect of multiplying the value of the logarithm by a fixed constant, namely log_c N = (log_c b) log_b N. Therefore, the choice of the basis b determines the unit used to measure information. In particular, if b is a positive integer, then the unit is the amount of information that can be stored in a system with N possible states.

When b is 2, the unit is the shannon, equal to the information content of one "bit" (a contraction of binary digit^[2]). A system with 8 possible states, for example, can store up to log₂8 = 3 bits of information. Other units that have been named include:

• Base b = 3: the unit is called "trit", and is equal to log₂ 3 (≈ 1.585) bits.^[3]
• Base b = 10: the unit is called decimal digit, hartley, ban, decit, or dit, and is equal to log₂ 10 (≈ 3.322) bits.^[1][4][5][6]
• Base b = e, the base of natural logarithms: the unit is called a nat, nit, or nepit (from Neperian), and is worth log₂ e (≈ 1.443) bits.^[1]

The trit, ban, and nat are rarely used to measure storage capacity; but the nat, in particular, is often used in information theory, because natural logarithms are sometimes more convenient than logarithms in other bases.

Units derived from bit

Several conventional names are used for collections or groups of bits.

Shave and a haircut

The name given to 2 bits. The name is a reference to the song, "Shave and a haircut".

Byte

Historically, a byte was the number of bits used to encode a character of text in the computer, which depended on computer hardware architecture; but today it almost always means eight bits — that is, an octet. A byte can represent 256 (2⁸) distinct values, such as the integers 0 to 255, or -128 to 127. The IEEE 1541-2002 standard specifies "B" (upper case) as the symbol for byte. Bytes, or multiples thereof, are almost always used to specify the sizes of computer files and the capacity of storage units. Most modern computers and peripheral devices are designed to manipulate data in whole bytes or groups of bytes, rather than individual bits.

Nibble

A group of four bits, or half a byte, is sometimes called a nibble or nybble. This unit is most often used in the context of hexadecimal number representations, since a nibble has the same amount of information as one hexadecimal digit.^[7]

Word, block, and page

Computers usually manipulate bits in groups of a fixed size, conventionally called words. The number of bits in a word is usually defined by the size of the registers in the computer's CPU, or by the number of data bits that are fetched from its main memory in a single operation. In the IA-32 architecture more commonly known as x86-32, a word is 16 bits, but other past and current architectures use words with 8, 24, 32, 36, 56, 64, 80 bits or others.

Some machine instructions and computer number formats use two words (a "double word" or "dword"), or four words (a "quad word" or "quad").

Computer memory caches usually operate on blocks of memory that consist of several consecutive words. These units are customarily called cache blocks, or, in CPU caches, cache lines.

Virtual memory systems partition the computer's main storage into even larger units, traditionally called pages.

Systematic multiples

Terms for large quantities of bits can be formed using the standard range of SI prefixes for powers of 10, e.g., kilo = 10³ = 1000 (as in kilobit or kbit), mega- = 10⁶ = 1000000 (as in megabit or Mbit) and giga = 10⁹ = 1000000000 (as in gigabit or Gbit). These prefixes are more often used for multiples of bytes, as in kilobyte (1 kB = 8000 bit), megabyte (1 MB = 8000000bit), and gigabyte (1 GB = 8000000000bit).

However, for technical reasons, the capacities of computer memories and some storage units are often multiples of some large power of two, such as 2²⁸ = 268435456 bytes. To avoid such unwieldy numbers, people have often misused the SI prefixes to mean the nearest power of two, e.g., using the prefix kilo for 2¹⁰ = 1024, mega for 2²⁰ = 1048576, and giga for 2³⁰ = 1073741824, and so on. For example, a random access memory chip with a capacity of 2²⁸ bytes would be referred to as a 256-megabyte chip. The table below illustrates these differences.

Multiples of bits
Decimal Value SI 1000 10^3 kbit kilobit 1000^2 10^6 Mbit megabit 1000^3 10^9 Gbit gigabit 1000^4 10^12 Tbit terabit 1000^5 10^15 Pbit petabit 1000^6 10^18 Ebit exabit 1000^7 10^21 Zbit zettabit 1000^8 10^24 Ybit yottabit	Binary Value IEC JEDEC 1024 2^10 Kibit kibibit Kbit kilobit 1024^2 2^20 Mibit mebibit Mbit megabit 1024^3 2^30 Gibit gibibit Gbit gigabit 1024^4 2^40 Tibit tebibit - 1024^5 2^50 Pibit pebibit - 1024^6 2^60 Eibit exbibit - 1024^7 2^70 Zibit zebibit - 1024^8 2^80 Yibit yobibit -
See also Nibble Byte Orders of magnitude of data

Symbol	Prefix	SI Meaning	Binary meaning	Size difference
k	kilo	10^3   = 1000^1	2^10 = 1024^1	2.40%
M	mega	10^6   = 1000^2	2^20 = 1024^2	4.86%
G	giga	10^9   = 1000^3	2^30 = 1024^3	7.37%
T	tera	10^12 = 1000^4	2^40 = 1024^4	9.95%
P	peta	10^15 = 1000^5	2^50 = 1024^5	12.59%
E	exa	10^18 = 1000^6	2^60 = 1024^6	15.29%
Z	zetta	10^21 = 1000^7	2^70 = 1024^7	18.06%
Y	yotta	10^24 = 1000^8	2^80 = 1024^8	20.89%

In the past, uppercase K has been used instead of lowercase k to indicate 1024 instead of 1000. However, this usage was never consistently applied.

On the other hand, for external storage systems (such as optical discs), the SI prefixes were commonly used with their decimal values (powers of 10). There have been many attempts to resolve the confusion by providing alternative notations for power-of-two multiples. In 1998 the International Electrotechnical Commission (IEC) issued a standard for this purpose, namely a series of binary prefixes that use 1024 instead of 1000 as the main radix:^[8]

Symbol	Prefix
Ki	kibi, binary kilo	1 kibibyte (KiB)	2^10 bytes	1024 B
Mi	mebi, binary mega	1 mebibyte (MiB)	2^20 bytes	1024 KiB
Gi	gibi, binary giga	1 gibibyte (GiB)	2^30 bytes	1024 MiB
Ti	tebi, binary tera	1 tebibyte (TiB)	2^40 bytes	1024 GiB
Pi	pebi, binary peta	1 pebibyte (PiB)	2^50 bytes	1024 TiB
Ei	exbi, binary exa	1 exbibyte (EiB)	2^60 bytes	1024 PiB

The JEDEC memory standards however define uppercase K, M, and G for the binary powers 2¹⁰, 2²⁰ and 2³⁰ to reflect common usage.^[9]

Size examples

• 1 bit – answer to a yes/no question
• 1 byte – a number from 0 to 255.
• 90 bytes: enough to store a typical line of text from a book.
• 512 bytes = ½ KiB: the typical sector of a hard disk.
• 1024 bytes = 1 KiB: the classical block size in UNIX filesystems.
• 2048 bytes = 2 KiB: a CD-ROM sector.
• 4096 bytes = 4 KiB: a memory page in x86 (since Intel 80386).
• 4 kB: about one page of text from a novel.
• 120 kB: the text of a typical pocket book.
• 1 MB – a 1024×1024 pixel bitmap image with 256 colors (8 bpp color depth).
• 3 MB – a three-minute song (133 kbit/s)
• 650-900 MB – a CD-ROM
• 1 GB – 114 minutes of uncompressed CD-quality audio at 1.4 Mbit/s
• 8/16 GB – size of a normal flash drive
• 4 TB – the size of a $150 hard disk (as of late 2014)
• 1.3 ZB – prediction of the volume of the whole internet in 2016.

Obsolete and unusual units

Several other units of information storage have been named.^[7]

• 1 bit: sniff.
• 2 bits: crumb,^[10] quad, quarter, tayste, tydbit, semi-nibble.
• 3 bits: triad, triade^[11][12]
• 5 bits: nickel, nyckle.
• 6 bits: byte (in early IBM machines using BCD alphamerics).
• 10 bits: declet,^{[13][14][15][16]} decle,^[17] deckle, dyme.
• 12 bits: slab^[18][19][20]
• 16 bits: wyde,^[21] doublet,^[22] plate, playte, chomp, chawmp (on a 32-bit machine).
• 18 bits: chomp, chawmp (on a 36-bit machine).
• 32 bits: quadlet,^[22] dinner, dynner, gawble (on a 32-bit machine).
• 48 bits: gobble, gawble (under circumstances that remain obscure).
• 64 bits: octlet.^[22]
• 128 bits: hexlet.^[22]
• 16 bytes: paragraph.
• 6 trits: tryte^[23]
• combit, comword^[24][25][26]

Most of these names are jargon, obsolete, or used only in very restricted contexts.

References

1. Norman Abramson (1963), Information theory and coding. McGraw-Hill.
2. Mackenzie, Charles E. (1980). Coded Character Sets, History and Development. The Systems Programming Series (1 ed.). Addison-Wesley Publishing Company, Inc. p. xii. ISBN 0-201-14460-3. LCCN 77-90165. ISBN 978-0-201-14460-4. Retrieved 2016-05-22.
3. Donald E. Knuth, The Art of Computer Programming, vol.2: Seminumerical algorithms.
4. Shanmugam (2006), Digital and Analog Computer Systems.
5. Gregg Jaeger (2007), Quantum information: an overview
6. I. Ravi Kumar (2001), Comprehensive Statistical Theory of Communication.
7. Nybble at dictionary reference.com; sourced from Jargon File 4.2.0, accessed 2007-08-12
8. ISO/IEC standard is ISO/IEC 80000-13:2008. This standard cancels and replaces subclauses 3.8 and 3.9 of IEC 60027-2:2005. The only significant change is the addition of explicit definitions for some quantities. ISO Online Catalogue
9. JEDEC Solid State Technology Association (December 2002). "Terms, Definitions, and Letter Symbols for Microcomputers, Microprocessors, and Memory Integrated Circuits" (PDF). JESD 100B.01. Retrieved 2009-04-05
10. Weisstein, Eric. W. "Crumb". MathWorld. Retrieved 2015-08-02.
11. Paul, Reinhold (2013). "Elektrotechnik und Elektronik für Informatiker - Grundgebiete der Elektronik". Leitfaden der Informatik. B.G. Teubner Stuttgart / Springer. ISBN 3322966526. 9783322966520. Retrieved 2015-08-03.
12. Böhme, Gert; Born, Werner; Wagner, B.; Schwarze, G. (2013-07-02) [1969]. Jürgen Reichenbach, ed. Programmierung von Prozeßrechnern. Reihe Automatisierungstechnik (in German). 79. VEB Verlag Technik Berlin, reprint: Springer Verlag. doi:10.1007/978-3-663-02721-8. ISBN 978-3-663-00808-8. 9/3/4185.
13. IEEE 754-2008 - IEEE Standard for Floating-Point Arithmetic. 2008-08-29. doi:10.1109/IEEESTD.2008.4610935. ISBN 978-0-7381-5752-8. Retrieved 2016-02-10.
14. Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010). Handbook of Floating-Point Arithmetic (1 ed.). Birkhäuser. doi:10.1007/978-0-8176-4705-6. ISBN 978-0-8176-4704-9. LCCN 2009939668.
15. Erle, Mark A. (2008-11-21). Algorithms and Hardware Designs for Decimal Multiplication (Thesis). Lehigh University: ProQuest (published 2009). ISBN 9781109042283. 1109042280. Retrieved 2016-02-10.
16. Kneusel, Ronald T. (2015). Numbers and Computers. Springer. ISBN 9783319172606. 3319172603. Retrieved 2016-02-10.
17. Joe Zbiciak. "AS1600 Quick-and-Dirty Documentation". Retrieved 2013-04-28.
18. "315 Electronic Data Processing System" (PDF). NCR. November 1965. NCR MPN ST-5008-15. Archived (PDF) from the original on 2016-05-24. Retrieved 2015-01-28.
19. Bardin, Hillel (1963). "NCR 315 Seminar" (PDF). Computer Usage Communique. 2 (3). Archived (PDF) from the original on 2016-05-24.
20. Schneider, Carl (2013) [1970]. Datenverarbeitungs-Lexikon [Lexicon of information technology] (in German) (softcover reprint of hardcover 1st ed.). Wiesbaden, Germany: Springer Fachmedien Wiesbaden GmbH / Betriebswirtschaftlicher Verlag Dr. Th. Gabler GmbH. pp. 201, 308. doi:10.1007/978-3-663-13618-7. ISBN 978-3-409-31831-0. ISBN 3-663-13618-3 / ISBN 978-3-663-13618-7 (ebook). Retrieved 2016-05-24. slab, Abk. aus syllable = Silbe, die kleinste adressierbare Informationseinheit für 12 bit zur Übertragung von zwei Alphazeichen oder drei numerischen Zeichen. (NCR) […] Hardware: Datenstruktur: NCR 315-100 / NCR 315-RMC; Wortlänge: Silbe; Bits: 12; Bytes: –; Dezimalziffern: 3; Zeichen: 2; Gleitkommadarstellung: fest verdrahtet; Mantisse: 4 Silben; Exponent: 1 Silbe (11 Stellen + 1 Vorzeichen) [slab, abbr. for syllable = syllable, smallest addressable information unit for 12 bits for the transfer of two alphabetical characters or three numerical characters. (NCR) […] Hardware: Data structure: NCR 315-100 / NCR 315-RMC; Word length: Syllable; Bits: 12; Bytes: –; Decimal digits: 3; Characters: 2; Floating point format: hard-wired; Significand: 4 syllables; Exponent: 1 syllable (11 digits + 1 prefix)]
21. The Art of Computer Programming.
22. IEEE Std 1754-1994 - IEEE Standard for a 32-bit Microcontroller Architecture. The Institute of Electrical and Electronic Engineers, Inc. pp. 5–7. doi:10.1109/IEEESTD.1995.79519. ISBN 1-55937-428-4. Retrieved 2016-02-10. (NB. The standard defines doublets, quadlets, octlets and hexlets as 2, 4, 8 and 16 bytes, giving the numbers of bits (16, 32, 64 and 128) only as a secondary meaning. This might be important given that bytes were not always understood to mean 8 bits (octets) historically.)
23. Brousentsov, N. P.; Maslov, S. P.; Ramil Alvarez, J.; Zhogolev, E.A. "Development of ternary computers at Moscow State University". Retrieved 2010-01-20.
24. US4319227, Malinowski, Christopher W.; Heinz Rinderle & Martin Siegle, "Three-state signaling system", issued 1982-03-09, assigned to Department of Research and Development, AEG-Telefunken, Heilbronn, Germany 
25. http://www.google.com/patents/US4319227
26. http://patentimages.storage.googleapis.com/pdfs/US4319227.pdf

External links

• Representation of numerical values and SI units in character strings for information interchanges
• Bit Calculator - Convert between bit, byte, kilobit, kilobyte, megabit, megabyte, gigabit, gigabyte
• Paper on standardized units for use in information technology
• Data Byte Converter