Binary prefix

In computing, binary prefixes can be used to quantify large numbers where powers of two are more useful than powers of ten. Each successive prefix is multiplied by 1024 (2¹⁰) rather than the 1000 (10³) used by the SI prefix system. Despite the ambiguity, binary prefixes are often written and pronounced identically to the SI prefixes, rather than using the system described below.

History

Using the prefixes kilo-, mega-, giga-, etc., and their symbols K, M, G, etc. (see below for the peculiarities of "K"), in the binary sense can cause serious confusion.

In January 1999, the International Electrotechnical Commission introduced the prefixes kibi-, mebi-, gibi-, etc., and the symbols Ki, Mi, Gi, etc. to specify binary multiples of a quantity. They have since been officially adopted by many other organizations, most notably the Institute of Electrical and Electronics Engineers, see standard IEEE 1541. However, they have not been widely adopted and many people continue to use the SI prefixes in a binary sense, despite the lack of support from official bodies. As a result, there is no unambiguous notation for decimal multiples of bits and bytes.

The names and values of the SI prefixes were defined in the 1960 SI standard, with powers-of-1000 values. As of 2005, standard dictionaries do not recognize the binary meanings for these prefixes.

Binary prefixes using SI symbols (non-standard, but common)

Name	Symbol	Value	Base 16	Base 10
kilo	k/K	210 = 1,024	= 162.5	> 103
mega	M	220 = 1,048,576	= 165	> 106
giga	G	230 = 1,073,741,824	= 167.5	> 109
tera	T	240 = 1,099,511,627,776	= 1610	> 1012
peta	P	250 = 1,125,899,906,842,624	= 1612.5	> 1015
exa	E	260 = 1,152,921,504,606,846,976	= 1615	> 1018
zetta	Z	270 = 1,180,591,620,717,411,303,424	= 1617.5	> 1021
yotta	Y	280 = 1,208,925,819,614,629,174,706,176	= 1620	> 1024

The one-letter abbreviations are identical to SI prefixes, except for "K", which is used interchangeably with "k" (in SI, "K" stands for the kelvin, and only "k" stands for 1,000). Some have suggested that "k" be used for 1,000, and "K" for 1,024, but this cannot be extended to the higher order prefixes and has never been widely recognised.

Notice that as the order of magnitude increases, the percentage difference between the binary and decimal values of a prefix increases, from 2.4 % (with the kilo prefix) to over 20 % (with the yotta prefix).

Approximate ratios between binary prefixes and their decimal homonyms

Name	Bin ÷ Dec	Dec ÷ Bin	Example	Percentage difference
kilobyte : kibibyte	1.024	0.976	100 KB ~ 97.6 KiB	+2.4 % or -2.3 %
megabyte : mebibyte	1.049	0.954	100 MB ~ 95.4 MiB	+4.9 % or -4.6 %
gigabyte : gibibyte	1.074	0.931	100 GB ~ 93.1 GiB	+7.4 % or -6.9 %
terabyte : tebibyte	1.100	0.909	100 TB ~ 90.9 TiB	+10 % or -9.1 %
petabyte : pebibyte	1.126	0.888	100 PB ~ 88.8 PiB	+12.6 % or -11.2 %
exabyte : exbibyte	1.153	0.867	100 EB ~ 86.7 EiB	+15.3 % or -13.3 %
zettabyte : zebibyte	1.181	0.847	100 ZB ~ 84.7 ZiB	+18.1 % or -15.3 %
yottabyte : yobibyte	1.209	0.827	100 YB ~ 82.7 YiB	+20.9 % or -17.3 %

Informally, the prefixes are often used on their own. Thus one might hear about "a 40K file" (40 binary kilobytes) or "a 2M Internet connection" (2 decimal megabits per second). What units are being used, and whether the multipliers are decimal or binary, depends on exactly what is being measured.

IEC standard prefixes

In 1999, the International Electrotechnical Commission (IEC) published Amendment 2 to "IEC 60027-2: Letter symbols to be used in electrical technology — Part 2: Telecommunications and electronics". This standard, which was approved in 1998, introduced the prefixes kibi-, mebi-, gibi-, tebi-, pebi-, exbi-, to be used in specifying binary multiples of a quantity. The names come from the first two letters of the original SI prefixes followed by bi which is short for "binary". It also clarifies that, from the point of view of the IEC, the SI prefixes only have their base-10 meaning and never have a base-2 meaning.

This amendment was included in the next edition of the standard: "IEC 60027-2 (2000-11) Ed. 2.0"

The second edition defined them only up to exbi-, but in 2005, the third edition of the standard added prefixes zebi- and yobi-, thus matching all standard SI prefixes with their binary counterparts.

As of 2005 this naming convention has not gained widespread use, but its use is growing. It is strongly supported by many standardization bodies, including IEEE and CIPM. In particular on March 19, 2005 the IEEE standard IEEE 1541-2002 (Prefixes for Binary Multiples) has been elevated to a full-use standard by the IEEE Standards Association after a two-year trial period.

Name	Symbol	Base 2	Base 16		Base 10
kibi	Ki	210	162.5	0x400	= 1,024	> 103
mebi	Mi	220	165	0x10 0000	= 1,048,576	> 106
gibi	Gi	230	167.5	0x4000 0000	= 1,073,741,824	> 109
tebi	Ti	240	1610	0x100 0000 0000	= 1,099,511,627,776	> 1012
pebi	Pi	250	1612.5	0x4 0000 0000 0000	= 1,125,899,906,842,624	> 1015
exbi	Ei	260	1615	0x1000 0000 0000 0000	= 1,152,921,504,606,846,976	> 1018
zebi	Zi	270	1617.5	0x40 0000 0000 0000 0000	= 1,180,591,620,717,411,303,424	> 1021
yobi	Yi	280	1620	0x1 0000 0000 0000 0000 0000	= 1,208,925,819,614,629,174,706,176	> 1024

Example: 300 GB ~ 279.5 GiB (= 0x117.6592E GiB = 0x45D96.4B8 MiB = 0x1176592E KiB = 0x45D964B800 bytes).

Usage notes

The phrase "decimal unit" will be used to denote "SI designation understood in its standard, decimal, power-of-1000 sense" and "binary unit" will mean "SI designation understood in its traditional computer-industry, binary, power-of-1024 sense." B will be used as the symbol for byte as per computer-industry standard (IEEE 1541 and IEC 60027; despite B being also the symbol for bel).

Certain units are always understood as decimal even in computing contexts. For example, hertz (Hz), which is used to measure clock rates of electronic components, and bit/s, used to measure bit rate. So a 1 GHz processor performs 1,000,000,000 clock ticks per second, a 128 kbit/s MP3 stream consumes 128,000 bits (16 kB, 15.625 KiB) per second, and a 1 Mbit/s Internet connection can transfer 1,000,000 bits (125 kB, approx 122 KiB) per second (assuming an 8-bit byte, and no overhead).

Measurements of most types of electronic memory such as RAM and ROM and Flash (large scale disk-like flash is sometimes an exception) are given in binary units because they are made in power-of-two sizes as this means that all combinations of their address lines map to a valid address allowing easy aggregation into a larger contiguous block of memory.

Hard disk drive manufacturers state capacity in decimal units. Since most computer operating systems report drive usage and capacity in binary units, the difference causes an apparent loss between the advertised capacity and the formatted, usable capacity. This usage has a long engineering tradition, predating consumer complaints about the apparent discrepancy, which began to surface in the mid-1990s. The decimal-based capacity in hard disk drives follows the method used for serially accessed storage media which predated direct access storage media like hard disk drives. Paper punch cards could only be used in a serial fashion, like the magnetic tapes that followed. When a stream of data is stored, it's more logical to indicate how many thousands, millions, or billions of bytes have been stored versus how many multiples of 1024, 1,048,576, or 1,073,741,824 bytes have been. When the first hard disk drives were being developed, the decimal measurement was only natural since the hard disk drive served essentially the same function as punch cards and tapes. Thus, today, any device that is addressed or seen as "storage" uses the decimal system to identify capacity.

Disk media are accessed by the sector, not the individual byte. Sectors are intended for direct transfer to RAM, which comes in powers of two, so sector size itself is almost always a power of two (some early 1960s sectors measured 100 bytes [3]). Common sector sizes range from 512 bytes (e.g. on floppy disks) to 2048 bytes (DVDs). Because of this, a very confusing hybrid system is sometimes used, in which a "megabyte" means a thousand 1024-byte "kilobytes". Thus, as of 2005, manufacturers universally use the designation "1.44 MB diskette" for a product which holds neither 1.44×2²⁰ bytes nor 1.44×10⁶ bytes, but rather 1.44×1000×1024 bytes (approximately 1.406 MiB, or 1.475 MB). Some manufacturers of disk-like flash memory seem to have adopted the somewhat dubious practice of selling drives measured in power of two multiples of decimal megabytes!

Modern-day PC users, of course, regard both RAM and disk as kinds of storage and expect their capacities to be measured in the same way. Operating systems usually report disk space using the binary version. To the purchaser of a "30 GB" hard drive, rather than reporting either "30 GB" or "28 GiB", Microsoft Windows reports "28 GB". This creates hard feelings that have even led to legal disputes, sometimes made worse by other technical issues such as failure to distinguish between unformatted and formatted capacities and to account for the overhead inherent in disk file systems.

CD capacities are always given in binary units. A "700 MB" (or "80 minute") CD has a nominal capacity of about 700 MiB. But DVD capacities are given in decimal units. A "4.7 GB" DVD has a nominal capacity of about 4.38 GiB.

Bus bandwidth is given in decimal units. This is not because hard drive capacities use the decimal versions, nor because bit rates do, but because clock speeds do. For example, "PC3200" memory runs on a double pumped 200 MHz bus, transferring 8 bytes per cycle, and hence has a bandwidth of 200,000,000×2×8 = 3,200,000,000 byte/s.

Pronunciation

It is suggested that in English, the first syllable of the name of the binary-multiple prefix should be pronounced in the same way as the first syllable of the name of the corresponding SI prefix, and that the second syllable should be pronounced as "bee."

Legal disputes

As a result of the confusion over decimal and binary capacity measures, there have been several lawsuits against companies who sell hard drives, flash memory devices, and computer systems that list the drive capacity. As consumers become less and less technical, the technical definitions of binary and decimal multiples have become meaningless and only the number reported by the operating system holds sway.

At least 3 significant lawsuits have been filed:

• On February 20, 2004, Willem Vroegh filed a lawsuit against Lexar Media, Dane–Elec Memory, Fuji Photo Film USA, Eastman Kodak Company, Kingston Technology Company, Inc., Memorex Products, Inc.; PNY Technologies Inc., SanDisk Corporation, Verbatim Corporation, and Viking InterWorks alleging that their descriptions of the capacity of their flash memory cards were false and misleading.

  As of early 2005, at least one of these companies has settled out of court.

• In September of 2003, Lanchau Dan, Adam Selkowitz, Tim Swan and John Zahabian filed a lawsuit against Dell, Inc., Apple Computer Inc., Gateway, Inc., Hewlett-Packard Co., IBM Corp., Sharp Corporation, Sony Corp. and Toshiba Corp claiming their advertising deceptively exaggerates the real capacity of their hard drives.
• On July 7, 2005, an action entitled Orin Safier v. Western Digital Corporation, et al., was filed in the Superior Court for the City and County of San Francisco, Case No. CGC-05-442812. The case was subsequently moved to the Northern District of California, Case No. 05-03353 BZ. The case was settled in March 2006 with June 14, 2006 as the Final Approval hearing date.

See also

• Integral data type
• Bit
• Nibble
• Byte
• Octet
• IEC 60027-2
• IEEE 1541
• SI prefix

Specific units of IEC 60027-2 A.2

Bit rates (data-rate units) Name Symbol Multiple bit per second bit/s 1 1 Metric prefixes (SI) kilobit per second kbit/s 103 10001 megabit per second Mbit/s 106 10002 gigabit per second Gbit/s 109 10003 terabit per second Tbit/s 1012 10004 Binary prefixes (IEC 80000-13) kibibit per second Kibit/s 210 10241 mebibit per second Mibit/s 220 10242 gibibit per second Gibit/s 230 10243 tebibit per second Tibit/s 240 10244
Multiple-byte units Decimal Value Metric 1000 kB kilobyte 10002 MB megabyte 10003 GB gigabyte 10004 TB terabyte 10005 PB petabyte 10006 EB exabyte 10007 ZB zettabyte 10008 YB yottabyte 10009 RB ronnabyte 100010 QB quettabyte Binary Value IEC Memory 1024 KiB kibibyte KB kilobyte 10242 MiB mebibyte MB megabyte 10243 GiB gibibyte GB gigabyte 10244 TiB tebibyte TB terabyte 10245 PiB pebibyte – 10246 EiB exbibyte – 10247 ZiB zebibyte – 10248 YiB yobibyte – 10249 – – 102410 – – Orders of magnitude of data	Multiple-bit units Decimal Value Metric 1000 kbit kilobit 10002 Mbit megabit 10003 Gbit gigabit 10004 Tbit terabit 10005 Pbit petabit 10006 Ebit exabit 10007 Zbit zettabit 10008 Ybit yottabit 10009 Rbit ronnabit 100010 Qbit quettabit Binary Value IEC Memory 1024 Kibit kibibit Kbit Kb kilobit 10242 Mibit mebibit Mbit Mb megabit 10243 Gibit gibibit Gbit Gb gigabit 10244 Tibit tebibit — 10245 Pibit pebibit — 10246 Eibit exbibit — 10247 Zibit zebibit — 10248 Yibit yobibit — — — Orders of magnitude of data

External links

• What is a Megabyte ...? Markus Kuhn's 1996–1999 paper on bits, bytes, prefixes and symbols
• Prefixes for binary multiples — NIST
• Get Ready for the mebi, gibi and tebi — NIST
• Description of binary prefixes — IEC
• Another description of binary prefixes
• White-paper on the controversy over drive capacities
• There is no such thing as a 1.44 MB standard format floppy disc
• A summary of the organizations, software, and so on that have implemented the new binary prefixes
• A plea for sanity
• KiloBytes vs. kilobits vs. Kibibytes (Binary prefixes)
• Here Come Zebi- and Yobi- IEC press release announcing new prefixes
• Decimal-to-Binary Prefixes and Binary-to-Decimal Prefixes Converter