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In computing, binary prefixes are often used to quantify large numbers where powers of two are more useful than powers of ten. They are commonly written and pronounced identically to the SI prefixes, but each successive prefix is multiplied by 1024 (210) rather than 1000 (103).
Using the prefixes kilo-, mega-, giga-, etc., and their symbols K, M, G, etc., in the binary sense can cause serious confusion. 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.
Binary prefixes using SI symbols
|Name||Symbol||Value||Base 16||Base 10|
k or K
|210 = 1 024||= 162.5||>= 103|
|220 = 1 048 576||= 165||>= 106|
|230 = 1 073 741 824||= 167.5||>= 109|
|240 = 1 099 511 627 776||= 1610||>= 1012|
|250 = 1 125 899 906 842 624||= 1612.5||>= 1015|
|260 = 1 152 921 504 606 846 976||= 1615||>= 1018|
|270 = 1 180 591 620 717 411 303 424||= 1617.5||>= 1021|
|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 1000.) Some have suggested that "k" be used for 1000, and "K" for 1024, but this is not 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% at kilo to over 20.8% at yotta.
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.
We will use the phrase "decimal unit" to mean "SI designation understood in its standard, decimal, power-of-1000 sense" and "binary unit" to mean "SI designation understood in its traditional computer-industry, binary, power-of-1024 sense." Also following the tradition of computer confusion, the symbol for byte used here is B, which in SI stands for the Bel. It has oft been recommended to use b for the byte, but that is also sometimes used for the bit (the other symbol used is bit). This issue is not about to be resolved, as SI does not deal with units of quantity of information.
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 bitrate. So a 1 GHz processor performs 1,000,000,000 clock ticks per second, a 128 kbit/s MP3 stream consumes 128,000 bits (15.625 KiB) per second, and a 1 Mbit/s internet connection can transfer 1,000,000 bits (approx. 122 KiB) per second.
Measurements of electronic memory such as RAM and ROM are given in binary units, because the physical structure of the device makes it naturally come in sizes that are powers of two. This is the case whether the capacity is given in bits or bytes.
Hard disk drive manufacturers state capacity in decimal units, so what is advertised as a "30 GB" hard drive will hold 30 × 109 bytes, roughly equal to 28×230 bytes (i.e. 28 GiB). This usage has a long engineering tradition, and was probably not influenced by marketing. It arose because nothing about the physical structure of the disk drives makes power-of-two capacities natural: the number of platters, tracks and sectors per track are all continuously variable (though the size of an individual sector is half a binary kilobyte).
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 HD 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, sometimes made worse by other technical issues such as failure to distinguish between unformatted and formatted capacities.
Some disk storage measurements—floppy disks, for example—use an even more confusing hybrid system. 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. Sector size may range from 512 bytes (floppy disks) to 2048 bytes (DVDs). A thousand binary-"kilobyte" sectors creates a 1,024,000-byte "megabyte." Thus a "1.44 MB" floppy holds neither 1.44 × 220 bytes nor 1.44 × 106 bytes, but rather 1.44 × 1000 × 1024 bytes (approximately 1.406 binary MB, or 1.475 decimal MB).
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.
Decimal versions are used when dealing with bus bandwidth (e.g. "Ultra SCSI has a bandwidth of 40 megabytes per second"). Interestingly, this is not because hard drive capacities use the decimal versions, or because bitrates do, but because clock speeds do.
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"
|Ki||kibi-||binary kilo||210||= 10001 × 1.024|
|Mi||mebi-||binary mega||220||= 10002 × 1.048 576|
|Gi||gibi-||binary giga||230||= 10003 × 1.073 741 824|
|Ti||tebi-||binary tera||240||= 10004 × 1.099 511 627 776|
|Pi||pebi-||binary peta||250||= 10005 × 1.125 899 906 842 624|
|Ei||exbi-||binary exa||260||= 10006 × 1.152 921 504 606 846 976|
Example : 300 GB = 279.5 GiB ( = 0x117.6592E GiB = 0x45D96.4B8 MiB = 0x1176592E KiB = 0x45D964B800 bytes).
Note that the IEC names are defined only up to exbi-, corresponding to the SI prefix exa-. The two SI prefixes zetta- (1021) and yotta- (1024) have no corresponding IEC binary prefixes, even though the obvious continuation would be zebi- (Zi = 270 = 10007 × 1.180 591 620 717 411 303 424) and yobi- (Yi = 280 = 10008 × 1.208 925 819 614 629 174 706 176).
|SI, bits||SI, bytes||IEC, bits||IEC, bytes|
- Markus Kuhn's 1996-1999 paper on bits, bytes, prefixes and symbols
- IEC prefixes and symbols for binary multiples
- Get Ready for the mebi, gibi and tebi (NIST TechBeat)
- WiebeTech (drive manufacturer) white-paper on the controversy over drive capacities
- "There is no such thing as a a 1.44MB standard format floppy disc.", by Jonathan de Boyne Pollard
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