** For every non-empty set S there is a binary operation defined on S that makes it a group.

For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the list being searched, or in O ( log ( n )), colloquially " in logarithmic time ".

For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log < sub > 2 </ sub > n + 1 time units are needed to return an answer.

For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.

For example, the IBM 701 ( 1952 ) used binary and could address 2048 36-bit words, while the IBM 702 ( 1953 ) used decimal and could address 7-bit words.

For example, an 8-bit CPU deals with a range of numbers that can be represented by eight binary digits ( each digit having two possible values ), that is, 2 < sup > 8 </ sup > or 256 discrete numbers.

For example, if a binary CPU uses 32 bits to represent a memory address, and each memory address represents one octet ( 8 bits ), the maximum quantity of memory that CPU can address is 2 < sup > 32 </ sup > octets, or 4 GiB.

For example, binary string 1011 is represented by the vector ( 1, 0, 1, 1 ).

For example, Duncan Kennedy, in explicit reference to semiotics and deconstruction procedures, maintains that various legal doctrines are constructed around the binary pairs of opposed concepts, each of which with a claim upon intuitive and formal forms of reasoning that must be made explicit, not only in their meaning but also its relative value, and criticized.

For each of the binary fields, one elliptic curve and one Koblitz curve was selected.

: For all a, b in F, both a + b and a · b are in F ( or more formally, + and · are binary operations on F ).

For instance, 1 / 5 cannot be represented exactly as a floating point number using a binary base but can be represented exactly using a decimal base ( 0. 2, or 2x10 ^- 1 ).

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms.

For example, when mapping character strings between upper and lower case, one can use the binary encoding of each character, interpreted as an integer, to index a table that gives the alternative form of that character (" A " for " a ", " 8 " for " 8 ", etc .).

For example, one may sort the list and use binary search, or build any efficient search data structure from it.

For example, to retrieve the contents of cell 123, we would load the value 123 ( in binary, of course ) into the MAR and perform a fetch operation.

For example, the binary representation for decimal 74 is 1001010, which can be grouped into ( 00 ) 1 001 010 – so the octal representation is 112.

For a bacterial cell to reproduce through binary fission, more than a million peptidoglycan subunits ( NAM-NAG + oligopeptide ) must be attached to existing subunits.

For moderate volumes of values, insertions and deletions in a colored binary tree are faster compared to B-trees because colored trees don't attempt to maximize the fill factor of each horizontal cluster of nodes ( only the minimum fill factor is guaranteed in colored binary trees, limiting the number of splits or junctions of clusters ).

For large volume scientific datasets, such as satellite data and output of numerical climate, weather, or ocean models, specific binary serialization standards have been developed, e. g. HDF, netCDF and the older GRIB.

For example, CORBA, GIOP, ICE, and DCOM use much shorter, binary message formats.

For instance, the problem of integer factorization is suspected to require more than a polynomial function of the length of the input as run-time if the input is given in binary, but it only needs linear runtime if the input is presented in unary.

For example, in actual combat, vehicle armor is generally a binary proposition.

For AMI, each mark pulse had the opposite polarity of the previous one and each space was at a level of zero, resulting in a three level signal which however only carried binary data.

For example, raw uncompressed PCM audio ( 44. 1 kHz, 16 bit stereo, as represented on an audio CD or in a. wav or. aiff file ) has long been a standard across multiple platforms, but its transmission over networks is slow and expensive compared with more modern compressed formats, such as MP3.

For more complex or proprietary file formats, file viewers are usually provided by the same companies that make the editing software using those formats ( viewers may be distributed free of charge, while editors have to be bought ).

Sundazed Records reissued Torrence's Save for a Rainy Day in 1996 in CD and vinyl formats, as well as the collector's vinyl 45 " companion EP, " Sounds For A Rainy Day ," featuring four instrumentals versions of the album's tracks.

For example CP / M machines used many different floppy disk formats, which meant that one machine could not normally read disks from another CP / M machine, and Kermit was used as part of a process to enable the transfer of applications and data between CP / M machines and other machines with different operating systems.

For example, the Beautiful music format that developed into today's Easy listening and Soft rock formats is nearly extinct due to a lack of interest from younger generations, whereas classic rock has become popular over the last 20 years or so and Jack FM has arisen only since 2000 or so.

For each class of application ( e. g., audio, video ), RTP defines a profile and one or more associated payload formats.

For bank A to send a message to bank B with a copy or authorization with institution C, it formats the message according to standard and securely sends it to SWIFT.

* For decimal formats, whether binary or decimal encoded, a NaN is identified by having the top five bits of the combination field after the sign bit set to ones.

For amateur use, several smaller and therefore less expensive formats were introduced.

For high-quality studio recording, all of these formats were effectively made obsolete in the early 1980s by two competing reel-to-reel formats with stationary heads: Sony's DASH format and Mitsubishi's continuation of the X-80 recorder, which was improved upon to become the ProDigi format.

For music storage, digital formats offer an absence of clicks, pops, wow, flutter, acoustic feedback, and rumble, compared to vinyl records.

For camcorders, Sony adapted the Betacam system with its Digital Betacam format, later following it up with the cheaper Betacam SX and MPEG IMX formats, and the semiprofessional DV-based DVCAM system.

For those formats where no open source decoder has been made yet MPlayer relies on binary codecs.

For QuickTime formats, Apple has defined two types of coding: MJPEG-A and MJPEG-B.

For instance, in EIAS version 8. 0, Animator supports five of the most popular model input formats, while Transporter supports 29 formats.

: For the film formats associated with the Instamatic and Pocket Instamatic camera ranges, see 126 film and 110 film respectively.

For some multimedia container formats the resulting file is valid and this provides an effective means of appending files, particularly video streams.

For a time, a commercial compiler called CM3 maintained by one of the chief implementors prior at DEC SRC who was hired before DEC being sold to Compaq, an integrated development environment called Reactor and an extensible Java Virtual Machine ( licensed in binary and source formats and buildable with Reactor ) were offered by Critical Mass, Inc., but that company ceased active operations in 2000 and gave some of the sources of its products to elego Software Solutions GmbH.

For these formats, the digitizer keeps the original file for the purposes of editing.

For many years this was the only disc-based release of the film, until widescreen DVD formats were released with extras.