101 Sentences With "binary digits" | Random Sentence Generator

Classical computers encode and manipulate information as strings of binary digits—503 or 250.

It took around 55 million photons and 10 minutes in order to generate 1,024 random binary digits.

The researchers found humans can process a maximum of 8 bits (binary digits) of information when planning a trip.

The "memory athlete" Munkhshur Narmandakh once employed a similar combination of mnemonics to commit more than 23,249 binary digits to memory in just 3413 minutes.

Computers store their data in binary code, rather than in base 10, so 1,024 random binary digits could be any one of 21024 numbers (its's a kilobit).

For a 2001 bit floating-point number, it turns out that we really only get 210 binary digits to represent the numerical content of a number, with the rest reserved for representing the position of the decimal point within the number (as an exponent, as in scientific notion).

Digital usually refers to something using digits, particularly binary digits.

Each octal digit corresponds to three binary digits, rather than four. Therefore we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.

In 1998 Simon Plouffe gave a ruler and compass algorithm that can be used to compute binary digits of certain numbers. The algorithm involves the repeated doubling of an angle and becomes physically impractical after about 20 binary digits.

The Ehrenfeucht–Mycielski sequence is a recursively defined sequence of binary digits with pseudorandom properties, defined by .

In public-key cryptography, keys are defined by their length expressed in binary digits - their bit length.

The codeword for symbol i is chosen to be the first l_i binary digits in the binary expansion of c_i.

It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark function. Such repeating sequences correspond to periodic orbits of the dyadic transformation (for the binary digits) and the Gauss map h(x)=1/x-\lfloor 1/x \rfloor for continued fractions.

Definition: The mobile identification number (MIN) is a number that is derived from the 10-digit directory telephone number assigned to a mobile station. The rules for deriving the MIN from the 10-digit telephone number are given in the IS-95 standard. MIN1 is the first or least significant 24 binary digits of the MIN. MIN2 is the second part of the MIN containing the 10 most significant binary digits.

The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers. In computers, the main numeral systems are based on the positional system in base 2 (binary numeral system), with two binary digits, 0 and 1. Positional systems obtained by grouping binary digits by three (octal numeral system) or four (hexadecimal numeral system) are commonly used. For very large integers, bases 232 or 264 (grouping binary digits by 32 or 64, the length of the machine word) are used, as, for example, in GMP.

In 1998, Broadhurst gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained in nearly linear time, and logarithmic space.

Although the digits of Ω cannot be determined, many properties of Ω are known; for example, it is an algorithmically random sequence and thus its binary digits are evenly distributed (in fact it is normal).

The length of each word is 60 binary digits (bits). The highly efficient address and data control mechanisms involved permit a word to be moved into or out of central memory in as little as 100 nanoseconds.

DC-nets are readily generalized to allow for transmissions of more than one bit per round, for groups larger than three participants, and for arbitrary "alphabets" other than the binary digits 0 and 1, as described below.

With sufficient drill, people found it possible to remember as many as forty binary digits. Miller wrote: > It is a little dramatic to watch a person get 40 binary digits in a row and > then repeat them back without error. However, if you think of this merely as > a mnemonic trick for extending the memory span, you will miss the more > important point that is implicit in nearly all such mnemonic devices. The > point is that recoding is an extremely powerful weapon for increasing the > amount of information that we can deal with.

The maximum number of binary digits that can be processed within a unit time by a computer system is called the maximum parallelism degree P. If a processor is processing P bits in unit time, then P is called the maximum degree of parallelism.

There are currently no mainstream general-purpose processors built to operate on 256-bit integers or addresses, though a number of processors do operate on 256-bit data. CPUs feature SIMD instruction sets (Advanced Vector Extensions and the FMA instruction set etc.) where 256-bit vector registers are used to store several smaller numbers, such as eight 32-bit floating-point numbers, and a single instruction can operate on all these values in parallel. However, these processors do not operate on individual numbers that are 256 binary digits in length, only their registers have the size of 256-bits. Binary digits are found together in 128-bit collections.

A quantum computer is a device that uses quantum mechanisms for computation. In this device the data are stored as qubits (quantum binary digits). That gives a quantum computer in comparison with a conventional computer the opportunity to solve complicated problems in a short time, e.g. discrete logarithm problem or factorization.

Each minor cycle is to be addressed as a unit (word addressing, Sec. 12.8). Instructions are to be executed sequentially, with a special instruction to switch to a different point in memory (i.e. a jump instruction). Binary digits in a delay line memory pass through the line and are fed back to the beginning.

Each signal (symbol) carries three bits of information. It takes three binary digits to encode eight states. The data rate is three bits per second. In the Navy, more than one flag pattern and arm can be used at once, so the combinations of these produce many symbols, each conveying several bits, a higher data rate.

It weighed 1000 pounds. The approximate cost of the basic system was $500,000 to the USAF PAFB. The system was fixed-point binary and used 45 binary digits per word (44 numerical, plus one for the sign). Instruction words were the same length as data words, and the computer used 19 total instructions and three-address code instruction type.

18 binary digits have (1000000 octal, 40000 hexadecimal) distinct combinations. 18 bits was a common word size for smaller computers in the 1960s, when large computers often used 36 bit words and 6-bit character sets, sometimes implemented as extensions of BCD, were the norm. There were also 18-bit teletypes experimented with in the 1940s.

Intuitively, an algorithmically random sequence (or random sequence) is a sequence of binary digits that appears random to any algorithm running on a (prefix-free or not) universal Turing machine. The notion can be applied analogously to sequences on any finite alphabet (e.g. decimal digits). Random sequences are key objects of study in algorithmic information theory.

Any form of signal that carries information in the form of variable physical values, such as amplitude or frequency modulation. A signal which moves through a continuous range of settings or levels. An adjective describing any signal that varies continuously as opposed to a digital signal that contains discrete levels representing the binary digits 0 and 1.

This is a list of some binary codes that are (or have been) used to represent text as a sequence of binary digits "0" and "1". Fixed-width binary codes use a set number of bits to represent each character in the text, while in variable-width binary codes, the number of bits may vary from character to character.

The second is that if any of the items is forgotten, the entire list may be in jeopardy. The third is the potential for confusing repeated segments of the list, a common problem when memorizing binary digits. This limitation can be resolved either through bundling or by using either the peg system or the method of loci.Bremer, Rod.

The Global Positioning System (GPS) Week Number Rollover is a phenomenon that happens every 1024 weeks, which is about 19.6 years. The Global Positioning system broadcasts a date, including a weekly counter that is stored in only ten binary digits. The range is therefore 0–1023. After 1023 the internal value "rolls over", changing to zero again.

The section below codifies this procedure. It is obvious that a similar method can be used to compute the square root in number systems other than the decimal number system. For instance, finding the digit-by-digit square root in the binary number system is quite efficient since the value of a_i is searched from a smaller set of binary digits {0,1}.

Similar to torque and energy in physics; information-theoretic information and data storage size have the same dimensionality of units of measurement, but there is in general no meaning to adding, subtracting or otherwise combining the units mathematically. Other units of information, sometimes used in information theory, include the natural digit also called a nat or nit and defined as log2 e (≈ 1.443) bits, where e is the base of the natural logarithms; and the dit, ban, or hartley, defined as log2 10 (≈ 3.322) bits. This value, slightly less than 10/3, may be understood because 103 = 1000 ≈ 1024 = 210: three decimal digits are slightly less information than ten binary digits, so one decimal digit is slightly less than 10/3 binary digits. Conversely, one bit of information corresponds to about ln 2 (≈ 0.693) nats, or log10 2 (≈ 0.301) hartleys.

As with the inverse ratio, this value, approximately 3/10, but slightly more, corresponds to the fact that 210 = 1024 ~ 1000 = 103: ten binary digits are slightly more information than three decimal digits, so one binary digit is slightly more than 3/10 decimal digits. Some authors also define a binit as an arbitrary information unit equivalent to some fixed but unspecified number of bits.

The word 'Wikipedia' represented in ASCII binary code, made up of 9 bytes (72 bits). A binary code represents text, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often "0" and "1" from the binary number system. The binary code assigns a pattern of binary digits, also known as bits, to each character, instruction, etc.

Turing also notes that we need to determine which "machines" we wish to consider. He points out that a human clone, while man-made, would not provide a very interesting example. Turing suggested that we should focus on the capabilities of digital machinery—machines which manipulate the binary digits of 1 and 0, rewriting them into memory using simple rules. He gave two reasons.

The digipeater is used in channels that transmit data by binary digital signals, in which the data is in the form of pulses with only two possible values, representing the binary digits 1 and 0. A digital repeater amplifies the signal, and it also may retime, resynchronize, and reshape the pulses. A repeater that performs the retiming or resynchronizing functions may be called a regenerator.

More so, the article argues that it is not so much the "bits" that are essential for the information superhighway, but rather the electron. It is the electrification systems that prove to be more efficient in the twentieth century. Yet, it did not decrease the use of resources, or the atomic component. Computers are able to function without the "binary digits", yet they are useless without the electricity, or electron.

The results of computer calculations are normally an approximation expressed in a limited number of significant digits, although they can be programmed to produce more precise results. Approximation can occur when a decimal number cannot be expressed in a finite number of binary digits. Related to approximation of functions is the asymptotic value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large.

Equivalently, expressed as strings of binary digits, the codewords are: :00000,\quad 00111,\quad 01110,\quad 01001, :11100,\quad 11011,\quad 10010,\quad 10101. This, as every polynomial code, is indeed a linear code, i.e., linear combinations of code words are again code words. In a case like this where the field is GF(2), linear combinations are found by taking the XOR of the codewords expressed in binary form (e.g.

In 1974 the Arecibo message was sent with a radio signal aimed at a star cluster. It consisted of 1679 binary digits intended to be interpreted as a 23 \times 73 bitmap image. The number 1679=23\cdot 73 was chosen because it is a semiprime and therefore can be arranged into a rectangular image in only two distinct ways (23 rows and 73 columns, or 73 rows and 23 columns).

PSK uses a finite number of phases, each assigned a unique pattern of binary digits. Usually, each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular phase. The demodulator, which is designed specifically for the symbol-set used by the modulator, determines the phase of the received signal and maps it back to the symbol it represents, thus recovering the original data.

In some contexts it is desirable to round a given number to a "neat" fraction — that is, the nearest fraction = / whose numerator and denominator do not exceed a given maximum. This problem is fairly distinct from that of rounding a value to a fixed number of decimal or binary digits, or to a multiple of a given unit . This problem is related to Farey sequences, the Stern–Brocot tree, and continued fractions.

In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers.

PSK uses a finite number of phases, each assigned a unique pattern of binary digits. Usually, each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular phase. The demodulator, which is designed specifically for the symbol-set used by the modulator, determines the phase of the received signal and maps it back to the symbol it represents, thus recovering the original data.

Memory span refers to the longest list of items (e.g., digits, letters, words) that a person can repeat back in correct order on 50% of trials immediately after presentation. Miller observed that memory span of young adults is approximately seven items. He noticed that memory span is approximately the same for stimuli with vastly different amount of information—for instance, binary digits have 1 bit each; decimal digits have 3.32 bits each; words have about 10 bits each.

Although quaternary (base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to a pair of quaternary digits and each quaternary digit corresponds to a pair of binary digits. In the above example 5 E B 5 216 = 11 32 23 11 024. The octal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4.

In a simplification of the thought experiment, the monkey could have a typewriter with just two keys: 1 and 0. The infinitely long string thusly produced would correspond to the binary digits of a particular real number between 0 and 1. A countably infinite set of possible strings end in infinite repetitions, which means the corresponding real number is rational. Examples include the strings corresponding to one-third (010101...), five-sixths (11010101...) and five- eighths (1010000...).

In binary (base-2), a full space can be used between groups of four digits, corresponding to a nibble, or equivalently to a hexadecimal digit. For integer numbers, dots are used as well to separate groups of four bits.As an example, the DR-DOS DEBUG `H` command displays the entered number in hexadecimal, decimal, octal and binary notation: -h 1234 1234 #4660 \011064 %0001.0010.0011.0100 Alternatively, binary digits may be grouped by threes, corresponding to an octal digit.

For easier entry and readout, on some computers (such as the DEC PDP-8 or MITS Altair 8800) binary digits were grouped into threes or fours on the front panel, with each group of lights or switches representing a single octal (between 0 and 7) or hexadecimal (between 0 and F) digit. Some decimal computers, e.g., IBM 1620, used binary-coded decimal for memory addresses. Next the operator would enter the value intended for that address.

Parallel versus serial communication In data transmission, parallel communication is a method of conveying multiple binary digits (bits) simultaneously. It contrasts with serial communication, which conveys only a single bit at a time; this distinction is one way of characterizing a communications link. The basic difference between a parallel and a serial communication channel is the number of electrical conductors used at the physical layer to convey bits. Parallel communication implies more than one such conductor.

This formula, unlike others before it, can produce any individual hexadecimal digit of without calculating all the preceding digits. Individual binary digits may be extracted from individual hexadecimal digits, and octal digits can be extracted from one or two hexadecimal digits. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found that rapidly produces decimal digits.. Plouffe did create a decimal digit extraction algorithm, but it is slower than full, direct computation of all preceding digits.

Memory architecture describes the methods used to implement electronic computer data storage in a manner that is a combination of the fastest, most reliable, most durable, and least expensive way to store and retrieve information. Depending on the specific application, a compromise of one of these requirements may be necessary in order to improve another requirement. Memory architecture also explains how binary digits are converted into electric signals and then stored in the memory cells. And also the structure of a memory cell.

A modern digital computer represents data using the binary numeral system. Text, numbers, pictures, audio, and nearly any other form of information can be converted into a string of bits, or binary digits, each of which has a value of 1 or 0. The most common unit of storage is the byte, equal to 8 bits. A piece of information can be handled by any computer or device whose storage space is large enough to accommodate the binary representation of the piece of information, or simply data.

Along with Bennet and Dr D.G. Prinz, Woods was involved in writing interpretive subroutines that were used by the Ferranti group. Errors with the programs were one problem, but errors caused by the computer were another. The computer frequently misread the binary digits it was given. The engineers thought the mathematicians could compensate for this by programming arithmetic checks, and the mathematicians would too readily assume that a wrong program result was due to a computer error when it was due to a program error.

Whereas common digital computing requires that the data be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits, which can be in superpositions of states. One of the greatest challenges is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. Some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence.

In computing, the binary (base-2), octal (base-8) and hexadecimal (base-16) bases are most commonly used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as "shorthand" for binary—every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).

In proving "perfect secrecy", Shannon determined that this could only be obtained with a secret key whose length given in binary digits was greater than or equal to the number of bits contained in the information being encrypted. Furthermore, Shannon developed the "unicity distance", defined as the "amount of plaintext that… determines the secret key." Shannon's work influenced further cryptography research in the 1970s, as the public-key cryptography developers, M. E. Hellman and W. Diffie cited Shannon's research as a major influence. His work also impacted modern designs of secret-key ciphers.

Chen noted that the digits zero through seven were simply encoded using three binary digits of the corresponding octal group. He also postulated that one could use a flag to identify a different encoding for the digits eight and nine, which would be encoded using a single bit. In practice, a series of Boolean transformations are applied to the stream of input bits, compressing BCD encoded digits from 12 bits per three digits to 10 bits per three digits. Reversed transformations are used to decode the resulting coded stream to BCD.

754px An N-OFDM carrier signal is the sum of a number of not-orthogonal subcarriers, with baseband data on each subcarrier being independently modulated commonly using some type of quadrature amplitude modulation (QAM) or phase-shift keying (PSK). This composite baseband signal is typically used to modulate a main RF carrier. \scriptstyle s[n] is a serial stream of binary digits. By inverse multiplexing, these are first demultiplexed into \scriptstyle N parallel streams, and each one mapped to a (possibly complex) symbol stream using some modulation constellation (QAM, PSK, etc.).

754px An OFDM carrier signal is the sum of a number of orthogonal subcarriers, with baseband data on each subcarrier being independently modulated commonly using some type of quadrature amplitude modulation (QAM) or phase-shift keying (PSK). This composite baseband signal is typically used to modulate a main RF carrier. \scriptstyle s[n] is a serial stream of binary digits. By inverse multiplexing, these are first demultiplexed into \scriptstyle N parallel streams, and each one mapped to a (possibly complex) symbol stream using some modulation constellation (QAM, PSK, etc.).

Arithmetic operations are to be performed one binary digit at a time. He estimates addition of two binary digits as taking one microsecond and that therefore a 30-bit multiplication should take about 302 microseconds or about one millisecond, much faster than any computing device available at the time. Von Neumann's design is built up using what he call "E elements," which are based on the biological neuron as model,Von Neumann credits this model to Warren McCulloch and Walter Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophysics, Vol.

When Lucassen began to write new music, he thought that would result in a solo album. Eventually, he realized the music was too heavy for him to sing, so he thought of making it an album by Star One, another of his projects. Finally, he noticed some folk elements and decided it would be an Ayreon release. Mike Mills wrote his character's melodies for "The Day That the World Breaks Down", in which his lyrics are the ASCII encoded binary digits for "Trust TH1", TH1 being his character in the album.

Whereas expressions denote mainly numbers in elementary algebra, in Boolean algebra, they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1, for which 1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction), respectively, with disjunction x∨y (inclusive-or) definable as x + y - xy.

Lorenz SZ cipher machine as used by the German military during World War II In a synchronous stream cipher a stream of pseudo-random digits is generated independently of the plaintext and ciphertext messages, and then combined with the plaintext (to encrypt) or the ciphertext (to decrypt). In the most common form, binary digits are used (bits), and the keystream is combined with the plaintext using the exclusive or operation (XOR). This is termed a binary additive stream cipher. In a synchronous stream cipher, the sender and receiver must be exactly in step for decryption to be successful.

Characterization by a cutting sequence with a line of slope 1/\varphi or \varphi-1, with \varphi the golden ratio. A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. It is a paradigmatic example of a Sturmian word and specifically, a morphic word. The name “Fibonacci word” has also been used to refer to the members of a formal language L consisting of strings of zeros and ones with no two repeated ones.

The correspondence between these values and the physical states of the underlying storage or device is a matter of convention, and different assignments may be used even within the same device or program. It may be physically implemented with a two-state device. The symbol for the binary digit is either bit per recommendation by the IEC 80000-13:2008 standard, or the lowercase character b, as recommended by the IEEE 1541-2002 and IEEE Std 260.1-2004 standards. A contiguous group of binary digits is commonly called a bit string, a bit vector, or a one- or more- dimensional bit array.

A group of eight binary digits is called one byte, but historically the size of the byte is not strictly defined. Frequently, half-, full-, double- and quad-words consist of a number of bytes which is a low power of two. In information theory, one bit is the information entropy of a binary random variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known. As a unit of information, the bit is also known as a shannon, named after Claude E. Shannon.

After programmable general purpose computers were invented, machine languages (consisting of strings of the binary digits 0 and 1 on punched paper tape) were introduced that sped up the programming process (Stern, 1981). OS/360 was used on most IBM mainframe computers beginning in 1966, including computers used by the Apollo program. In the early 1950s, a computer could execute only one program at a time. Each user had sole use of the computer for a limited period and would arrive at a scheduled time with their program and data on punched paper cards or punched tape.

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 does not need to be 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 towards the width of the significand. 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.

This form of distortion, sometimes called granular or quantization distortion, has been pointed to as a fault of some digital systems and recordings particularly some early digital recordings, where the digital release was said to be inferior to the analog version. The range of possible values that can be represented numerically by a sample is determined by the number of binary digits used. This is called the resolution, and is usually referred to as the bit depth in the context of PCM audio. The quantization noise level is directly determined by this number, decreasing exponentially (linearly in dB units) as the resolution increases.

Metrics are usually described in terms of variables that are a function of the input. For example, the statement that insertion sort requires O(n2) comparisons is meaningless without defining n, which in this case is the number of elements in the input list. Because many different contexts use the same letters for their variables, confusion can arise. For example, the complexity of primality tests and multiplication algorithms can be measured in two different ways: one in terms of the integers being tested or multiplied, and one in terms of the number of binary digits (bits) in those integers.

Another definition of the Moser–de Bruijn sequence is that it is the ordered sequence of numbers whose binary representation has nonzero digits only in the even positions. For instance, 69 belongs to the sequence, because its binary representation 10001012 has nonzero digits in the positions for 26, 22, and 20, all of which have even exponents. The numbers in the sequence can also be described as the numbers whose base-4 representation uses only the digits 0 or 1. For a number in this sequence, the base-4 representation can be found from the binary representation by skipping the binary digits in odd positions, which should all be zero.

Experimental mathematics as a separate area of study re-emerged in the twentieth century, when the invention of the electronic computer vastly increased the range of feasible calculations, with a speed and precision far greater than anything available to previous generations of mathematicians. A significant milestone and achievement of experimental mathematics was the discovery in 1995 of the Bailey–Borwein–Plouffe formula for the binary digits of π. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous proof found.The Quest for Pi by David H. Bailey, Jonathan M. Borwein, Peter B. Borwein and Simon Plouffe.

The CDC STAR-100 is a vector supercomputer that was designed, manufactured, and marketed by Control Data Corporation (CDC). It was one of the first machines to use a vector processor to improve performance on appropriate scientific applications. It was also the first supercomputer to use integrated circuits and the first to be equipped with one million words of computer memory. The name STAR was a construct of the words STrings of binary digits that made up ARrays, referring to the vector concept. The 100 came from 100 million floating point operations per second (MFLOPS), the speed at which the machine was designed to operate.

In mathematics and computing, the hexadecimal (also base 16 or hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the common way of representing numbers using 10 symbols, hexadecimal uses 16 distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values 10 to 15. Hexadecimal numerals are widely used by computer system designers and programmers because they provide a human-friendly representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble), which is half a byte.

However, these processors do not operate on individual numbers that are 128 binary digits in length; only their registers have the size of 128 bits. The DEC VAX supported operations on 128-bit integer ('O' or octaword) and 128-bit floating-point ('H-float' or HFLOAT) datatypes. Support for such operations was an upgrade option rather than being a standard feature. Since the VAX's registers were 32 bits wide, a 128-bit operation used four consecutive registers or four longwords in memory. The ICL 2900 Series provided a 128-bit accumulator, and its instruction set included 128-bit floating-point and packed decimal arithmetic.

It is important to disambiguate algorithmic randomness with stochastic randomness. Unlike algorithmic randomness, which is defined for computable (and thus deterministic) processes, stochastic randomness is usually said to be a property of a sequence that is a priori known to be generated (or is the outcome of) by an independent identically distributed equiprobable stochastic process. Because infinite sequences of binary digits can be identified with real numbers in the unit interval, random binary sequences are often called (algorithmically) random real numbers. Additionally, infinite binary sequences correspond to characteristic functions of sets of natural numbers; therefore those sequences might be seen as sets of natural numbers.

A quantum computer is a computation system that makes direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data."Quantum Computing with Molecules" article in Scientific American by Neil Gershenfeld and Isaac L. Chuang Quantum computers are different from digital computers based on transistors. Whereas digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses qubits (quantum bits), which can be in superpositions of states. A theoretical model is the quantum Turing machine, also known as the universal quantum computer.

If a computer file that uses n bits of storage contains only m < n bits of information, then that information can in principle be encoded in about m bits, at least on the average. This principle is the basis of data compression technology. Using an analogy, the hardware binary digits refer to the amount of storage space available (like the number of buckets available to store things), and the information content the filling, which comes in different levels of granularity (fine or coarse, that is, compressed or uncompressed information). When the granularity is finer—when information is more compressed—the same bucket can hold more.

In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or ‘Ars generalis’ based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence.(see Bonner 2007 , Fidora et al. 2011 ) In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text.

He notes that multiplication and division could be done with logarithm tables, but to keep the tables small enough, interpolation would be needed and this in turn requires multiplication, though perhaps with less precision. Numbers are to be represented in binary notation. He estimates 27 binary digits (he did not use the term "bit," which was coined by Claude Shannon in 1948) would be sufficient (yielding 8 decimal place accuracy) but rounds up to 30-bit numbers with a sign bit and a bit to distinguish numbers from orders, resulting in 32-bit word he calls a minor cycle. Two’s complement arithmetic is to be used, simplifying subtraction.

The World Championships consist of ten different disciplines, where the competitors have to memorize as much as they can in a period of time: # One hour numbers (23712892....) # 5-minute numbers # Spoken numbers, read out one per second # 30-minute binary digits (011100110001001....) # One hour playing cards (as many decks of cards as possible) # 15-minute random lists of words (house, playing, orphan, encyclopedia....) # 15-minute names and faces # 5-minute historic dates (fictional events and historic years) # 15-minute abstract images (WMSC, black and white randomly generated spots) / 5-minute random images (IAM, concrete images) # Speed cards - Always the last discipline. Memorize the order of one shuffled deck of 52 playing cards as fast as possible.

The term digitization is often used when diverse forms of information, such as an object, text, sound, image or voice, are converted into a single binary code. The core of the process is the compromise between the capturing device and the player device so that the rendered result represents the original source with the most possible fidelity, and the advantage of digitization is the speed and accuracy in which this form of information can be transmitted with no degradation compared with analog information. Digital information exists as one of two digits, either 0 or 1. These are known as bits (a contraction of binary digits) and the sequences of 0s and 1s that constitute information are called bytes.

As RSA Laboratories is a provider of RSA-based products, the challenge was used by them as an incentive for the academic community to attack the core of their solutions -- in order to prove its strength. The RSA numbers were generated on a computer with no network connection of any kind. The computer's hard drive was subsequently destroyed so that no record would exist, anywhere, of the solution to the factoring challenge. The first RSA numbers generated, RSA-100 to RSA-500 and RSA-617, were labeled according to their number of decimal digits; the other RSA numbers (beginning with RSA-576) were generated later and labelled according to their number of binary digits.

Originally an acronym of Video Audio Integrated Operation, this was amended to Visual Audio Intelligent Organizer in 2008 to celebrate the brand's 10th anniversary. The logo concept was created by Teiyu Goto, supervisor of product design from the Sony Creative Center in Tokyo. He incorporated many meanings into the logo and acronym: the pronunciation is similar to "bio", which is symbolic of life and the product's future evolution; it's also near "violet", which is why most early Vaios were purple or included purple components. Additionally, the logo is stylized to make the "VA" look like a sine wave and the "IO" like binary digits 1 and 0, the combination representing the merging of analog and digital signals.

In addition to accuracy and precision, measurements may also have a measurement resolution, which is the smallest change in the underlying physical quantity that produces a response in the measurement. In numerical analysis, accuracy is also the nearness of a calculation to the true value; while precision is the resolution of the representation, typically defined by the number of decimal or binary digits. In military terms, accuracy refers primarily to the accuracy of fire (justesse de tir), the precision of fire expressed by the closeness of a grouping of shots at and around the centre of the target.North Atlantic Treaty Organization, Nato Standardization Agency AAP-6 - Glossary of terms and definitions, p 43.

Bit-length or bit width is the number of binary digits, called bits, necessary to represent an integer as a binary number. Formally, the number of bits of zero is 1 and any other natural number n>0 is a function, bitLength(n), of the binary logarithm of n: :bitLength(n)= \lfloor log_2(n) + 1 \rfloor = \lceil log_2(n+1) \rceil At their most fundamental level, digital computers and telecommunications devices (as opposed to analog devices) can process only data that has been expressed in binary format. The binary format expresses data as an arbitrary length series of values with one of two choices: Yes/No, 1/0, True/False, etc., all of which can be expressed electronically as On/Off.

When the information capacity of a storage system or a communication channel is presented in bits or bits per second, this often refers to binary digits, which is a computer hardware capacity to store binary data (0 or 1, up or down, current or not, etc.). Information capacity of a storage system is only an upper bound to the quantity of information stored therein. If the two possible values of one bit of storage are not equally likely, that bit of storage contains less than one bit of information. Indeed, if the value is completely predictable, then the reading of that value provides no information at all (zero entropic bits, because no resolution of uncertainty occurs and therefore no information is available).

An advantage of digital circuits when compared to analog circuits is that signals represented digitally can be transmitted without degradation caused by noise.Paul Horowitz and Winfield Hill, The Art of Electronics 2nd Ed. Cambridge University Press, Cambridge, 1989 page 471 For example, a continuous audio signal transmitted as a sequence of 1s and 0s, can be reconstructed without error, provided the noise picked up in transmission is not enough to prevent identification of the 1s and 0s. In a digital system, a more precise representation of a signal can be obtained by using more binary digits to represent it. While this requires more digital circuits to process the signals, each digit is handled by the same kind of hardware, resulting in an easily scalable system.

The International Bureau of Weights and Measures states that "when there are only four digits before or after the decimal marker, it is customary not to use a space to isolate a single digit". Likewise, some manuals of style state that thousands separators should not be used in normal text for numbers from 1,000 to 9,999 inclusive where no decimal fractional part is shown (in other words, for four-digit whole numbers), whereas others use thousands separators, and others use both. For example, APA style stipulates a thousands separator for "most figures of 1,000 or more" except for page numbers, binary digits, temperatures, etc. There are always "common-sense" country-specific exceptions to digit grouping, such as year numbers, postal codes and ID numbers of predefined nongrouped format, which style guides usually point out.

W. S. Anglin and J. Lambek, The Heritage of Thales, Springer, 1995, George Boole The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. In the 11th century, scholar and philosopher Shao Yong developed a method for arranging the hexagrams which corresponds, albeit unintentionally, to the sequence 0 to 63, as represented in binary, with yin as 0, yang as 1 and the least significant bit on top. The ordering is also the lexicographical order on sextuples of elements chosen from a two-element set. In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text.

Also, giving an angle in degrees, minutes and seconds (with decimals), or a time in days, hours, minutes and seconds, can be interpreted as mixed-radix systems. Sequences where each weight is not an integral multiple of the previous weight may also be used, but then every integer may not have a unique representation. For example, Fibonacci coding uses the digits 0 and 1, weighted according to the Fibonacci sequence (1, 2, 3, 5, 8, ...); a unique representation of all non-negative integers may be ensured by forbidding consecutive 1s. Binary-coded decimal (BCD) are mixed base systems where bits (binary digits) are used to express decimal digits. E.g., in 1001 0011, each group of four bits may represent a decimal digit (in this example 9 and 3, so the eight bits combined represent decimal 93).

The Baby was designed to show that the system was a practical storage device, by testing that data held within it could be read and written at the speed necessary for use in a computer. For use in a binary digital computer, the tube had to be capable of storing either one of two states at each of its memory locations, corresponding to the binary digits (bits) 0 and 1\. It exploited the positive or negative electric charge generated by displaying either a dash or a dot at any position on the CRT screen, a phenomenon known as secondary emission. A dash generated a positive charge, and a dot a negative charge, either of which could be picked up by a detector plate in front of the screen; a negative charge represented 0, and a positive charge 1\.

Directional drillers rely on receiving accurate, quality tested data from the MWD operator to allow them to keep the well safely on the planned trajectory. Directional survey measurements are taken by three orthogonally mounted accelerometers to measure inclination, and three orthogonally mounted magnetometers which measure direction (azimuth). Gyroscopic tools may be used to measure azimuth where the survey is measured in a location with disruptive external magnetic influences, inside "casing", for example, where the hole is lined with steel tubulars (tubes). These sensors, as well as any additional sensors to measure rock formation density, porosity, pressure or other data, are connected, physically and digitally, to a logic unit which converts the information into binary digits which are then transmitted to surface using "mud pulse telemetry" (MPT, a binary coding transmission system used with fluids, such as, combinatorial, Manchester encoding, split-phase, among others).

Nor did it implement the stored program architecture, first implemented in the Manchester Baby of 1948, required for fully general-purpose practical computing machines. Add-subtract module (reconstructed) from Atanasoff–Berry Computer The machine was, however, the first to implement three critical ideas that are still part of every modern computer: #Using binary digits to represent all numbers and data #Performing all calculations using electronics rather than wheels, ratchets, or mechanical switches #Organizing a system in which computation and memory are separated. The memory of the Atanasoff–Berry Computer was a system called regenerative capacitor memory, which consisted of a pair of drums, each containing 1600 capacitors that rotated on a common shaft once per second. The capacitors on each drum were organized into 32 "bands" of 50 (30 active bands and two spares in case a capacitor failed), giving the machine a speed of 30 additions/subtractions per second.

Though the simple DC-nets protocol uses binary digits as its transmission alphabet, and uses the XOR operator to combine cipher texts, the basic protocol generalizes to any alphabet and combining operator suitable for one-time pad encryption. This flexibility arises naturally from the fact that the secrets shared between the many pairs of participants are, in effect, merely one-time pads combined together symmetrically within a single DC-net round. One useful alternate choice of DC- nets alphabet and combining operator is to use a finite group suitable for public-key cryptography as the alphabet—such as a Schnorr group or elliptic curve—and to use the associated group operator as the DC-net combining operator. Such a choice of alphabet and operator makes it possible for clients to use zero-knowledge proof techniques to prove correctness properties about the DC-net ciphertexts that they produce, such as that the participant is not "jamming" the transmission channel, without compromising the anonymity offered by the DC-net.

A means of converting digital audio into video format was necessary. Such an audio recording system includes two devices: the PCM adaptor, which converts audio into pseudo-video, and the videocassette recorder. A PCM adaptor performs an analog-to-digital conversion producing series of binary digits, which, in turn, is coded and modulated into a black and white video signal, appearing as a vibrating checkerboard pattern, which can then be recorded as a video signal. Most video-based PCM adaptors record audio at 14 bits per sample, and a sampling frequency of 44.056 kHz for EIAN countries (or 44.1 kHz for CCIR countries.) However, some of the earlier models, such as the Sony PCM-100, recorded 16-bits per sample as well, but used only 14 of the bits for the audio, with the remaining 2 bits used for error correction for the case of dropouts or other anomalies being present on the videotape.

Consider the sum: 12345678 \+ 87654322 = 100000000 Using basic arithmetic, we calculate right to left, "8 + 2 = 0, carry 1", "7 + 2 + 1 = 0, carry 1", "6 + 3 + 1 = 0, carry 1", and so on to the end of the sum. Although we know the last digit of the result at once, we cannot know the first digit until we have gone through every digit in the calculation, passing the carry from each digit to the one on its left. Thus adding two n-digit numbers has to take a time proportional to n, even if the machinery we are using would otherwise be capable of performing many calculations simultaneously. In electronic terms, using bits (binary digits), this means that even if we have n one-bit adders at our disposal, we still have to allow a time proportional to n to allow a possible carry to propagate from one end of the number to the other.

In metrology, such as when performed in support of science, engineering or manufacturing objectives, dynamic range refers to the range of values that can be measured by a sensor or metrology instrument. Often this dynamic range of measurement is limited at one end of the range by saturation of a sensing signal sensor or by physical limits that exist on the motion or other response capability of a mechanical indicator. The other end of the dynamic range of measurement is often limited by one or more sources of random noise or uncertainty in signal levels that may be described as defining the sensitivity of the sensor or metrology device. When digital sensors or sensor signal converters are a component of the sensor or metrology device, the dynamic range of measurement will be also related to the number of binary digits (bits) used in a digital numeric representation in which the measured value is linearly related to the digital number.