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Digital

Digital

:For other uses, see Digital (disambiguation) A digital system is one that uses numbers, especially binary numbers, for input, processing, transmission, storage, or display, rather than a continuous spectrum of values (an analog system) or non-numeric symbols such as letters or icons. The distinction of "digital" versus "analog" or "symbolic" can refer to method of input, data storage and transfer, the internal working of an instrument, and the kind of display. The word comes from the same source as the word digit and digitus: the Latin word for finger (counting on the fingers) as these are used for discrete counting. The word digital is most commonly used in computing and electronics, especially where real-world information is converted to binary numeric form as in digital audio and digital photography. Such data-carrying signals carry either one of two electronic or optical pulses, logic 1 (pulse present) or 0 (pulse absent). The term is often meant by the prefix "e-", as in e-mail and ebook, even though not all electronics systems are digital.

Digital noise

When data is transmitted using analog methods, a certain amount of noise enters into the signal. This can have myriad causes: data transmitted by radio may be received badly, suffer interference from other radio sources, or pick up background radio noise from the rest of the universe. Electric pulses being sent down wires are attenuated by the resistance of the wire, and dispersed by its capacitance, and heat variations can increase or reduce these effects. While digital transmissions are also degraded, any slight variations can be safely ignored. Any variance could provide a great amount of distortion in an analog signal. In a digital signal, these variances can be overcome, as any signal close to a particular value will be interpreted as that value. Care must be taken when connected digital and analog systems; tolerable variances for the digital part can leak into the analog part and become intolerable.

Analog, symbolic, and digital displays; ease of reading

For human readable information, digital, analog, and symbol display methods can all be useful. Should an instant impression be required, analog meters and indicator lights often give information quickly. Many people glance quickly at their analog watch and know roughly what the time is or at an automobile dashboard and know that a door is ajar. When accuracy is required, however, digital displays are preferred. Reading analog meters requires time and a little bit of skill, whereas writing down the value on a digital display is merely a case of copying down the numbers. In cases where both accuracy and quick reckoning are both required, dual displays are often used. A needle (analog) just touching onto the bottom of an orange shaded area is much different to a needle almost touching into the red area, but an indicator lamp (symbol) would just glow orange and a numeric (digital) display, although it could be colored orange, would not indicate the relative level of danger to an untrained operator.

Analog to digital conversion

:Main article: Analog-to-digital converter Converting an analog source to digital data is done with two steps: sampling, which changes the source to a series of discrete values (called samples), and quantization, which converts each sample to a number. For example, the sensor of a digital camera contains millions of sensing elements (one for each pixel). When an exposure is made, the light focused on the array is converted into millions of electric charges (sampled). These charges are then amplified and converted to numbers (quantized). The resulting digital image is then processed and stored in the camera's memory card. The samples in this case are spatial. In contrast, converting an audio source to digital requires temporal samples: it is converted to an electrical signal using a microphone, and the voltage of this signal is sampled thousands of times per second (the sampling frequency). Each sample is then quantized to form the digital audio data. Both sampling and quantization will result in a loss of data. Changes in the original data that occur between the samples will not appear in the digital data (or worse, will cause aliasing, the appearance of data not present in the original source). And while a voltage can be any of a seemingly unlimited number of values between its minimum and maximum (limited only by quantum mechanics), a digital representation using

n

bits can have only

2^n

possible values. While this information will be preserved in future transmission, the data has been lost. The amount of information that can be stored in a digital representation is called its resolution. And since the conversion to digital is a two step process, there are two types of resolution: sampling resolution and quantization resolution. Sampling resolution can be either spatial (expressed in pixels per inch) or temporal (expressed as samples per second) or both (for example, a video). Quantization resolution is usually expressed as the number of bits used to represent each sample and is thus often called the bit depth or (for pictures) the color depth. The best resolution for a given set of digital data depends on the processing it will undergo and its ultimate purpose. For example, compact discs use a sampling resolution of 44,100 samples/second, which is sufficient for audio in the range of human hearing. Most digital photographs use a bit depth of 8 bits/color, which produces more colors than the human eye can discern. However many photographers use camera raw with 12 bits/color to allow for more accuracy during processing before producing a final photograph at 8 bits/color for display or printing. Scientific photography may also require greater bit depth. If sufficient resolution is used, the data loss caused by the conversion to digital is offset by the accuracy of digital processing. When analog signals are transmitted and stored, accuracy is lost due to noise and distortion. So neither digital nor analog offer perfect fidelity; resolution is sacrificed for accuracy with digital and vice versa for analog. When both high resolution and high accuracy are needed, either a high resolution digital system or a high accuracy analog system must be used (with a correspondingly high cost).

Symbol to digital conversion

Since symbols are not continuous, converting symbols to digital is simpler and less prone to data loss than analog to digital conversion. Instead of sampling and quantization, similar steps are used: polling and encoding. A symbol input device usually consists of a number of switches that are polled at regular intervals to see which switches are pressed. Data will be lost if, within a single polling interval, two switches are pressed, or a switch is pressed, released, and pressed again. This polling can be done by a specialized processor in the device to prevent burdening the main CPU. When a new symbol has been entered, the device sends an interrupt to alert the CPU to read it. For devices with just a few switches (such as the buttons on a joystick), the status of each can be encoded as bits (usually 0 for released and 1 for pressed) in a single word. This is very useful when combinations of key presses are meaningful, and is sometimes used for passing the status of modifier keys on a keyboard (such as shift and control). But it does not scale to support more keys than the number of bits in a single byte or word. Devices with many switches (such as a computer keyboard) usually arrange these switches in a scan matrix, with the individual switches on the intersections of x and y lines. When a switch is pressed, it connects the corresponding x and y lines together. Polling (often called scanning in this case) is done by activating each x line in sequence and detecting which y lines then have a signal, thus which keys are pressed. When the keyboard processor detects that a key has changed state, it sends a signal to the CPU indicating the scan code of the key and its new state. The symbol is then encoded, or converted into a number, based on the status of modifier keys and the desired character encoding. Using a custom encoding for a specific application can be done with no loss of data. However, using a standard encoding such as ASCII is problematic if a symbol such as 'ß' needs to be converted but is not in the standard.

Historical digital systems

Although digital signals are generally associated with the binary electronic digital systems used in modern electronics and computing, digital systems are actually ancient, and need not be binary nor electronic.
- A beacon is perhaps the simplest non-electronic digital signal, with just two states (on and off). In particular, smoke signals are one of the oldest examples of a digital signal, where an analog "carrier" (smoke) is modulated with a blanket to generate a digital signal (puffs) that conveys information.
- DNA comprises a long sequence of four digits (denoted A, C, G, and T), effectively a base-four numeral system. (In fact, in the double helix structure, there are two strands, but one of them is never read.) Each of these digits is an organic molecule, known as a nucleotide. DNA is the major system of information transfer from one generation to another, and evolution has developed its digital properties into a robust method of communication.
- Morse code uses five digital states—dot, dash, short gap (between each letter), medium gap (between words), and long gap (between sentences)—to send messages via a variety of potential carriers such as electricity or light, for example using an electrical telegraph or a flashing light.
- Semaphore signalling uses rods or flags held in particular positions to send messages to the receiver watching them some distance away.
- International maritime signal flags have distinctive markings that represent letters of the alphabet to allow ships to send messages to each other.
- More recently invented, a modem modulates an analog "carrier" signal (such as sound) to encode binary electrical digital information, as a series of binary digital sound pulses. A slightly earlier, surprisingly reliable version of the same concept was to bundle a sequence of audio digital "signal" and "no signal" information (i.e. "sound" and "silence") on magnetic cassette tape for use with early home computers.

Digital (disambiguation)

Digital has multiple meanings. All of the following meanings are based on digital:
- For the electric circuit, see Digital circuit.
- For the company commonly known as "Digital", see Digital Equipment Corporation.
- In control theory, digital control uses a digital computer in the control path.
- In information theory, a digital signal is a signal of digital values.
- Digital library: a library in the "digital world".
- Digital management: management of digital assets.
- Digital Revolution: the revolution of rapid expansion and reduced cost of digital devices.
- Digitalism
- Digital is the nickname of football (soccer) player Vitalis Takawira.

Number

: This article is about numbers such as counting numbers and measurements. For other uses of the term, see Number (disambiguation). A number originally was a count or a measurement. Mathematicians have extended this concept to include abstractions such as the square root of minus one. In common usage, number symbols are often used as labels (highway numbers) or to indicate order (serial numbers).

Examples

The most familiar numbers are the counting numbers or natural numbers. Some writers include 0, thus: . Others do not: . In the base ten number system, now in almost universal use worldwide, the symbols for natural numbers are written using ten digits, 0 through 9. The symbol for the set of all natural numbers is N. If the negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers Z (from the German word "zahlen"). (Some authors use W for the whole numbers, but other authors use W for the natural numbers, so the W symbol is ambiguous.) Negative numbers are used to indicate an opposite. If a positive number is used to indicate distance to the right of some fixed point, a negative number indicates distance to the left. If a positive number indicates a bank deposit, a negative number indicates a withdrawal. Rational numbers are made up of all numbers that can be expressed as a fraction, with integer numerator and non-zero natural number denominator. The fraction m/n represents the quantity arrived at when a whole is divided into n equal parts, and m of those equal parts are chosen. If m is greater than n, the fraction is greater than one. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face Q (for "quotient"). The real numbers are made up of all numbers that can be expressed as a decimal. These are the measuring numbers, and in the base ten number system are written as a string of digits, with a dot (US) or a comma (Europe) to the right of the ones place. The symbol for the real numbers is R. All measurements are necessarily approximations; the accuracy of the approximation depends on the accuracy of the measuring device. Therefore all measurements are properly represented by decimals that end, the last decimal place indicating the accuracy of the measurement. For example, 1.23 inches indicates a measurement accurate to the nearest hundredth of an inch. However, mathematically, when a rational number is expressed as a decimal, it may never end. Thus 1/3 becomes 0.3333... (unending threes). Mathematicians, therefore, consider both decimals that end and decimals that go on forever. The latter represent an infinite series. Some real numbers can be written as fractions, 0.3333... for example. Others cannot, 0.1010010001... for example. A decimal that can be written as a fraction is called rational, a decimal that cannot be written as a fraction is called irrational. A decimal is rational when it either ends or repeats forever. There is a technical sense in which the real numbers are the ideal set of numbers. They are the only complete ordered field. Moving to a greater level of abstraction, and away from counting and measuring, the real numbers can be extended to the complex numbers C. This set of numbers arose, historically, from consideration of the question of whether or not there was any sense in which negative numbers can have a square root. A new number was invented, the square root of negative one, denoted by i, a symbol assigned to this new number by Leonhard Euler. The complex numbers consist of all numbers of the form a + bi, where a and b are real numbers. If b is zero, then a + bi is real. If a is zero, then a + bi is called imaginary. The complex numbers are an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors with complex coefficients. The above symbols are often written in blackboard bold, thus: :

\mathbb\sub\mathbb\sub\mathbb\sub\mathbb\sub\mathbb

While the natural numbers and the real numbers suffice for most everyday purposes, mathematicians have invented many other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example the roots of polynomials with rational coefficients are called the algebraic numbers. Real numbers that are not algebraic are called transcendental numbers. The Gaussian integers are complex numbers a + bi where a and b are integers. Sets of numbers that are not subsets of the complex numbers include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative.

Further generalizations

Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as a kind of number by number theorists.

Numerals and numbering

Numbers should be distinguished from numerals, the symbols used to represent numbers. The number five can be represented by both the base ten numeral 5 and by the Roman numeral V. Notations used to represent numbers are discussed in the article numeral systems. Numbers are often used to give objects unique names. Examples are telephone numbers, social security numbers, and ISBNs.

Extensions

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While real numbers may have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left, with digits in base p, where p is prime. This leads to the p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; but they differ in the infinite case.) The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.

External links

- [http://freepages.history.rootsweb.com/~catshaman/13comp/0numer.htm Mesopotamian and Germanic numbers]

References

- Erich Friedman, [http://www.stetson.edu/~efriedma/numbers.html What's special about this number?]
- [http://www.cut-the-knot.org/do_you_know/numbers.shtml What's a Number?] at cut-the-knot Category:Group theory Category:Numbers __NOTOC__ ko:수 (수학) ja:数 simple:Number th:จำนวน

Binary numbers

The binary numeral system represents numeric values using two symbols, typically 0 and 1. More specifically, binary is a positional notation with a radix of two. Owing to its relatively straightforward implementation in electronic circuitry, the binary system is used internally by virtually all modern computers.

History

Ancient Indian mathematician Pingala presented the first known description of a binary numeral system in the 3rd century BC, which coincided with his discovery of the concept of zero. The modern binary number system was fully documented by Gottfried Leibniz in the 17th century in his article Explication de l'Arithmétique Binaire. Leibniz's uses 0 and 1, like the modern binary numeral system. In 1854, British mathematician George Boole published a landmark paper detailing a system of logic that would become known as Boolean algebra. His logical system proved instrumental in the development of the binary system, particularly in its implementation in electronic circuitry. In 1937, Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits, Shannon's thesis essentially founded practical digital circuit design. In November of 1937, George Stibitz, then working at Bell Labs, completed a relay-based computer he dubbed the "Model K" (for "kitchen", where he had assembled it), which calculated using binary addition. Bell Labs thus authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed January 8, 1940, was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on September 11, 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John Von Neumann, John Mauchly, and Norbert Wiener, who wrote about it in his memoirs.

Representation

A binary number can be represented by any sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. The following sequences of symbols could all be interpreted as different binary numeric values: 1 0 1 0 0 1 1 - | | - - x x x o x o o n y y n bit to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time. ]] The numeric value represented in each case is dependent upon the value assigned to each symbol. In a computer, the numeric values may be represented by two different voltages; on a magnetic disk, magnetic polarities may be used. A "positive", "yes", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use. In keeping with customary representation of numerals using arabic numerals, binary numbers are commonly written using the symbols 0 and 1. When written, binary numerals are often subscripted or suffixed in order to indicate their base, or radix. The following notations are equivalent: :100101 binary (explicit statement of format) :100101b (a suffix indicating binary format) :bin 100101 (a prefix indicating binary format) :100101₂ (a subscript indicating base-2 notation) When spoken, binary numerals are usually pronounced by pronouncing each individual digit, in order to distinguish them from decimal numbers. For example, the binary numeral "100" is pronounced "one zero zero", rather than "one hundred", to make its binary nature explicit, and for purposes of correctness. Since the binary numeral "100" is equal to the decimal value four, it would be confusing, and numerically incorrect, to refer to the numeral as "one hundred."

Counting in binary

Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Decimal counting uses the symbols 0 through 9, while binary only uses the symbols 0 and 1. When the symbols for the first digit are exhausted, the next-higher digit (to the left) is incremented, and counting starts over at 0. In decimal, counting proceeds like so: :00, 01, 02, ... 07, 08, 09 (rightmost digit starts over, and the 0 is incremented) :10, 11, 12, ... 17, 18, 19 (rightmost digit starts over, and the 1 is incremented) :20, 21, 22, ... When the rightmost digit reaches 9, counting returns to 0, and the second digit is incremented. In binary, counting is similar, with the exception that only the two symbols 0 and 1 are used. When 1 is reached, counting begins at 0 again, with the digit to the left being incremented: :000, 001 (rightmost digit starts over, and the second 0 is incremented) :010, 011 (middle and rightmost digits start over, and the first 0 is incremented) :100, 101 (rightmost digit starts over again, middle 0 is incremented) :110, 111...

Binary Simplified!

Okay, for those of us, like me, who have trouble understanding binary, think of it like this... We use a base ten system. What this means, exactly, is that the value of each position in a numerical value can be represented by one of ten possible symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 We are all familiar with these and how the decimal system works using these ten symbols. When we begin counting values, we should start with the symbol 0, and proceed to 9 when counting. We call this the "ones" place. The "ones" place, with those digits, might be thought of as a multiplication problem. 5 can be thought of as 5 X 10^0 (10 to the zeroeth power, which equals 5 X 1, since any number to the zero power is one). [10^0 = 1] As we move to the left of the ones place, we increase the power of 10 by one. Thus, to represent 50 in this same manner, it can be thought of as 5 X 10^1, or 5 X 10.

500 = (5  -  10^2) + (0  -  10^1) + (0  -  10^0) =  5  -  100

$5000 = (5 - 10^3) + (0 - 10^2) + (0 - 10^1) + (0 - 10^0) = 5 - 1000$

When we run out of symbols in the decimal numeral system, we "move to the left" one place and use a '1' to represent the "tens" place. Then we reset the symbol in the "ones" place back to the first symbol, zero. Binary is a base two system which works just like our decimal system, however with only two symbols which can be used to represent numerical values: 0 and 1. We begin in the "ones" place with 0, then go up to 1. Now we are out of symbols, so to represent a higher value, we must place a '1' in the "twos" place, since we don't have a symbol we can use in the binary system for 2, like we do in the decimal system. In the binary numeral system, the value represented as 10 is 0 times (1
- 2^1) + (0
- 2^0). Thus, it equals '2' in our decimal system.

Binary - to - Decimal equivalence:

$1 = 1 - 2^0 = 1 - 1 = 1$

$10 = (1 - 2^1) + (0 - 2^0) = 2 + 0 = 2$

$100 = (1 - 2^2) + (0 - 2^1) + (0 - 2^0) = 4 + 0 + 0 = 4$

While this is in the conversion guide, it is 'hidden', and the conversion guide makes it a little more complicated.

Binary arithmetic

Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.

Addition

decimal, which adds two bits together, producing sum and carry bits.]] The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple: :0 + 0 = 0 :0 + 1 = 1 :1 + 0 = 1 :1 + 1 = 10 (the 1 is carried) Adding two "1" values produces the value "10", equivalent to the decimal value 2. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result exceeds the value of the radix (10), the digit to the left is incremented: :5 + 5 = 10 :7 + 9 = 16 This is known as carrying in most numeral systems. When the result of an addition exceeds the value of the radix, the procedure is to "carry the one" to the left, adding it to the next positional value. Carrying works the same way in binary: 1 1 1 1 1 (carry) 0 1 1 0 1 + 1 0 1 1 1 ------------- = 1 0 0 1 0 0 In this example, two numerals are being added together: 01101 (13 decimal) and 10111 (23 decimal). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 10. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 10 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 11. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100.

Subtraction

Subtraction works in much the same way: :0 - 0 = 0 :0 - 1 = 1 (with borrow) :1 - 0 = 1 :1 - 1 = 0 One binary numeral can be subtracted from another as follows:
-
-
-
- (starred columns are borrowed from) 1 1 0 1 1 1 0 - 1 0 1 1 1 ---------------- = 1 0 1 0 1 1 1 Subtracting a positive number is equivalent to adding a negative number of equal absolute value; computers typically use the two's complement notation to represent negative values. This notation eliminates the need for a separate "subtract" operation. For further details, see two's complement.

Multiplication

Multiplication in binary is similar to its decimal counterpart. Two numbers A and B can be multiplied by partial products: for each digit in B, the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used. The sum of all these partial products gives the final result. Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:
- If the digit in B is 0, the partial product is also 0
- If the digit in B is 1, the partial product is equal to A For example, the binary numbers 1011 and 1010 are multiplied as follows: 1 0 1 1 (A) × 1 0 1 0 (B) --------- 0 0 0 0 ← Corresponds to a zero in B 1 0 1 1 ← Corresponds to a one in B 0 0 0 0 + 1 0 1 1 --------------- = 1 1 0 1 1 1 0 See also Booth's multiplication algorithm.

Division

Binary division is again similar to its decimal counterpart: __________ 1 0 1 | 1 1 0 1 1 Here, the divisor is 101, or 5 decimal, while the dividend is 11011, or 27 decimal. The procedure is the same as that of decimal long division; here, the divisor 101 goes into the first three digits 110 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence: 1 __________ 1 0 1 | 1 1 0 1 1 - 1 0 1 ----- 0 1 1 The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted: 1 0 1 __________ 1 0 1 | 1 1 0 1 1 - 1 0 1 ----- 0 1 1 - 0 0 0 ----- 1 1 1 - 1 0 1 ----- 1 0 Thus, the quotient of 11011 divided by 101 is 101₂, as shown on the top line, while the remainder, shown on the bottom line, is 10₂. In decimal, 27 divided by 5 is 5, with a remainder of 2.

Bitwise logical operations

Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using Boolean logical operators. When a string of binary symbols is manipulated in this way, it is called a bitwise operation; the logical operators AND, OR, and XOR may be performed on corresponding bits in two binary numerals provided as input. The logical NOT operation may be performed on individual bits in a single binary numeral provided as input. Sometimes, such operations may be used as arithmetic short-cuts, and may have other computational benefits as well. For example, discarding the last bit of a binary number (also known as binary shifting), is the decimal equivalent of division by two. See Bitwise operation.

Conversion to and from other numeral systems

Decimal

This method works for conversion from any base, but there are better methods for bases which are powers of two, such as octal and hexadecimal given below. In place-value numeral systems, digits in successively lower, or less significant, positions represent successively smaller powers of the radix. The starting exponent is one less than the number of digits in the number. A five-digit number would start with an exponent of four. In the decimal system, the radix is 10 (ten), so the left-most digit of a five-digit number represents the 10⁴ (ten thousands) position. Consider: :97352₁₀ is equal to: ::9 times 10⁴ (9 × 10000 = 90000) plus ::7 times 10³ (7 × 1000 = 7000) plus ::3 times 10² (3 × 100 = 300) plus ::5 times 10¹ (5 × 10 = 50) plus ::2 times 10⁰ (2 × 1 = 2) Multiplication by the radix is simple. The digits are shifted left, and a 0 is appended to the right end of the number. For example, 9735 times 10 is equal to 97350. So one way to interpret a string of digits is as the last digit added to the radix times all but the last digit. 97352 equals 9735 times 10 plus 2. An example in binary is 1101100111₂ equals 110110011₂ times 2 plus 1. This is the essence of the conversion method. At each step, write the number to be converted as 2
- k + 0 or 2
- k + 1 for an integer k, which becomes the new number to be converted. :118₁₀ equals ::59 x 2 + 0 ::(29 x 2 + 1) x 2 + 0 ::((14 x 2 + 1) x 2 + 1) x 2 + 0 ::(((7 x 2 + 0) x 2 + 1) x 2 + 1) x 2 + 0 ::((((3 x 2 + 1) x 2 + 0) x 2 + 1) x 2 + 1) x 2 + 0 ::(((((1 x 2 + 1) x 2 + 1) x 2 + 0) x 2 + 1) x 2 + 1) x 2 + 0 ::1 x 2⁶ + 1 x 2⁵ + 1 x 2⁴ + 0 x 2³ + 1 x 2² + 1 x 2¹ + 0 x 2⁰ ::1110110₂ So in the algorithm to convert from an integer decimal numeral to its binary equivalent, the number is divided by two, and the remainder written in the ones-place. The result is again divided by two, its remainder written in the next place to the left. This process repeats until the number becomes zero. For example, 118₁₀, in binary, is: Reading the sequence of remainders from the bottom up gives the binary numeral 1110110₂. To convert from binary to decimal is the reverse algorithm. Starting from the left, double the result and add the next digit until there are no more. For example to convert 110010101101₂ to decimal: and the result is 3245₁₀. The fractional parts of a numbers are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving. In a fractional binary number such as .11010110101₂, the first digit is 1/2, the second 1/2², etc. So if there is a 1 in the first place after the decimal, then the number is at least 1/2, and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part. For example, (1/3)₁₀, in binary, is: which is the repeating fraction 0.0101...₂ Or for example, 0.1₁₀, in binary, is: which is also a repeating fraction 0.000110011...₂ It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 0.1 + ... + 0.1, (10 additions) differs from 1 in floating point arithmetic. In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not. The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base. For example, Another way, perhaps quicker and more efficient than the previous, of converting from binary to decimal, is to do so indirectly- first converting (x binary) or (x decimal) to (x hexidecimal) and then converting (x hexidecimal) to the opposite of the former, respectively.

Hexadecimal

Binary may be converted to and from hexadecimal somewhat more easily. This is due to the fact that the radix of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 2⁴, so it takes exactly four digits of binary to represent one digit of hexadecimal. The following table shows each hexadecimal digit along with the equivalent four-digit binary sequence: To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits: :3A₁₆ = 0011 1010₂ :E7₁₆ = 1110 0111₂ To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra 0 bits at the left (called padding). For example: :1010010₂ = 0101 0010 grouped with padding = 52₁₆ :11011101₂ = 1101 1101 grouped = DD₁₆

Octal

Binary is also easily converted to the octal numeral system, since octal uses a radix of 8, which is a power of two (namely, 2³, so it takes exactly three binary digits to represent an octal digit). The correspondence between octal and binary numerals is the same as for the first eight digits of hexadecimal in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so on. Converting from octal to binary proceeds in the same fashion as it does for hexadecimal: :65₈ = 110 101₂ :17₈ = 001 111₂ And from binary to octal: :101100₂ = 101 100₂ grouped = 54₈ :10011₂ = 010 011₂ grouped with padding = 23₈

Representing real numbers

Non-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point (called a decimal point in the decimal system). For example, the binary number 11.01₂ thus means: :1 times 2¹ (1 × 2 = 2) plus :1 times 2⁰ (1 × 1 = 1) plus :0 times 2^-1 (0 × (1/2) = 0) plus :1 times 2^-2 (1 × (1/4) = 0.25) For a total of 3.25 decimal. All dyadic rational numbers p/2^a have a terminating binary numeral -- the binary representation has only finitely many terms after the radix point. Other rational numbers have binary representation, but instead of terminating, they recur, with a finite sequence of digits repeating indefinitely. For instance :1/3₁₀ = 1/11₂ = 0.0101010101...₂ :12₁₀/17₁₀ = 1100₂ / 10001₂ = 0.10110100 10110100 10110100...₂ The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in Decimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the geometric series 2^-1 + 2^-2 + 2^-3 + ... which is 1. Binary numerals which neither terminate nor recur represent irrational numbers. For instance,
- 0.10100100010000100000100.... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
- 1.0110101000001001111001100110011111110... is the binary representation of √2, the square root of 2, another irrational. It has no discernible pattern, although a proof that √2 is irrational requires more than this. See irrational number.

Binary humor

- "Binary is as easy as 1, 10, 11."
- "There are 10 kinds of people in the world - those who understand binary numbers, and those who don't."

External links

- [http://www.insidereality.net/site/content/math/base_conversion.php Simple Conversion Methods]
- [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/index.html Indian mathematics]
- [http://www.cut-the-knot.org/binary.shtml Base Converter] at cut-the-knot
- [http://www.cut-the-knot.org/do_you_know/BinaryHistory.shtml Binary System] at cut-the-knot
- [http://www.cut-the-knot.org/blue/frac_conv.shtml Conversion of Fractions] at cut-the-knot
- [http://leetkey.mozdev.org This FireFox extension supports ASCII/Binary conversions and typing]
- [http://www.permadi.com/tutorial/numHexToDec/ Converting Hexadecimal to Decimal]
- [http://www.permadi.com/tutorial/numDecToHex/ Converting Decimal to Hexadecimal]
- [http://www.paulschou.com/tools/xlate/ Online tool] to translate ASCII to/from Binary, Hex, Decimal and Base64 Category:Computer arithmetic Category:Elementary arithmetic Category:Numeration 2 ja:2進数 th:เลขฐานสอง

Analog

Analog or Analogue may refer to:
- In chemistry, a structural derivative of a parent compound that often differs from it by a single element, see analog (chemistry).
- An analog signal,
- An analog circuit,
- An analog computer,
- Analog Science Fiction and Science Fact magazine,
- A.N.A.L.O.G. (Atari News And Lots Of Games), a [http://www.cyberroach.com/analog/ magazine] focusing on Atari computers,
- A computer program entitled Analog,
- The Federal Analog Act.
- Analog Devices, a semiconductor company.
- See analog at the wiktionary.
- The Analogs - polish street punk band See analogy.

Usage

The spelling analog is predominant in American English, whilst analogue is used in Commonwealth English; see og/ogue. However, the spellings given above should be retained in cases where it forms part of a name or is an acronym. th:แอนะล็อก

Symbol

:For the Romanian choir, see Symbol (choir) A symbol, in its basic sense, is a conventional representation of a concept or quantity; i.e., an idea, object, concept, quality, etc. In more psychological and philosophical terms, all concepts are symbolic in nature, and representations for these concepts are simply token artifacts that are allegorical to (but do not directly codify) a symbolic meaning, or symbolism. Spoken language, for example, consists of distinct auditory tokens for representing symbolic concepts (words), arranged in an order which further suggests their meaning.

Nature of symbols

word]A symbol can be a material object whose shape or origin is related, by nature or convention, to the thing it represents: for instance, the cross is the main symbol of Christianity, and the scepter is a traditional symbol of royal power. A symbol can also be a more or less conventional image (i.e. an icon), or a detail of an image, or even a pattern or color: for example, the olive branch in heraldry represents peace, the halo is a conventional symbol of sainthood in Christian imagery, tartans are symbols of Scottish clans, and the color red is often used as a symbol for socialist movements, especially communism. More often, a symbol is a conventional written or printed sign (specifically, a glyph), usually standing for anything other than a sound (symbols for sounds are usually called graphemes, letters, logograms, diacritics, etc.). Thus mathematical symbols such as π and + represent quantities and operations, currency symbols represent monetary units, chemical symbols represent elements, and so forth. Symbols can also be immaterial entities like sounds, words and gestures. The ringing of gongs and bells, and the banging of a judge's gavel, often have conventional meanings in certain contexts; and bowing is a common way to indicate respect. In fact, every word in a natural language is a symbol for some concept or relationship between concepts. A symbol is usually recognized only within some specific culture, religion, or discipline, but a few hundred symbols are now recognized internationally. See list of common symbols and List of symbols.

Use of symbols

Human beings' ability to manipulate symbols allows them to explore the relationships between ideas, things, concepts, and qualities - far beyond the explorations of which any other species on earth is capable. The discipline of semiotics studies symbols and symbol systems in general; semantics is specifically concerned with the main meaning of words or other linguistic units. Literary works are often admired for their artful use of symbolism, i.e. the use of words, phrases and situations to evoke ideas and feelings beyond their plain interpretations; these uses are the subject of literary semiotics. Religious and metaphysical writings are also known for their use of esoteric symbolism. Alchemical writings made extensive use of symbols for spiritual and chemical processes (which they also saw as symbols of each other). The interpretation of dreams as symbols of one's experiences is a main feature of Freudian psychoanalysis and Jungian analytical psychology.

Etymology

The word "symbol" came to the English language, by way of Middle English, Old French, and Latin, from the Greek σύμβολον súmbolon from the root words σύμ- (sym-) meaning "together" and βολή bolḗ "a throw", having the approximate meaning of "to throw together", so "sign, ticket, or contract".

External links

- [http://www.symbols.com Symbol search engine]
- [http://altreligion.about.com/library/glossary/blsymbols.htm Religious and Cultural Symbols]
- ja:シンボル simple:Symbol

Icon

An icon (from Greek , eikon, "image") is an image, picture, or representation; it is a sign or likeness that stands for an object by signifying or representing it, or by analogy, as in semiotics; in computers an icon is a symbol on the monitor used to signify a command; by extension, icon is also used, particularly in modern popular culture, in the general sense of symbol — i.e. a name, face, picture or even a person readily recognized as having some well-known significance or embodying certain qualities. In Eastern Orthodoxy and other icon painting Christian traditions, the icon is generally a flat panel painting depicting a holy being or object such as Jesus, Mary, saints, angels, or the cross. Icons may also be cast in metal, carved in stone, embroidered on cloth, done in mosaic work, printed on paper or metal, etc.

Images in Religion

Throughout history religion has often made use of images, whether in two dimensions or three. Some, such as Hinduism, have a very rich iconography called murti, while others, such as Islam, severely limit the use of visual representations. The function and degree to which images are used or permitted, and whether they are for purposes of ornament, instruction, inspiration, or treated as sacred objects of veneration or worship, thus depends upon the tenets of a given religion.

Icons in Christianity

Christianity originated as a movement within Judaism during a time when there was great concern about idolatry. There is no evidence of the making and use of painted icons or of similar religious images by Christians within the New Testament writings. However, Eastern Orthodox theologian Rev. Dr. Steven Bigham writes (Early Christian Attitudes Toward Images, Orthodox Research Institute, 2004), "The first thing to note is that there is a total silence about Christian and non-idolatrous images. It is important to note that the silence is in the New Testament texts, and this silence should not be interpreted as describing all the activities of the Apostles or 1st century Christians. St. John himself said that 'Jesus did many other signs in the presence of the disciples, which are not written in this book...' (Jn 20.30). We could easily add that the Apostles also did and said many things not recorded in the New Testament. It is obvious, therefore, that we do not have a complete account of the activities and sayings of the Apostles. So, if we want to find out if the first Christians made or ordered any kind of figurative art, the New Testament is of no use whatsoever. The silence is a fact, but the reason given for the silence varies from exegete to exeget depending on his assumptions." In other words, relying only upon the New Testament as evidence of no painted icons amounts to an argument from silence. Though the word eikon is found in the New Testament (see below), it is never in the context of painted icons. The earliest written records available of Christian images treated like icons are in a pagan or Gnostic context. Alexander Severus (A.D. 222–235) kept a domestic chapel for the veneration of images of deified emperors, of portraits of his ancestors, and of Christ, Apollonius, Orpheus and Abraham (Lampridius, Life of Alexander Severus xxix.). Irenaeus, in his Against Heresies 1:25;6, says of the Gnostic Carpocratians, “They also possess images, some of them painted, and others formed from different kinds of material; while they maintain that a likeness of Christ was made by Pilate at that time when Jesus lived among them. They crown these images, and set them up along with the images of the philosophers of the world that is to say, with the images of Pythagoras, and Plato, and Aristotle, and the rest. They have also other modes of honouring these images, after the same manner of the Gentiles [pagans].” A criticism of image veneration is found in the apocryphal Acts of John (generally considered a gnostic work), in which the Apostle John discovers that one of his followers has had a portrait made of him, and is venerating it: (27) “...he [John] went into the bedchamber, and saw the portrait of an old man crowned with garlands, and lamps and altars set before it. And he called him and said: Lycomedes, what do you mean by this matter of the portrait? Can it be one of thy gods that is painted here? For I see that you are still living in heathen fashion.” Later in the passage John says, "But this that you have now done is childish and imperfect: you have drawn a dead likeness of the dead." In addition to the legend that Pilate had made an image of Christ, the 4th Century bishop Eusebius, in his Church History, provides another reference to a “first” icon of Jesus. He relates that King Abgar of Edessa sent a letter to Jesus at Jerusalem, asking Jesus to come and heal him of an illness. In this version there is no image. Then, in the later account found in the Syriac Doctrine of Addai, a painted image of Jesus is mentioned in the story; and even later, in the account given by Evagrius, the painted image is transformed into an image that miraculously appeared on a towel when Christ pressed the cloth to his wet face (Veronica and her Cloth, Kuryluk, Ewa, Basil Blackwell, Cambridge, 1991). Further legends relate that the cloth remained in Edessa until the 10th century, when it was taken to Constantinople. In 1204 it was lost when Constantinople was sacked by Crusaders. Elsewhere in his Church History, Eusebius reports seeing what he took to be portraits of Jesus, Peter and Paul, and also mentions a bronze statue at Banias / Paneas, of which he wrote, "They say that this statue is an image of Jesus" (H.E. 7:18); further, he relates that locals thought the image to be a memorial of the healing of the woman with an issue of blood by Jesus (Luke 8:43-48), because it depicted a standing man wearing a double cloak and with arm outstretched, and a woman kneeling before him with arms reaching out as if in supplication. Some scholars today think it possible to have been a misidentified pagan statue whose true identity had been forgotten; some have thought it to be Aesculapius, the God of healing, but the description of the standing figure and the woman kneeling in supplication is precisely that found on coins depicting the bearded emperor Hadrian reaching out to a female figure symbolizing a province kneeling before him (see John Francis Wilson's Caesarea Philippi: Banias, the Lost City of Pan; I.B Taurus, London, 2004). When Christianity was legalized by the emperor Constantine within the Roman Empire in the early 4th Century, huge numbers of pagans became converts. This created the opportunity for the transfer of allegiance and practice from the old gods and heroes to the new religion, and for the gradual adaptation of the old system of image making and veneration to a Christian context. "By the early fifth century, we know of the ownership of private icons of saints; by c. 480-500, we can be sure that the inside of a saint's shrine would be adorned with images and votive portraits, a practice which had probably begun earlier" (Pagans and Christians, Robin Lane Fox, Alfred A. Knopf, New York, 1989).

Images from Constantine to Justinian

Constantine to Justinian (337-430) After the legalization of Christianity under Constantine, and its adoption as the Roman state religion under Theodosius I, Christian art began to change remarkably not only in quality and sophistication, but also in nature. This was in no small part due to Christians being free for the first time to express their faith openly without persecution from the state, in addition to the faith spreading to the non-poor segments of society. Paintings of martyrs and their feats began to appear, and early writers commented on their lifelike effect, one of the elements a few Christian writers criticized in pagan art — the ability to imitate life. The writers mostly criticized that the pagan works of art pointed to false gods, and thusly constituted idolatry. Nilus of Sinai, in his Letter to Heliodorus Silentiarius, records a miracle in which St. Plato of Ankyra appeared to a Christian in a dream. The Saint was recognized because the young man had often seen his portrait. This recognition of a religious figure from likeness to an image was also a characteristic of pagan pious accounts of appearances of gods to humans. However, in the Old Testament we read of prophets having dreams of various heavenly figures, including a vision of God who appeared to Daniel as an elderly man, the "Ancient of Days". It is also in this period that the first mention of an image of Mary painted from life appears, though earlier paintings on cave walls bear resemblance to modern icons of Mary. Theodorus Lector, in the History of the Church 1:1 (excerpted by Nicephorus Callistus Xanthopoulos) stated that Eudokia (wife of Theodosius II , died 460) sent an image of “the Mother of God” from Jerusalem to Pulcheria, daughter of the Emperor Arcadius (this is by some considered a later interpolation). The image was specified to have been “painted by the Apostle Luke.” In later tradition the number of icons of Mary attributed to Luke would greatly multiply. The first depictions of Jesus were generic rather than portrait images, generally representing him as a beardless young man. It was some time before the earliest examples of the long-haired, bearded face that was later to become standardized as the image of Jesus appeared. And when they began to appear there was still variation. Augustine of Hippo (354-430) said that no one knew the appearance of Jesus or that of Mary (De Trinitatis 8:4-5), though it should be noted that Augustine wasn't a resident of the Holy Lands and therefore wasn't familiar with the local populations and their oral traditions. Gradually, paintings of Jesus took on characteristics of portrait images. At this time the manner of depicting Jesus was not yet uniform, and there was some controversy over which of the two most common forms was to be favored. The first or “Semitic” form showed Jesus with short and “frizzy” hair; the second showed a bearded Jesus with hair parted in the middle, the manner in which the god Zeus was depicted. Theodorus Lector remarked (Church History 1:15) that of the two, the one with short and frizzy hair was “more authentic.” He also relates a story (excerpted by John of Damascus) that a pagan commissioned to paint an image of Jesus used the “Zeus” form instead of the “Semitic” form, and that as punishment his hands withered. Though their development was gradual, we can date the full-blown appearance and general ecclesiastical (as opposed to simply popular or local) acceptance of Christian images as venerated and miracle-working objects to the 6th century, when, as Hans Belting writes, "We first hear of the church's use of religious images...(Likeness and Presence, University of Chicago Press,1994). "...As we reach the second half of the sixth century, we find that images are attracting direct veneration and some of them are credited with the performance of miracles" (Patricia Karlin-Hayter, The Oxford History of Byzantium, Oxford, 2002). Cyril Mango writes, "In the post-Justinianic period the icon assumes an ever increasing role in popular devotion, and there is a proliferation of miracle stories connected with icons, some of them rather shocking to our eyes" (The Art of the Byzantine Empire 312-1453, University of Toronto Press, 1986). However, the earlier references by Eusebius and Irenaeus indicate veneration of images and reported miracles associated with them as early as the second century. It must also be noted that what might be shocking to our contemporary eyes may not have been viewed as such by the early Christians. In Acts 5:15 of the New Testament, it is written that "people brought the sick into the streets and laid them on beds and mats so that at least Peter's shadow might fall on some of them as he passed by."

The Iconoclast Period

Main article: Iconoclasm There was a continuing opposition to misuse of images within Christianity from very early times. "Whenever images threatened to gain undue influence within the church, theologians have sought to strip them of their power" (Belting, Hans; Likeness and Presence, Chicago and London, 1994). Further,"there is no century between the fourth and the eighth in which there is not some evidence of opposition to images even within the Church (Kitzinger, Ernst; The Cult of Images in the Age before Iconoclasm, Dumbarton Oaks, 1954; repeated by Pelikan, Jaroslav; The Spirit of Eastern Christendom 600-1700, University of Chicago Press, 1974). Nonetheless, popular favoritism for icons guaranteed their continued existence, while as yet no systematic apologia for or against icons, or doctrinal authorization or condemnation of icons existed. The use of icons was seriously challenged by Byzantine Imperial authority in the 8th century. Though by this time opposition to images was strongly entrenched in Judaism and in the rising religion of Islam, attribution of the impetus toward an iconoclastic movement in Eastern Orthodoxy to Muslims or Jews "seems to have been highly exaggerated, both by contemporaries and by modern scholars" (see Pelikan, The Spirit of Eastern Christendom). Though significant in the history of religious doctrine, the Byzantine controversy over images is not seen as of primary importance in Byzantine history. "Few historians still hold it to have been the greatest issue of the period..." (Patricia Karlin-Hayter, Oxford History of Byzantium, Oxford University Press, 2002). The Iconoclastic Period began when images were banned by Emperor Leo III sometime between 726 and 730. Under his son Constantine V, an ecumenical council forbidding image veneration was held at Hieria near Constantinople in 754. Image veneration was later reinstated by the Empress Regent Irene, under whom another ecumenical council was held reversing the decisions of the previous iconoclast council and taking its title as Seventh Ecumenical Council. The council anathemized all who hold to iconoclasm, i.e. those who held that veneration of images constitutes idolatry. Then the ban was enforced again by Leo V in 815. And finally icon veneration was decisively restored by Empress Regent Theodora. Empress Regent Theodora

Icons in Greek-speaking regions

Icons are used particularly among Eastern Orthodox, Oriental Orthodox, Coptic and Eastern-rite Catholic populations. The icon painting tradition developed in Byzantium, with Constantinople as the chief city. Few icons from early Constantinople have survived, first because of the Iconoclastic reforms during which many were destroyed, second because of plundering by Venetians in 1204 during the Crusades, and finally the taking of the city by the Islamic Turks in 1453. Still, both some panel paintings and mosaics, etc. still exist. Early icons such as those preserved at the Monastery of St. Catherine at Sinai are realistic in appearance, in contrast to the later stylization. They are very similar to the mummy portraits done in encaustic wax and found at Faiyum in Egypt. In the Comnenian Period (1081-1185), religious sculpture was abandoned in favor of panel painting. The style of the time was severe, hieratic and distant. In the late Comnenian period this severity softened, and emotion, formerly avoided, entered icon painting. This was particularly evident in outlying regions influenced by Byzantine culture, now in Macedonia and the former Yugoslavia. The tendency toward emotionalism in icons continued in the Paleologan Period, which began in 1261. Paleologan art reached its pinnacle in paintings such as those of the Kariye Camii (former Chora Monastery). In the last half of the 1300s, Paleologan saints were painted in an exaggerated manner, very slim and in contorted positions. After the fall of Constantinople to the Turks in 1453, the Byzantine tradition was carried on in regions previously influenced by its religion and culture--the Balkans and Russia, Georgia, and in the Greek-speaking realm, on Crete. Crete, at that time, was under Venetian control and became a thriving center of art of the Scuola di San Luca, the "School of St. Luke," an organized guild of painters. Cretan painting was heavily patronized both by Catholics of Venetian territories and by Eastern Orthodox. For ease of transport, Cretan iconographers specialized in panel paintings, and developed the ability to work in many styles to fit the taste of various patrons. In 1669 the city of Heraklion, on Crete, which at one time boasted at least 120 painters, finally fell to the Turks, and from that time Greek icon painting went into a decline, with a revival attempted in the 20th century by art reformers such as Photios Kontoglou, who emphasized a return to earlier styles.

Icons in Russia

Main article: Russian icons Russian icons are typically paintings on wood, often small, though some in churches and monasteries may be as large as a table top. Many religious homes in Russia have icons hanging on the wall in the krasny ugol, the "red" or "beautiful" corner. There is a rich history and elaborate religious symbolism associated with icons. In Russian churches, the nave is typically separated from the sanctuary by an iconostasis (Russian ikonostás) a wall of icons. The use and making of icons entered Kievan Rus' (which later expanded to become the Russian Empire) following its conversion to Orthodox Christianity in 988 A.D. As a general rule, these icons strictly followed models and formulas hallowed by usage, some of which had originated in Constantinople. As time passed, the Russians widened the vocabulary of types and styles far beyond anything found elsewhere. The personal, improvisatory and creative traditions of Western European religious art are largely lacking in Russia before the 17th century, when Russian icon painting became strongly influenced by religious paintings and engravings from both Protestant and Catholic Europe. In the mid-1600s changes in liturgy and practice instituted by Patriarch Nikon resulted in a split in the Russian Orthodox Church. The traditionalists, the persecuted "Old Ritualists" or Old Believers," continued the traditional stylization of icons, while the State Church modified its practice. From that time icons began to be painted not only in the traditional stylized and nonrealistic mode, but also in a mixture of Russian stylization and Western European realism, and in a Western European manner very much like that of Catholic religious art of the time.

Icon Traditions in Other Regions

In Romania, icons painted as reversed images on glass and set in frames were common in the 19th century and are still made. "In the Transylvanian countryside, the expensive icons on panels imported from Moldavia, Wallachia, and Mt. Athos were gradually replaced by small, locally produced icons on glass, which were much less expensive and thus accessible to the Transylvanian peasants..." (Romanian Icons on Glass, Dancu, Juliana and Dumitru Dancu, Wayne State University Press, 1982). The Egyptian Coptic Church and the Ethiopian Church also have distinctive, living icon painting traditions.

The Protestant Reformation

The abundant use and veneration historically accorded images in the Roman Catholic Church was a point of contention for Protestant reformers, who varied in their attitudes toward images. In the consequent religious struggles many statues were removed from churches, and there was also destruction of images in some cases. Though followers of Zwingli and Calvin were more severe in their rejection, Lutherans tended to be moderate with many of there parishes having displays of statues and crucifixes. A joint Lutheran-Orthodox statement in Helsinki reaffirmed the Ecumenical Council decisions on the nature of Christ and the veneration of images: "The Seventh Ecumenical Council, the Second Council of Nicaea in 787, which rejected iconoclasm and restored the veneration of icons in the churches, was not part of the tradition received by the Reformation. Lutherans, however, rejected the iconoclasm of the 16th century, and affirmed the distinction between adoration due to the Triune God alone and all other forms of veneration. Through historical research this council has become better known. Nevertheless it does not have the same significance for Lutherans as it does for the Orthodox. Yet, Lutherans and Orthodox are in agreement that the Second Council of Nicaea confirms the christological teaching of the earlier councils and in setting forth the role of images (icons) in the lives of the faithful reaffirms the reality of the incarnation of the eternal Word of God, when it states: "The more frequently, Christ, Mary, the mother of God, and the saints are seen, the more are those who see them drawn to remember and long for those who serve as models, and to pay these icons the tribute of salutation and respectful veneration. Certainly this is not the full adoration in accordance with our faith, which is properly paid only to the divine nature, but it resembles that given to the figure of the honored and life?giving cross, and also to the holy books of the gospels and to other sacred objects" (Definition of the Second Council of Nicaea)."

Icons and Images in Contemporary Christianity

Today attitudes can vary even from church to church within a given denomination, whether Catholic or Protestant. Protestants generally use religious art for teaching and for inspiration, but such images are not venerated as in Orthodoxy, and many Protestant church sanctuaries contain no imagery at all. After the Second Vatican Council declared in the 1960s that the use of statues and pictures in churches should be moderate, most statuary was removed from many Catholic Churches. Eastern Orthodoxy, however, continues to give such strong importance to the use and veneration of icons that they are often seen as the chief symbol of Orthodoxy. Catholicism has a long tradition of valuing the arts and patronized a significant number of famous artists. Present-day imagery within Roman Catholicism varies in style from traditional to modern, and is often affected by trends in the art world in general. Ethiopian Ethiopian Icons are often illuminated with a candle or jar of oil with a wick. (Beeswax for candles and olive oil for oil lamps are preferred because they burn very cleanly, although other materials are sometimes used.) The illumination of religious images with lamps or candles is an ancient practice pre-dating Christianity. Historically and even today among conservative Eastern Orthodox there are reports of miraculous icons that exude a fragrant, healing oil. When these reports are verified by Orthodox clergy, they are still explained as miracles performed by God through the prayers of the saint, rather than being magical properties of the painted wood itself.

Eastern Orthodox and Roman Catholic teaching about Icons

Icons are used particularly in Eastern Orthodox, Coptic Orthodox, Oriental Orthodox, and Eastern-rite Catholic churches. The Eastern Orthodox view of the origin of icons is quite different from that of some secular scholars and from some in contemporary Roman Catholic circles: "The Orthodox Church maintains and teaches that the sacred image has existed from the beginning of Christianity" (Leonid Ouspensky, Theology of the Icon," St. Vladimir's Seminary Press, 1978). Accounts that some non-Orthodox writers consider legends are, within Eastern Orthodoxy, accepted as history, because they are a part of Church Tradition. Thus accounts such as that of the miraculous "Image Not Made by Hands," and the weeping and moving "Mother of God of the Sign" of Novgorod are accepted as fact: "Church Tradition tells us, for example, of the existence of an Icon of the Savior during His lifetime (the "Icon-Made-Without-Hands") and of Icons of the Most-Holy Theotokos [Mary] immediately after Him." (These Truths we Hold, St. Tikhon's Seminary Press, 1986). Eastern Orthodox further believe that "a clear understanding of the importance of Icons" was part of the church from its very beginning, and has never changed, although explanations of their importance may have developed over time. This is due to the fact that iconography is rooted in the theology of the Incarnation (Christ being the eikon of God) which didn't change, though its subsequent clarification within the Church occurred over the period of the first seven Ecumenical Councils. Also, icons served as tools of edification for the faithful during most of the history of Christendom when most couldn't read nor write. Eastern Orthodox find the first instance of an image or icon in the Bible when God made man in His own image (Septuagint Greek eikona), recorded in Genesis 1:26-27. In Exodus, God commanded that the Israelites not make any graven image; but soon afterwards, he commanded that they make graven images of cherubim and other like things, both as statues and woven on tapestries. Later, Solomon included still more such imagery when he built the first temple. Eastern Orthodox believe these qualify as icons, in that they were visible images depicting heavenly beings and, in the case of the cherubim, used to indirectly indicate God's presence above the Ark. In Numbers it is written that God told Moses to make a bronze serpent and hold it up, so that anyone looking at the snake would be healed of their snakebites. In John 3, Jesus refers to the same serpent, saying that he must be lifted up in the same way that the serpent was. John of Damascus also regarded the brazen serpent as an icon. Further, Jesus Christ himself is called the "image of the invisible God" in Colossians 1:15, and is therefore in one sense an icon. As people are also made in God's images, people are also considered to be living icons, and are therefore "censed" along with painted icons during Orthodox prayer services. According to John of Damascus, anyone who tries to destroy icons "is the enemy of Christ, the Holy Mother of God and the saints, and is the defender of the Devil and his demons." This is because the theology behind icons is closely tied to the Incarnational theology of the humanity and divinity of Jesus, so that attacks on icons typically have the effect of undermining or attacking the Incarnation of Jesus himself as elucidated in the Ecumenical Councils. The Eastern Orthodox teaching regarding veneration of icons is that the praise and veneration shown to the icon passes over to the archetype (Basil of Caesarea,On the Holy Spirit 18:45: "The honor paid to the image passes to the prototype"). Thus to kiss an icon of Christ, in the Eastern Orthodox view, is to show love towards Christ Jesus himself, not mere wood and paint making up the physical substance of the icon. Worship of the icon as somehow entirely separate from its prototype is expressly forbidden by the Seventh Ecumenical Council; standard teaching in the Eastern Orthodox and Roman Catholic churches alike conforms to this principle. The Catholic Church accepts the same Councils and the canons therein which codified the teaching of icon veneration. The Latin Church of the West, which after 1054 was to become separate as the Roman Catholic Church, accepted the decrees of the iconodule Seventh Ecumenical Council regarding images. There is some minor difference, however, in the Catholic attitude to images from that of the Orthodox. Following Gregory the Great, Catholics emphasize the role of images as the Biblia Pauperum, the “Bible of the Poor,” from which those who could not read could nonetheless learn. This view of images as educational is shared by most Protestants. Catholics also, however, accept in principle the Eastern Orthodox veneration of images, believing that whenever approached, images of the cross, saints, etc. are to be reverenced. Though using both flat wooden panel and stretched canvas paintings, Catholics traditionally have also favored images in the form of three-dimensional statuary, whereas in the East statuary is much less widely employed.

Eikon in the Septuagint

The Greek word eikon means an image or likeness of any kind. Anything that represents something else is an eikon. Nothing is implied about sanctity or its absence, or veneration or its absence by the word itself. The Septuagint is the Greek translation of the Hebrew Scriptures used by the early Christians, and Eastern Orthodox consider it the only authoritative text of those Scriptures. In it the word eikon is used for everything from man being made in the divine image to the "molten idol" placed by Manasses in the Temple. The word eikon is found in: #Genesis 1:26-27; #Genesis 5:1-3; #Genesis 9:6; #Deuteronomy 4:16 #1 Samuel (1 Kings) 6:11 (Alexandrian manuscript); #2 Kings 11:18; #2 Chronicles 33:7; #Psalm 38:7 #Psalm 72:20; #Isaiah 40, 19-20; #Ezekiel 7:20; #Ezekiel 8:5 (Alexandrian manuscript); #Ezekiel 16:17; Ezekiel 23:14; Daniel 2:31,32,34,35; Daniel 3:1,2,3,5,7,11,12,14,15,18; Hosea 13:2 Be aware that Septuagint numberings and names and the English Bible numberings and names are not uniformly identical.

Eikon in the New Testament

In the New Testament the term is used for everything from Jesus as the image of the invisible God (Colossians 1:15) to the image of Caesar on a Roman coin (Matthew 22:20) to the image of the Beast in the Apocalypse (Revelation 14:19). Here is a complete listing: #Matthew 22:20; #Mark 12:16 #Luke 20:24 #Romans 1:23 #Romans 8:29; #1 Corinthians 11:7; #1 Corinthians 15:49 #2 Corinthings 3:18; #2 Corinthians 4:4; #Colossians 1:15; #Colossians 3:10; #Hebrews 10:1; #Revelation 13:13; #Revelation 13:15; #Revelation 14:9; #Revelation 14:11 #Revelation 15:2 #Revelation 16:2 #Revelation 19:20; #Revelation 20:4.
Icons in Hinduism
In Hinduism, the icon is called murti.
See also

- Simulacrum
- Crucifix
- Templon
- Proskynetarion
- John Climacus
- Eastern Orthodoxy
- Georgian Orthodox and Apostolic Church
- Andrei Rublev
- Symbolism
- Religious symbolism
- Jewish symbolism
- Christian symbolism
- Anthropology of religion
- List of religious topics
- Emblem
- Iconoclasm
- Iconography
- Iconostasis
- Ideogram
- Idolatry
- Image
- Ishta-Deva
- Lingam
- Logotype
- Masking
- Murti
- Saligrama
- Sign
- Symbol
- Veneration
External links; various points of view

- [http://www.angelfire.com/pa3/OldWorldBasic/NewQ_Anew.html#Icon What is an icon?]
- [http://www.newadvent.org/cathen/07664a.htm Catholic Encyclopedia:]"Veneration of Images" (The C.E. avoids an entry "Icon")
- [http://www.arcadianart.fi/eng/ Religious Icons Gallery]
- [http://passmoore.com/pictures.php?S=ic Some examples of orthodox iconography]
- [http://www.fisheaters.com/images.html The Use of Statues and Other Icons in the Latin Church]
- [http://www.instaplanet.com/icon.html Icons and links to Russian lacquerware, etc.]
- [http://www.xpucmoc.org/icon.htm Theology of Icons: a Protestant Perspective]
- [http://www.orthodoxinfo.com/general/icon_faq.aspx The Icon Faq; an Eastern Orthodox perspective]
- [http://www.icon-art.info/index.php?lng=en Russian Icons Gallery] and a [http://www.icon-art.info/phpBB2/viewforum.php?f=9 forum] on this site (both links in English).
- [http://www.pressiechurch.org/Theol_2/eastern_orthodoxy.htm Eastern Orthodoxy, Icons, and Christology; a Protestant perspective]
- [http://www.monasteryicons.com/ Contemporary Byzantine style Icons and articles on iconography] at monasteryicons.com. Category:Religious objects Category:Art genres ja:イコン

Digit

A digit is:
- In anatomy, a finger or toe.
- In mathematics and computer science, a numerical digit is a symbol used in the representation of integers or real numbers in positional numeral systems.
- In metrology, a digit (unit) is an ancient unit based on the size of an human finger.
- Digit (magazine) is an Indian information technology magazine.

Finger

:For the network protocol, see finger protocol. For the hand gesture, see the finger. ----

Human fingers; 15kb
Fingers of the human left hand

The finger is any of the digits of the hand in humans and other species such as the great apes. The grace of the fingers is not sacrificed to their dexterity due to the placement of their muscles in the forearm, with motion communicated via long tendons which may be observed on the back of the hand. A notable exception is the thumb, with its flexor and rotators comprised in the hand itself. The bones of the fingers are called phalanges (singular phalanx); the thumb has two phalanges, and the other fingers have three. The fingers' names in English are shown below, from the most radial to the most ulnar: # thumb # index finger, pointer finger, or forefinger # middle finger or long finger # ring finger # little finger or pinky finger

Anatomy of the finger

In anatomy, the thumb is the first finger and the little finger is the fifth finger. Thus the third finger means the middle finger in anatomy, not the ring finger as in daily English. Relative to much of the skin of the human body, the fingertips have a high concentration of nerve endings, equipping them as centers of tactile sensation; touching something or someone is often done with the hands and in particular the fingers. There are three bones in each finger called the proximal phalanx, the middle phalanx and the distal phalanx. Each finger has three joints. The first joint is where the finger joins the hand. This joint is where the bones that form the palm of the hand, the metacarpals, join with the first bone of the finger, called the proximal phalanx. The second joint is the proximal interphalangeal joint, sometimes called the PIP joint for short. The last joint of the finger is called the distal interphalangeal joint, or DIP. Each of these joints is covered with articular cartilage. Articular cartilage is the smooth spongy material that covers the end of bones that make up a joint. The cartilage allows the bones to slide easily against one another as the joint moves through its range of motion. Another important example of this capacity is in the ability to read Braille. Additionally, prehension is enhanced by the presence of the ridges and whorls known as fingerprints. Each finger is protected at its dorsal terminus by a fingernail (Latin unguis, unguiculus). In young children, the fingertip is one of the few tissues in the human body capable of full regeneration, although this ability disappears after about age 6. [http://www.straightdope.com/mailbag/mnewlimb.html]

Binary

Binary may mean:
- Binary (novel), a 1972 novel by Michael Crichton (writing as John Lange).
- Binary (music), a musical form that consists of two parts.
- Binary (comics), a superheroine in the Marvel Universe.
- Binary (chemical weapon), one that contains two capsules, each of which contains a chemical, that when combined with the contents of the other, will react to make a toxic agent.
- Binary explosive, composed of two non-explosive substances which only become explosive when mixed.
- Binomial nomenclature, in the taxonomy of living organisms, is sometimes called binary nomenclature.
- Binary star, a stellar system consisting of two nearby stars that revolve around a common center of mass.
- Binary thinking, a term to describe dichotomous perspectives on an issue.
- Binary numeral system, a representation for numbers that uses only zeroes and ones as digits.
- Binary and text files, a computer file that comprises a sequence of encoded numerical values rather than human-readable text
- Executable, a computer file containing machine code that can be executed by the operating system.
- In higher education, an education system that includes both polytechnic, or college, and university style institutions. These institutions are often intended to complement each other and form an important basis in the overall education policy and infrastructure of a country or region. ja:バイナリ

Digital audio

Digital audio describes sound recording and reproduction systems which work by using a digital representation of the audio waveform.

Technology overview

Sampling

To convert a signal from continuous time to discrete time, the value of the signal is measured at certain intervals in time. This process is known as sampling. One can think of sampling as taking "snapshots" of a certain signal as it moves continuously in time. The rate at which the values of the signal level is taken is known as the sampling rate. Modern digital audio is based on two fundamental theorems on sampling: the Nyquist Theorem and frequency analysis based on Fourier transforms. The Nyquist Theorem states it is possible to recreate perfect bandwidth-limited signals by sampling at equal to or more than twice the highest frequency component. Fourier Transforms allow signals in time domain to be broken down into an integral of sinusoidal functions multiplied by its amplitude. Since the normal human range of hearing corresponds to no more than 20 KHz, the sampling rate will have to be higher than 40 kHz to fully represent the range of human audible frequencies. According to the Nyquist Theorem it is not necessary to sample at a higher rate for accuracy, as a human audible waveform could be as perfectly recreated at either 40 kHz or, say, 192 kHz. Any extra sampling information is superfluous. However, due to benefits such as less steep cut-off filters which can lead to better sound reproduction, a variety of current professional applications and storage capacities such as DVD-Audio are using higher sampling rates to store their data.

Quantization

A process known both from audio technology and computer aided music composition. In computer aided music composition the term refers to rounding off rhythm values to whole tone multiples of the beat speed (tempo) or other specified rhythm value, and serves the improvement of musical timing. It is a process which records or changes temporal phenomena of acoustic music.

Quantization Error

Quantization Error occus when the A/D converter tries to quantize a signal that is too quiet. When quantizing, the converter rounds decimels to the nearest whole bit. If the acoustic signal level equals anywhere from 0.5 to 1.49 Volts, it is quantized as 1 Volt. If the acoustic signal level equals anywhere from 0 to 0.49 Volts, it is quantized as 0 Volts. That original acoustic signal is never quantized and is therefore lost. Quantization error = 1/2LSB (Least Significant Bit) The higher resolution you quantize at, the less noticeable the quantization error. For example: If you quantize at 1 bit: -The LSB is 1 Volt and thus the quantization error = 0.5 Volts -The maximum resolution for a 1 bit recording is 1 Volt -The quantization error is half the maximum resolution which means that 50% of the original recording is vulnerable to quantization error. If you quantize at 2 bits: -The LSB is 1 Volt and thus the quantization error = 0.5 Volts -The maximum resolution for a 2 bit recording is 3 Volts -This leaves 16.67% of the original signal vulnerable to quantization error. This obviously has a lot to do with the dynamic range of the original acoustic signal. If the signal is always at least 1/2LSB, the signal will always be quantized, and no information will be lost.

Digitization

Dither

Noise-shaping

Jitter

For more information, please see the Wikipedia article Jitter.

Methods of digital encoding

Pulse-code modulation

The most common method of creating digital audio is Pulse-code modulation (PCM). PCM digital audio is typically sampled at 44.1 kHz for CD recordings, or higher for professional audio applications. For comparison, speech signals for telephony are only sampled at 8 kHz. Higher sample rates for professional recording are becoming popular. These include 88.2 kHz, 96 kHz, and 192 kHz. The amplitude of each sample is a numeric value that is represented by a certain number of bits. The more bits that are used to represent the amplitude, the greater the dynamic range that can be represented, with each bit providing a gain of approximately 6 dB. The dynamic range of 16 bit digital audio is therefore approximately 96 dB, whereas the dynamic range of 24 bit digital audio is 144 dB. 8 bit digital audio has a dynamic range of approximately 48 dB. The amount of data created by digital audio is quite large. 16 bits per sample at 44.1 kHz creates 705600 bits per second (8 bits = 1 byte). Thus for a stereo recording, approximately 10 MB will be generated per minute. 24 bit, 96 kHz digital audio has a bit rate of 2304000 bits per second, or around 33 MB per minute for stereo. Due to this, different forms of audio data compression have recently become more popular. Another method of creating a digital representation of the audio waveform is Direct Stream Digital or DSD. The Super audio compact disc uses this format. Since digital audio, unlike analog audio, is always accompanied implicitly or explicitly by a sample clock, synchronization is a crucial consideration in digital audio systems. This is usually accomplished by genlocking all the systems in a facility to a single master audio clock. Plesiochronous operation is not advisable, as it tends to result in widespread hard-to-debug problems.

Digital audio technologies

- Digital audio tape (DAT)
- DAB (Digital Audio Broadcasting)
- Compact disc (CD)
- DVD
- Minidisc
- Super audio compact disc
- Digital audio workstation
- and various audio file formats

Digital audio interfaces and interconnects

- AC97 (Audio Codec 1997) interface between Integrated circuits on PC motherboards
- ADAT interface
- AES/EBU interface with XLR connectors
- AES47, Professional AES3 digital audio over Asynchronous Transfer Mode networks
- I2S (Inter-IC sound) interface between Integrated circuits in consumer electronics
- MIDI -- low-bandwidth interconnect for carrying instrument data; cannot carry sound
- S/PDIF, either over coaxial cable or TOSLINK Audio signals can also be carried losslessly over general-purpose buses such as USB or FireWire.

External links

- [http://www.dmalham.freeserve.co.uk/adat.html Information on the ADAT interface]
- [http://www.studioathome.com Home audio recording forum] Category:Audio engineering
-

Digital photography

] ]] Digital photography, as opposed to film photography, uses an electronic sensor to record the image as binary data. This facilitates storage and editing of the images on personal computers. Digital cameras now outsell film cameras and include features not found in film cameras such as the ability to shoot video and record audio. Some other devices, such as mobile phones, now include digital photography features.

History

The first video tape recorder (VTR) was made in 1951 that captured live images from television cameras and saved the information in digital form onto magnetic tapes. Bing Crosby labs pioneered the VTR research, and was put in common use in the television industry. During 1960s, governments took an interest in digital imaging due to its application in spy satellites. Government use of digital technology helped advance the science of digital imaging, however, the private sector also made significant contributions. Texas Instruments patented a film-less electronic camera in 1972, the first to do so. In August, 1981, Sony released the Sony Mavica electronic still camera, the camera which was the first commercial electronic camera. Images were recorded onto a mini disc and then put into a video reader that was connected to a television monitor or color printer. However, the early Mavica cannot be considered a true digital camera even though it started the digital camera revolution. It was a video camera that took video freeze-frames. Kodak took an interest in digital cameras, and since mid-1970s, it invented several solid-state sensors to convert light and pictures directly to digital form. In 1986, Kodak scientists invented the world's first megapixel sensor, capable of recording 1.4 million pixels that could produce a 5x7-inch digital photo-quality print. In 1987, Kodak released seven products for recording, storing, manipulating, transmitting and printing electronic still video images. In 1990, Kodak developed the Photo CD system and proposed "the first worldwide standard for defining color in the digital environment of computers and computer peripherals." In 1991, Kodak released the first professional digital camera system (DCS), aimed at photojournalists. It was a Nikon F-3 camera equipped by Kodak with a 1.3 megapixel sensor. The first digital cameras for the consumer-level market that worked with a home computer via a serial cable were th

Digital

Digital noise

Analog, symbolic, and digital displays; ease of reading

Analog to digital conversion

Symbol to digital conversion

Historical digital systems

See also

Digital (disambiguation)

Number

Examples

Further generalizations

Numerals and numbering

Extensions

See also

External links

References

Binary numbers

History

Representation

Counting in binary

Binary Simplified!

Binary arithmetic

Addition

Subtraction

Multiplication

Division

Bitwise logical operations

Conversion to and from other numeral systems

Decimal

Hexadecimal

Octal

Representing real numbers

Binary humor

See also

External links

Analog

Usage

Symbol

Nature of symbols

Use of symbols

Etymology

See also

External links

Icon

Images in Religion

Icons in Christianity

Images from Constantine to Justinian

The Iconoclast Period

Icons in Greek-speaking regions

Icons in Russia

Icon Traditions in Other Regions

The Protestant Reformation

Icons and Images in Contemporary Christianity

Eastern Orthodox and Roman Catholic teaching about Icons

Eikon in the Septuagint

Eikon in the New Testament

Icons in Hinduism

See also

External links; various points of view

Digit

Finger

Anatomy of the finger

See also:

Binary

Digital audio

Technology overview

Sampling

Quantization

Quantization Error

Digitization

Dither

Noise-shaping

Jitter

Methods of digital encoding

Pulse-code modulation

Digital audio technologies

Digital audio interfaces and interconnects

See also

External links

Digital photography