Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: "0" and "1". The base-2 numeral system is a positional notation with a radix of 2; each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by all modern computers and computer-based devices; the modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt and India. Leibniz was inspired by the Chinese I Ching; the scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions. Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, 1/64.

Early forms of this system can be found in documents from the Fifth Dynasty of Egypt 2400 BC, its developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt 1200 BC. The method used for ancient Egyptian multiplication is closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value is either doubled or has the first number added back into it; this method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC. The I Ching dates from the 9th century BC in China; the binary notation in the I Ching is used to interpret its quaternary divination technique. It is based on taoistic duality of yin and yang.eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.

Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. The Indian scholar Pingala developed a binary system for describing prosody, he used binary numbers in the form of long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter; the binary representations in Pingala's system increases towards the right, not to the left like in the binary numbers of the modern, Western positional notation. The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Asia. Sets of binary combinations similar to the I Ching have been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy.

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. In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could be encoded as scarcely visible variations in the font in any random text. For the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only. John Napier in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results.

The first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700. Leibniz studied binary numbering in 1679. Leibniz's system uses 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 0 0 0 1 numerical value 20 0 0 1 0 numerical value 21 0 1 0 0 numerical value 22 1 0 0 0 numerical value 23Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus; as a Sinophile, Leibniz was aware of

Natural number

In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.

A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.

Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.

Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a

Philadelphia

Philadelphia, sometimes known colloquially as Philly, is the largest city in the U. S. state and Commonwealth of Pennsylvania, the sixth-most populous U. S. city, with a 2017 census-estimated population of 1,580,863. Since 1854, the city has been coterminous with Philadelphia County, the most populous county in Pennsylvania and the urban core of the eighth-largest U. S. metropolitan statistical area, with over 6 million residents as of 2017. Philadelphia is the economic and cultural anchor of the greater Delaware Valley, located along the lower Delaware and Schuylkill Rivers, within the Northeast megalopolis; the Delaware Valley's population of 7.2 million ranks it as the eighth-largest combined statistical area in the United States. William Penn, an English Quaker, founded the city in 1682 to serve as capital of the Pennsylvania Colony. Philadelphia played an instrumental role in the American Revolution as a meeting place for the Founding Fathers of the United States, who signed the Declaration of Independence in 1776 at the Second Continental Congress, the Constitution at the Philadelphia Convention of 1787.

Several other key events occurred in Philadelphia during the Revolutionary War including the First Continental Congress, the preservation of the Liberty Bell, the Battle of Germantown, the Siege of Fort Mifflin. Philadelphia was one of the nation's capitals during the revolution, served as temporary U. S. capital while Washington, D. C. was under construction. In the 19th century, Philadelphia became a railroad hub; the city grew from an influx of European immigrants, most of whom came from Ireland and Germany—the three largest reported ancestry groups in the city as of 2015. In the early 20th century, Philadelphia became a prime destination for African Americans during the Great Migration after the Civil War, as well as Puerto Ricans; the city's population doubled from one million to two million people between 1890 and 1950. The Philadelphia area's many universities and colleges make it a top study destination, as the city has evolved into an educational and economic hub. According to the Bureau of Economic Analysis, the Philadelphia area had a gross domestic product of US$445 billion in 2017, the eighth-largest metropolitan economy in the United States.

Philadelphia is the center of economic activity in Pennsylvania and is home to five Fortune 1000 companies. The Philadelphia skyline is expanding, with a market of 81,900 commercial properties in 2016, including several nationally prominent skyscrapers. Philadelphia has more outdoor murals than any other American city. Fairmount Park, when combined with the adjacent Wissahickon Valley Park in the same watershed, is one of the largest contiguous urban park areas in the United States; the city is known for its arts, culture and colonial history, attracting 42 million domestic tourists in 2016 who spent US$6.8 billion, generating an estimated $11 billion in total economic impact in the city and surrounding four counties of Pennsylvania. Philadelphia has emerged as a biotechnology hub. Philadelphia is the birthplace of the United States Marine Corps, is the home of many U. S. firsts, including the first library, medical school, national capital, stock exchange and business school. Philadelphia contains 67 National Historic Landmarks and the World Heritage Site of Independence Hall.

The city became a member of the Organization of World Heritage Cities in 2015, as the first World Heritage City in the United States. Although Philadelphia is undergoing gentrification, the city maintains mitigation strategies to minimize displacement of homeowners in gentrifying neighborhoods. Before Europeans arrived, the Philadelphia area was home to the Lenape Indians in the village of Shackamaxon; the Lenape are a Native American tribe and First Nations band government. They are called Delaware Indians, their historical territory was along the Delaware River watershed, western Long Island, the Lower Hudson Valley. Most Lenape were pushed out of their Delaware homeland during the 18th century by expanding European colonies, exacerbated by losses from intertribal conflicts. Lenape communities were weakened by newly introduced diseases smallpox, violent conflict with Europeans. Iroquois people fought the Lenape. Surviving Lenape moved west into the upper Ohio River basin; the American Revolutionary War and United States' independence pushed them further west.

In the 1860s, the United States government sent most Lenape remaining in the eastern United States to the Indian Territory under the Indian removal policy. In the 21st century, most Lenape reside in Oklahoma, with some communities living in Wisconsin, in their traditional homelands. Europeans came to the Delaware Valley in the early 17th century, with the first settlements founded by the Dutch, who in 1623 built Fort Nassau on the Delaware River opposite the Schuylkill River in what is now Brooklawn, New Jersey; the Dutch considered the entire Delaware River valley to be part of their New Netherland colony. In 1638, Swedish settlers led by renegade Dutch established the colony of New Sweden at Fort Christina and spread out in the valley. In 1644, New Sweden supported the Susquehannocks in their military defeat of the English colony of Maryland. In 1648, the Dutch built Fort Beversreede on the west bank of the Delaware, south of the Schuylkill near the present-day Eastwick neighborhood, to reassert their dominion over the area.

The Swedes responded by building Fort Nya Korsholm, or New Korsholm, named after a town in Finland with a Swedish majority. In 1655, a

Hexadecimal

In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most the symbols "0"–"9" to represent values zero to nine, "A"–"F" to represent values ten to fifteen. Hexadecimal numerals are used by computer system designers and programmers, as they provide a more human-friendly representation of binary-coded values; each hexadecimal digit represents four binary digits known as a nibble, half a byte. For example, a single byte can have values ranging from 0000 0000 to 1111 1111 in binary form, which can be more conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is used to specify the radix. For example the decimal value 10,995 would be expressed in hexadecimal as 2AF316. In programming, a number of notations are used to support hexadecimal representation involving a prefix or suffix; the prefix 0x is used in C and related languages, which would denote this value by 0x2AF3. Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.

In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript can give the base explicitly: 15910 is decimal 159; some authors prefer a text subscript, such as 159decimal and 159hex, or 159h. In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen: In URIs, character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space character, ASCII code point 20 in hex, 32 in decimal. In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation ode, thus ’. In the Unicode standard, a character value is represented with U+ followed by the hex value, e.g. U+20AC is the Euro sign. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #: white, for example, is represented #FFFFFF.

CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33. Unix shells, AT&T assembly language and the C programming language use the prefix 0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits:'\x1B' represents the Esc control character. To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In MIME quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits prefixed by an equal to sign =, as in Espa=F1a to send "España". In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h: FFh or 05A3H; some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh instead of FFh Other assembly languages, Delphi, some versions of BASIC, GameMaker Language and Forth use $ as a prefix: $5A3.

Some assembly languages use the notation H'ABCD'. Fortran 95 uses Z'ABCD'. Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. For bit vector constants VHDL uses the notation x"5A3". Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant; the Smalltalk language uses the prefix 16r: 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#: 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC... Common Lisp uses the prefixes # 16r. Setting the variables *read-base* and *print-base* to 16 can be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers, thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16. MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3 BBC BASIC and Locomotive BASIC use & for hex.

TI-89 and 92 series uses a 0h prefix: 0h5A3 ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3. Binary and octal numbers can be specified similarly; the most common format for hexadecimal on IBM mainframes and midrange computers running the traditional OS's is X'5A3', is used in Assembler, PL/I, COBOL, JCL, scripts and other places. This format was common on

Divisor

In mathematics, a divisor of an integer n called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one says that n is a multiple of m. An integer n is divisible by another integer m. If m and n are nonzero integers, more nonzero elements of an integral domain, it is said that m divides n, m is a divisor of n, or n is a multiple of m, this is written as m ∣ n, if there exists an integer k, or an element k of the integral domain, such that m k = n; this definition is sometimes extended to include zero. This does not add much to the theory, as 0 does not divide any other number, every number divides 0. On the other hand, excluding zero from the definition simplifies many statements. In ring theory, an element a is called a "zero divisor" only if it is nonzero and ab = 0 for a nonzero element b. Thus, there are no zero divisors among the integers. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors.

For example, there are six divisors of 4. 1 and −1 divide every integer. Every integer is a divisor of itself. Integers divisible by 2 are called and integers not divisible by 2 are called odd. 1, −1, n and −n are known as the trivial divisors of n. A divisor of n, not a trivial divisor is known as a non-trivial divisor. A non-zero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors. There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits. 7 is a divisor of 42 because 7 × 6 = 42, so we can say 7 ∣ 42. It can be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42; the non-trivial divisors of 6 are 2, −2, 3, −3. The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42; the set of all positive divisors of 60, A = ordered by divisibility, has the Hasse diagram: There are some elementary rules: If a ∣ b and b ∣ c a ∣ c, i.e. divisibility is a transitive relation.

If a ∣ b and b ∣ a a = b or a = − b. If a ∣ b and a ∣ c a ∣ holds, as does a ∣. However, if a ∣ b and c ∣ b ∣ b does not always hold. If a ∣ b c, gcd = 1 a ∣ c; this is called Euclid's lemma. If p is a prime number and p ∣ a b p ∣ a or p ∣ b. A positive divisor of n, different from n is called a proper divisor or an aliquot part of n. A number that does not evenly leaves a remainder is called an aliquant part of n. An integer n > 1 {\displ

E number

E numbers are codes for substances that are permitted to be used as food additives for use within the European Union and EFTA. The "E" stands for "Europe". Found on food labels, their safety assessment and approval are the responsibility of the European Food Safety Authority. Having a single unified list for food additives was first agreed upon in 1962 with food colouring. In 1964, the directives for preservatives were added, 1970 for antioxidants and 1974 for the emulsifiers, stabilisers and gelling agents; the numbering scheme follows that of the International Numbering System as determined by the Codex Alimentarius committee, though only a subset of the INS additives are approved for use in the European Union as food additives. Outside the European continent plus Russia, E numbers are encountered on food labelling in other jurisdictions, including the Cooperation Council for the Arab States of the Gulf, South Africa, New Zealand and Israel, they are though still found on North American packaging on imported European products.

In some European countries, "E number" is sometimes used informally as a pejorative term for artificial food additives, products may promote themselves as "free of E numbers". This is incorrect, because many components of natural foods have assigned E numbers, e.g. vitamin C and lycopene, found in carrots. NB: Not all examples of a class fall into the given numeric range. Moreover, many chemicals in the E400–499 range, have a variety of purposes; the list shows all components that had an E-number assigned. Not all additives listed are still allowed in the EU, but are listed as they used to have an E-number. For an overview of allowed additives see information provided by the Food Standards Agency of the UK. Food Chemicals Codex List of food additives List of food additives, Codex Alimentarius Codex Alimentarius, the international foods standards, established by the Food and Agriculture Organization and the World Health Organization in 1963 See their document "Class Names and the International Numbering System for Food Additives" Joint FAO/WHO Expert Committee on Food Additives publications at the World Health Organization Food Additive Index, JECFA, Food and Agriculture Organization E-codes and ingredients search engine with details/suggestions for Muslims Current EU approved additives and their E Numbers Food Additives in the European Union Food Additives, Food Safety, website of the European Union.

Includes Lists of authorised food additives Food additives database The Food Additives and Ingredients Association, FAIA website, UK