1.
Subset
–
In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
Subset
–
Euler diagram showing A is a proper subset of B and conversely B is a proper superset of A
2.
Complex numbers
–
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
Complex numbers
–
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an
Argand diagram, representing the
complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the
imaginary unit which satisfies i 2 = −1.
3.
Counting
–
Counting is the action of finding the number of elements of a finite set of objects. The related term refers to uniquely identifying the elements of a finite set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one, for example, when counting money, counting out change, counting by twos, there is archeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track of social and economic data such as number of members, prey animals, property. The development of counting led to the development of mathematical notation, numeral systems, counting can occur in a variety of forms. Counting can be verbal, that is, speaking every number out loud to keep track of progress and this is often used to count objects that are present already, instead of counting a variety of things over time. Counting can also be in the form of tally marks, making a mark for each number and this is useful when counting objects over time, such as the number of times something occurs during the course of a day. Tallying is base 1 counting, normal counting is done in base 10, counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations, finger-counting uses unary notation, and is thus limited to counting 10. Older finger counting used the four fingers and the three bones in each finger to count to the number twelve, other hand-gesture systems are also in use, for example the Chinese system by which one can count 10 using only gestures of one hand. By using finger binary, it is possible to keep a count up to 1023 =210 −1. Various devices can also be used to facilitate counting, such as tally counters. Inclusive counting is usually encountered when dealing with time in the Romance languages, in exclusive counting languages such as English, when counting 8 days from Sunday, Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day. When counting inclusively, the Sunday will be day 1 and therefore the following Sunday will be the eighth day, for example, the French phrase for fortnight is quinzaine, and similar words are present in Greek, Spanish and Portuguese. In contrast, the English word fortnight itself derives from a fourteen-night, as the archaic sennight does from a seven-night, learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a very first step into mathematics. However, some cultures in Amazonia and the Australian Outback do not count, many children at just 2 years of age have some skill in reciting the count list. They can also answer questions of ordinality for small numbers, e. g and they can even be skilled at pointing to each object in a set and reciting the words one after another
Counting
–
Counting using
tally marks at
Hanakapiai Beach
4.
Measurement
–
Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of a measurement is dependent on the context, however, in other fields such as statistics as well as the social and behavioral sciences, measurements can have multiple levels, which would include nominal, ordinal, interval, and ratio scales. Measurement is a cornerstone of trade, science, technology, historically, many measurement systems existed for the varied fields of human existence to facilitate comparisons in these fields. Often these were achieved by local agreements between trading partners or collaborators, since the 18th century, developments progressed towards unifying, widely accepted standards that resulted in the modern International System of Units. This system reduces all physical measurements to a combination of seven base units. The science of measurement is pursued in the field of metrology, the measurement of a property may be categorized by the following criteria, type, magnitude, unit, and uncertainty. They enable unambiguous comparisons between measurements, the type or level of measurement is a taxonomy for the methodological character of a comparison. For example, two states of a property may be compared by ratio, difference, or ordinal preference, the type is commonly not explicitly expressed, but implicit in the definition of a measurement procedure. The magnitude is the value of the characterization, usually obtained with a suitably chosen measuring instrument. A unit assigns a mathematical weighting factor to the magnitude that is derived as a ratio to the property of a used as standard or a natural physical quantity. An uncertainty represents the random and systemic errors of the measurement procedure, errors are evaluated by methodically repeating measurements and considering the accuracy and precision of the measuring instrument. Measurements most commonly use the International System of Units as a comparison framework, the system defines seven fundamental units, kilogram, metre, candela, second, ampere, kelvin, and mole. Instead, the measurement unit can only ever change through increased accuracy in determining the value of the constant it is tied to and this directly influenced the Michelson–Morley experiment, Michelson and Morley cite Peirce, and improve on his method. With the exception of a few fundamental quantum constants, units of measurement are derived from historical agreements, nothing inherent in nature dictates that an inch has to be a certain length, nor that a mile is a better measure of distance than a kilometre. Over the course of history, however, first for convenience and then for necessity. Laws regulating measurement were originally developed to prevent fraud in commerce.9144 metres, in the United States, the National Institute of Standards and Technology, a division of the United States Department of Commerce, regulates commercial measurements. Before SI units were adopted around the world, the British systems of English units and later imperial units were used in Britain, the Commonwealth. The system came to be known as U. S. customary units in the United States and is still in use there and in a few Caribbean countries. S
Measurement
–
A typical
tape measure with both
metric and
US units and two
US pennies for comparison
Measurement
–
A
baby bottle that measures in three
measurement systems,
Imperial (U.K.),
U.S. customary, and
metric.
Measurement
–
Four measuring devices having metric calibrations
5.
Nominal number
–
Nominal numbers or categorical numbers are numeric codes, meaning numerals used for labelling or identification only. The values of the numerals are irrelevant, and they do not indicate quantity, rank, the term nominal number is quite recent and of limited use. It appears to have originated as a usage in school textbooks derived from the term nominal data. This usage comes from the sense of nominal as name, nominal number can be broadly defined as any numeral used for identification, however it was assigned, or narrowly as a numeral with no information other than identification. For the purposes of naming, the number is often used loosely to refer to any string. For instance, National Insurance numbers, some drivers license numbers, similarly, one can add or subtract ZIP codes, but this is meaningless,12345 −11111 does not have any meaning as a ZIP code. In general, the only operation with nominal numbers is to compare two nominal numbers to see whether they are identical or not. Postal codes, such as ZIP codes Telephone numbers, assigned by various telephone numbering plans, such as the ITU-T E.164, numbers of train or bus routes in public transport Car model names from some car manufacturers, such as BMW or Peugeot, are plain numbers. Numerical identifiers that are nominal numbers narrowly defined, viz, convey no information other than identity, are quite rare and these must be defined either arbitrarily or randomly, and most commonly arise in computer applications, such as dynamic IP addresses assigned by Dynamic Host Configuration Protocol. A more everyday example are sports squad numbers, which do not in general have any meaning beyond identity. In some settings, these are based on position, but in others they are associated with an individual, farber, William, Germain-Williams, Terri L. Paris, Elaine, Thaller, Bernd, Lehmann, Ingmar. 100 Commonly Asked Questions in Math Class
Nominal number
–
Numbers 102 and 400: bus route 102 in
London, run by an
Alexander Dennis Enviro400 double-decker bus.
Nominal number
–
Number 107: the car
Peugeot 107.
6.
Natural number
–
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, 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. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the 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, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, 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, and 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, and 6 ones, and similarly for the number 4,622. A much later 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, 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, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the 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, 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 really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Natural number
–
The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
7.
Symbol
–
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different concepts and experiences, all communication is achieved through the use of symbols. Symbols take the form of words, sounds, gestures, ideas or visual images and are used to other ideas. For example, a red octagon may be a symbol for STOP, on a map, a blue line might represent a river. Alphabetic letters may be symbols for sounds, personal names are symbols representing individuals. A red rose may symbolize love and compassion, the variable x, in a mathematical equation, may symbolize the position of a particle in space. In cartography, a collection of symbols forms a legend for a map The word derives from the Greek symbolon meaning token or watchword. It is an amalgam of syn- together + bole a throwing, a casting, the sense evolution in Greek is from throwing things together to contrasting to comparing to token used in comparisons to determine if something is genuine. The meaning something which stands for something else was first recorded in 1590, later, expanding on what he means by this definition Campbell says, a symbol, like everything else, shows a double aspect. We must distinguish, therefore between the sense and the meaning of the symbol. The term meaning can only to the first two but these, today, are in the charge of science – which is the province as we have said, not of symbols. The ineffable, the unknowable, can be only sensed. Heinrich Zimmer gives an overview of the nature, and perennial relevance. Concepts and words are symbols, just as visions, rituals, through all of these a transcendent reality is mirrored. They are so many metaphors reflecting and implying something which, though thus variously expressed, is ineffable, though thus rendered multiform, Symbols hold the mind to truth but are not themselves the truth, hence it is delusory to borrow them. Each civilisation, every age, must bring forth its own, in the book Signs and Symbols, it is stated that A symbol. Is a visual image or sign representing an idea -- a deeper indicator of a universal truth, Symbols are a means of complex communication that often can have multiple levels of meaning. This separates symbols from signs, as signs have only one meaning, human cultures use symbols to express specific ideologies and social structures and to represent aspects of their specific culture
Symbol
–
The square and compasses, symbol of the Freemasons
Symbol
–
A red
octagon symbolizes "stop" even without the word.
Symbol
–
Topics and terminology
8.
Numeral system
–
A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
Numeral system
–
Numeral systems
9.
Telephone number
–
A telephone number serves as an address for switching telephone calls using a system of destination code routing. Telephone numbers are entered or dialed by a party on the originating telephone set. The exchange completes the call either to another locally connected subscriber or via the PSTN to the called party, telephone numbers were first used in 1879 in Lowell, Massachusetts when they replaced the request for subscriber names by callers connecting to the switchboard operator. Telephone numbers are dialed in conjunction with other signaling code sequences, such as vertical service codes. When telephone numbers were first used they were short, from one to three digits, and were communicated orally to a switchboard operator when initiating a call. As telephone systems have grown and interconnected to encompass worldwide communication, in addition to telephones, they have been used to access other devices, such as computer modems, pagers, and fax machines. The number contains the necessary to identify uniquely the intended endpoint for the telephone call. Each such endpoint must have a number within the public switched telephone network. Most countries use fixed length numbers and therefore the number of endpoints determines the length of the telephone number. It is also possible for each subscriber to have a set of numbers for the endpoints most often used. These shorthand or speed calling numbers are translated to unique telephone numbers before the call can be connected. Some special services have their own short numbers The dialing plan in some areas permits dialing numbers in the local calling area without using area code or city code prefixes. For example, a number in North America consists of a three-digit area code, a three-digit central office code. If the area has no area code overlays, seven-digit dialing may be permissible for calls within the area, for each large metro area, all of these lines will share the same prefix, the last digits typically corresponding to the stations frequency, callsign, or moniker. In the international network, the format of telephone numbers is standardized by ITU-T recommendation E.164. This code specifies that the number should be 15 digits or shorter. For most countries, this is followed by a code or city code and the subscriber number. ITU-T recommendation E.123 describes how to represent an international telephone number in writing or print, starting with a plus sign and the country code
Telephone number
–
2008 photo shows a hairdressing shop in Toronto with an exterior sign showing the shop's telephone number in the old two-letters plus five-digits format.
Telephone number
–
Telephone numbers for sale in
Hong Kong, note how the prices are higher for 'luckier' numbers.
10.
Serial number
–
A serial code is a unique identifier assigned incrementally or sequentially to an item. It is also called a number, although it may be a character string that includes letters and other typographical symbols. Serial numbers identify otherwise identical individual units with many, obvious uses, Serial numbers are a deterrent against theft and counterfeit products, as they can be recorded, and stolen or otherwise irregular goods can be identified. Banknotes and other documents of value bear serial numbers to assist in preventing counterfeiting and tracing stolen ones. They are valuable in quality control, as once a defect is found in the production of a batch of product. Serial numbers may be used to identify individual physical or intangible objects, the purpose and application is different. A software serial number, otherwise called product key, is not embedded in the software. The software will function if a potential user enters a valid product code. The vast majority of codes are rejected by the software. If an unauthorised user is found to be using the software, the term serial number is sometimes used for codes which do not identify a single instance of something. It takes its name from the library use of the word serial to mean a periodical. Certificates and certificate authorities are necessary for use of cryptography. These depend on applying mathematically rigorous serial numbers and serial number arithmetic, the term serial number is also used in military formations as an alternative to the expression service number. Because of this, the number is sometimes called a tail number. LZ548/G—the prototype de Havilland Vampire jet fighter, or ML926/G—a de Havilland Mosquito XVI experimentally fitted with H2S radar, the serial number follows the aircraft throughout its period of service. In 2009 the U. S. FDA published draft guidance for the industry to use serial numbers on prescription drug packages. This measure will enhance the traceability of drugs and help to prevent counterfeiting, Serial numbers are often used in network protocols. However, most sequence numbers in computer protocols are limited to a number of bits
Serial number
–
Serial number from an
identity document
Serial number
–
Serial number on a semi-automatic pistol
11.
ISBN
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
ISBN
–
A 13-digit ISBN, 978-3-16-148410-0, as represented by an
EAN-13 bar code
12.
Numeral (linguistics)
–
In linguistics, a numeral is a member of a word class designating numbers, such as the English word two and the compound seventy-seven. Numerals function most typically as an adjective or a pronoun and express numbers and relations to numbers for example, quantity, sequence, frequency, numerals may be attributive, as in two dogs, or pronominal, as in I saw two. Many words of different parts of speech indicate number or quantity, quantifiers do not enumerate, or designate a specific number, but give another, often less specific, indication of amount. Examples are words such as every, most, least, some, some times a quantifier can have a definite amount. Examples are words such as five, ten, fifty, one hundred, etc. There are also number words which enumerate but are not a part of speech, such as dozen, which is a noun, first, which is an adjective, or twice. Numerals enumerate, but in addition have distinct grammatical behavior, when a numeral modifies a noun, it may replace the article, numerals may be simple, such as eleven, or compound, such as twenty-three. However, not all words for numbers are necessarily numerals. For example, million is grammatically a noun, and must be preceded by an article or numeral itself. In Old Church Slavonic, the cardinal numbers 5 to 10 were feminine nouns, when quantifying a noun, examples are ordinal numbers, multiplicative adverbs, multipliers, and distributive numbers. In other languages, there may be kinds of number words. For example, in Slavic languages there are numbers which describe sets. Georgian, Latin, and Romanian have regular distributive numbers, such as Latin singuli one-by-one, bini in pairs, two-by-two, terni three each, etc. Some languages have a limited set of numerals, and in some cases they arguably do not have any numerals at all. Other languages had a system but borrowed a second set of numerals anyway. An example is Japanese, which uses either native or Chinese-derived numerals depending on what is being counted, in many languages, such as Chinese, numerals require the use of numeral classifiers. Many sign languages, such as ASL, incorporate numerals, not all languages have numeral systems. Specifically, there is not much need for numeral systems among hunter-gatherers who do not engage in commerce, indeed, several languages from the Amazon have been independently reported to have no specific number words other than one
Numeral (linguistics)
–
Numeral systems
13.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
14.
Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
15.
Rational number
–
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d
Rational number
–
A diagram showing a representation of the equivalent classes of pairs of integers
16.
One half
–
One half is the irreducible fraction resulting from dividing one by two, or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or halving, conversely, division by one half is equivalent to multiplication by two, or doubling, one half appears often in mathematical equations, recipes, measurements, etc. Half can also be said to be one part of something divided into two equal parts, for instance, the area S of a triangle is computed S = 1⁄2 × base × perpendicular height. The Riemann hypothesis states that every nontrivial complex root of the Riemann zeta function has a part equal to 1⁄2. One half has two different decimal expansions, the familiar 0.5 and the recurring 0.49999999 and it has a similar pair of expansions in any even base. It is a trap to believe these expressions represent distinct numbers. Equals 1 for detailed discussion of a related case, in odd bases, one half has no terminating representation, only a single representation with a repeating fractional component, such as 0.11111111. in ternary. 1⁄2 is also one of the few fractions to usually have a key of its own on typewriters and it also has its own code point in some early extensions of ASCII at 171. In Unicode, it has its own unit at U+00BD in the C1 Controls and Latin-1 Supplement block. List of numbers Division by two
One half
–
Postal stamp, Ireland, 1940: one halfpenny postage due.
17.
Real number
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
Real number
–
A symbol of the set of real numbers (ℝ)
18.
Square root of 2
–
The square root of 2, or the th power of 2, written in mathematics as √2 or 2 1⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational, the rational approximation of the square root of two,665, 857/470,832, derived from the fourth step in the Babylonian algorithm starting with a0 =1, is too large by approx. 1. 6×10−12, its square is 2. 0000000000045… The rational approximation 99/70 is frequently used, despite having a denominator of only 70, it differs from the correct value by less than 1/10,000. The numerical value for the root of two, truncated to 65 decimal places, is,1. 41421356237309504880168872420969807856967187537694807317667973799….41421296 ¯. That is,1 +13 +13 ×4 −13 ×4 ×34 =577408 =1.4142156862745098039 ¯. This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as a secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras number or Pythagoras constant, for example by Conway & Guy, there are a number of algorithms for approximating √2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots. It goes as follows, First, pick a guess, a0 >0, then, using that guess, iterate through the following recursive computation, a n +1 = a n +2 a n 2 = a n 2 +1 a n. The more iterations through the algorithm, the approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits, starting with a0 =1 the next approximations are 3/2 =1.5 17/12 =1.416. The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanadas team in 1997, in February 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010, for a development of this record, see the table below. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely, such computations aim to check empirically whether such numbers are normal
Square root of 2
–
Babylonian clay tablet YBC 7289 with annotations. Besides showing the square root of 2 in
sexagesimal (1 24 51 10), the tablet also gives an example where one side of the square is 30 and the diagonal then is 42 25 35. The sexagesimal digit 30 can also stand for 1/2, in which case 42 25 35 is approximately 0.7071065.
Square root of 2
–
The square root of 2 is equal to the length of the
hypotenuse of a
right triangle with legs of length 1.
19.
Pi
–
The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
Pi
–
The constant π is represented in this
mosaic outside the Mathematics Building at the
Technical University of Berlin.
Pi
–
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
Pi
–
Archimedes developed the polygonal approach to approximating π.
Pi
–
Isaac Newton used
infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
20.
Imaginary unit
–
The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 +1 =0. The term imaginary is used there is no real number having a negative square. There are two square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used, in the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. For the history of the unit, see Complex number § History. The imaginary number i is defined solely by the property that its square is −1, with i defined this way, it follows directly from algebra that i and −i are both square roots of −1. In polar form, i is represented as 1eiπ/2, having a value of 1. In the complex plane, i is the point located one unit from the origin along the imaginary axis, more precisely, once a solution i of the equation has been fixed, the value −i, which is distinct from i, is also a solution. Since the equation is the definition of i, it appears that the definition is ambiguous. However, no ambiguity results as long as one or other of the solutions is chosen and labelled as i and this is because, although −i and i are not quantitatively equivalent, there is no algebraic difference between i and −i. Both imaginary numbers have equal claim to being the number whose square is −1, the issue can be a subtle one. See also Complex conjugate and Galois group, a more precise explanation is to say that the automorphism group of the special orthogonal group SO has exactly two elements — the identity and the automorphism which exchanges CW and CCW rotations. All these ambiguities can be solved by adopting a rigorous definition of complex number. For example, the pair, in the usual construction of the complex numbers with two-dimensional vectors. The imaginary unit is sometimes written √−1 in advanced mathematics contexts, however, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the square root function. Similarly,1 i =1 −1 =1 −1 = −11 = −1 = i, the calculation rules a ⋅ b = a ⋅ b and a b = a b are only valid for real, non-negative values of a and b. These problems are avoided by writing and manipulating expressions like i√7, for a more thorough discussion, see Square root and Branch point
Imaginary unit
–
i in the
complex or
cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis
21.
Arithmetical operations
–
Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
Arithmetical operations
–
Arithmetic tables for children, Lausanne, 1835
Arithmetical operations
–
A scale calibrated in imperial units with an associated cost display.
22.
Addition
–
Addition is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two together, making a total of five apples. This observation is equivalent to the mathematical expression 3 +2 =5 i. e.3 add 2 is equal to 5, besides counting fruits, addition can also represent combining other physical objects. In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others, in algebra, addition is studied more abstractly. It is commutative, meaning that order does not matter, and it is associative, repeated addition of 1 is the same as counting, addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication, performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers, the most basic task,1 +1, can be performed by infants as young as five months and even some members of other animal species. In primary education, students are taught to add numbers in the system, starting with single digits. Mechanical aids range from the ancient abacus to the modern computer, Addition is written using the plus sign + between the terms, that is, in infix notation. The result is expressed with an equals sign, for example, 3½ =3 + ½ =3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead, the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example, ∑ k =15 k 2 =12 +22 +32 +42 +52 =55. The numbers or the objects to be added in addition are collectively referred to as the terms, the addends or the summands. This is to be distinguished from factors, which are multiplied, some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an addend at all, today, due to the commutative property of addition, augend is rarely used, and both terms are generally called addends. All of the above terminology derives from Latin, using the gerundive suffix -nd results in addend, thing to be added. Likewise from augere to increase, one gets augend, thing to be increased, sum and summand derive from the Latin noun summa the highest, the top and associated verb summare
Addition
–
Part of Charles Babbage's
Difference Engine including the addition and carry mechanisms
Addition
–
The plus sign
Addition
–
A circular slide rule
23.
Subtraction
–
Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign, for example, in the picture on the right, there are 5 −2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. It is anticommutative, meaning that changing the order changes the sign of the answer and it is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction of 0 does not change a number, subtraction also obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers, general binary operations that continue these patterns are studied in abstract algebra. Performing subtraction is one of the simplest numerical tasks, subtraction of very small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the system, starting with single digits. Subtraction is written using the minus sign − between the terms, that is, in infix notation, the result is expressed with an equals sign. This is most common in accounting, formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend. All of this terminology derives from Latin, subtraction is an English word derived from the Latin verb subtrahere, which is in turn a compound of sub from under and trahere to pull, thus to subtract is to draw from below, take away. Using the gerundive suffix -nd results in subtrahend, thing to be subtracted, likewise from minuere to reduce or diminish, one gets minuend, thing to be diminished. Imagine a line segment of length b with the left end labeled a, starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition, a + b = c, from c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction, c − b = a, now, a line segment labeled with the numbers 1,2, and 3. From position 3, it takes no steps to the left to stay at 3 and it takes 2 steps to the left to get to position 1, so 3 −2 =1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3, to represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number, from 3, it takes 3 steps to the left to get to 0, so 3 −3 =0. But 3 −4 is still invalid since it leaves the line
Subtraction
–
Placard outside shop in Bordeaux advertising subtraction of 20% from the price of a second perfume
Subtraction
–
"5 − 2 = 3" (verbally, "five minus two equals three")
Subtraction
–
1 + … = 3
24.
Multiplication
–
Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
Multiplication
–
4 × 5 = 20, the rectangle is composed of 20 squares, having dimensions of 4 by 5.
Multiplication
–
Four bags of three
marbles gives twelve marbles (4 × 3 = 12).
25.
Division (mathematics)
–
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The division of two numbers is the process of calculating the number of times one number is contained within one another. For example, in the picture on the right, the 20 apples are divided into groups of five apples, Division can also be thought of as the process of evaluating a fraction, and fractional notation is commonly used to represent division. Division is the inverse of multiplication, if a × b = c, then a = c ÷ b, as long as b is not zero. Division by zero is undefined for the numbers and most other contexts, because if b =0, then a cannot be deduced from b and c. In some contexts, division by zero can be defined although to a limited extent, in division, the dividend is divided by the divisor to get a quotient. In the above example,20 is the dividend, five is the divisor, in some cases, the divisor may not be contained fully by the dividend, for example,10 ÷3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. Besides dividing apples, division can be applied to other physical, Division has been defined in several contexts, such as for the real and complex numbers and for more abstract contexts such as for vector spaces and fields. Division is the most mentally difficult of the four operations of arithmetic. Teaching the objective concept of dividing integers introduces students to the arithmetic of fractions, unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder, to complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called. When students advance to algebra, the theory of division intuited from arithmetic naturally extends to algebraic division of variables, polynomials. Division is often shown in algebra and science by placing the dividend over the divisor with a line, also called a fraction bar. For example, a divided by b is written a b This can be read out loud as a divided by b, a fraction is a division expression where both dividend and divisor are integers, and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus, common in arithmetic, in this manner, ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the operation itself. In some non-English-speaking cultures, a divided by b is written a, b and this notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae and later popularized by Gottfried Wilhelm Leibniz
Division (mathematics)
–
This article is about the arithmetical operation. For other uses, see
Division (disambiguation).
26.
Exponentiation
–
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
Exponentiation
–
Graphs of y = b x for various bases b: base 10 (green), base e (red), base 2 (blue), and base 1 / 2 (cyan). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
27.
Arithmetic
–
Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
Arithmetic
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Arithmetic tables for children, Lausanne, 1835
Arithmetic
–
A scale calibrated in imperial units with an associated cost display.
28.
Number theory
–
Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
Number theory
–
A
Lehmer sieve, which is a primitive
digital computer once used for finding
primes and solving simple
Diophantine equations.
Number theory
–
The Plimpton 322 tablet
Number theory
–
Title page of the 1621 edition of Diophantus' Arithmetica, translated into
Latin by
Claude Gaspard Bachet de Méziriac.
Number theory
–
Leonhard Euler
29.
Pseudoscience
–
Pseudoscience consists of claims, beliefs, or practices presented as being plausible scientifically, but which are not justifiable by the scientific method. The term pseudoscience is often considered pejorative because it suggests something is being presented as science inaccurately or even deceptively, accordingly, those termed as practicing or advocating pseudoscience often dispute the characterization. The demarcation between science and pseudoscience has philosophical and scientific implications, differentiating science from pseudoscience has practical implications in the case of health care, expert testimony, environmental policies, and science education. The word pseudoscience is derived from the Greek root pseudo meaning false and the English word science, an earlier recorded use of the term was during 1843 by the French physiologist François Magendie. From time to time, though, the usage of the word occurred in a formal, technical manner around a perceived threat to individual and institutional security in a social and cultural setting. A number of principles are accepted by scientists as standards for determining whether a body of knowledge, method. Experimental results should be reproducible and verified by other researchers, statistical quantification of significance, confidence, and error are also important tools for the scientific method. During the mid-20th century, Karl Popper emphasized the criterion of falsifiability to distinguish science from nonscience, falsifiability means a result can be disproved. For example, a statement such as God exists may be true or false, Popper used astrology and psychoanalysis as examples of pseudoscience and Einsteins theory of relativity as an example of science. The point is made there is no physical test to refute the claim of the presence of this dragon. No matter what test you think you can devise, there is then a reason why this does not apply to the invisible dragon, sagan concludes, Now, whats the difference between an invisible, incorporeal, floating dragon who spits heatless fire and no dragon at all. During 1942, Robert K. Merton identified a set of five norms which he characterized as what makes a real science, If any of the norms were violated, Merton considered the enterprise to be nonscience. These are not broadly accepted by the scientific community and his norms were, Originality, The tests and research done must present something new to the scientific community. Detachment, The scientists reasons for practicing this science must be simply for the expansion of their knowledge, the scientists should not have personal reasons to expect certain results. Universality, No person should be able to easily obtain the information of a test than another person. Social class, religion, ethnicity, or any other factors should not be factors in someones ability to receive or perform a type of science. Skepticism, Scientific facts must not be based on faith, one should always question every case and argument and constantly check for errors or invalid claims. Public accessibility, Any scientific knowledge one obtains should be available to everyone
Pseudoscience
–
A typical 19th century
phrenology chart: In the 1820s, phrenologists claimed the mind was located in areas of the brain, and were attacked for doubting that mind came from the nonmaterial soul. Their idea of reading "bumps" in the skull to predict personality traits was later discredited. Phrenology was first called a pseudoscience in 1843 and continues to be considered so.
30.
Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
Greek mathematics
–
Statue of Euclid in the
Oxford University Museum of Natural History
Greek mathematics
–
An illustration of
Euclid 's proof of the
Pythagorean Theorem
Greek mathematics
–
The
Antikythera mechanism, an ancient mechanical calculator.
31.
Ring (mathematics)
–
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices, the conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether, rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry. A ring is a group with a second binary operation that is associative, is distributive over the abelian group operation. By extension from the integers, the group operation is called addition. Whether a ring is commutative or not has profound implications on its behavior as an abstract object, as a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has greatly influenced by problems and ideas occurring naturally in algebraic number theory. The most familiar example of a ring is the set of all integers, Z, −5, −4, −3, −2, −1,0,1,2,3,4,5. The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings, a ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms 1. R is a group under addition, meaning that, + c = a + for all a, b, c in R. a + b = b + a for all a, b in R. There is an element 0 in R such that a +0 = a for all a in R, for each a in R there exists −a in R such that a + =0. R is a monoid under multiplication, meaning that, · c = a · for all a, b, c in R. There is an element 1 in R such that a ·1 = a and 1 · a = a for all a in R.3. Multiplication is distributive with respect to addition, a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many follow a alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless stated, a ring is assumed to have such an identity
Ring (mathematics)
–
Richard Dedekind, one of the founders of
ring theory.
Ring (mathematics)
–
Chapter IX of
David Hilbert 's Die Theorie der algebraischen Zahlkörper. The chapter title is Die Zahlringe des Körpers, literally "the number rings of the field". The word "ring" is the contraction of "Zahlring".
32.
Field (mathematics)
–
In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Associativity of addition and multiplication For all a, b, and c in F, the following hold, a + = + c. Commutativity of addition and multiplication For all a and b in F, the following hold, a + b = b + a. Existence of additive and multiplicative identity elements There exists an element of F, called the identity element and denoted by 0, such that for all a in F. Likewise, there is an element, called the identity element and denoted by 1, such that for all a in F. To exclude the trivial ring, the identity and the multiplicative identity are required to be distinct. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element −a in F, similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 =1. In other words, subtraction and division operations exist, distributivity of multiplication over addition For all a, b and c in F, the following equality holds, a · = +. A simple example of a field is the field of numbers, consisting of numbers which can be written as fractions a/b, where a and b are integers. The additive inverse of such a fraction is simply −a/b, to see the latter, note that b a ⋅ a b = b a a b =1. In addition to number systems such as the rationals, there are other. The following example is a field consisting of four elements called O, I, A and B, the notation is chosen such that O plays the role of the additive identity element, and I is the multiplicative identity. One can check that all field axioms are satisfied, for example, A · = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity. This field is called a field with four elements
Field (mathematics)
–
Given 0, 1, r 1 and r 2, the construction yields r 1 · r 2
33.
Indian mathematics
–
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama, the decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, in addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China and this was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved, all mathematical works were orally transmitted until approximately 500 BCE, thereafter, they were transmitted both orally and in manuscript form. A later landmark in Indian mathematics was the development of the series expansions for functions by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a theory of differentiation and integration. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of practical mathematics. The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4,2,1, considered favourable for the stability of a brick structure. They used a system of weights based on the ratios, 1/20, 1/10, 1/5, 1/2,1,2,5,10,20,50,100,200. They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, the inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts, bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. The religious texts of the Vedic Period provide evidence for the use of large numbers, by the time of the Yajurvedasaṃhitā-, numbers as high as 1012 were being included in the texts. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta, With three-fourths Puruṣa went up, the Satapatha Brahmana contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Śulba Sūtras list rules for the construction of fire altars. Most mathematical problems considered in the Śulba Sūtras spring from a single theological requirement, according to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. The diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately and they contain lists of Pythagorean triples, which are particular cases of Diophantine equations
Indian mathematics
Indian mathematics
–
The design of the domestic fire altar in the Śulba Sūtra
34.
Set (mathematics)
–
In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
Set (mathematics)
–
A set of polygons in a
Venn diagram
35.
Natural numbers
–
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, 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. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the 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, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, 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, and 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, and 6 ones, and similarly for the number 4,622. A much later 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, 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, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the 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, 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 really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Natural numbers
–
The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural numbers
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
36.
Real numbers
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
Real numbers
–
A symbol of the set of real numbers (ℝ)
37.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory
Group theory
38.
Complex number
–
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
Complex number
–
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an
Argand diagram, representing the
complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the
imaginary unit which satisfies i 2 = −1.
39.
Canonical isomorphism
–
In mathematics, an isomorphism is a homomorphism or morphism that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them, an automorphism is an isomorphism whose source and target coincide. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if, in topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a map that is an isomorphism. Two objects are said to be isomorphic if there is a canonical isomorphism between them. Isomorphisms are formalized using category theory, let R + be the multiplicative group of positive real numbers, and let R be the additive group of real numbers. The logarithm function log, R + → R satisfies log = log x + log y for all x, y ∈ R +, so it is a group homomorphism. The exponential function exp, R → R + satisfies exp = for all x, y ∈ R, the identities log exp x = x and exp log y = y show that log and exp are inverses of each other. Since log is a homomorphism that has an inverse that is also a homomorphism, because log is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to real numbers using a ruler. Consider the group, the integers from 0 to 5 with addition modulo 6 and these structures are isomorphic under addition, if you identify them using the following scheme, ↦0 ↦1 ↦2 ↦3 ↦4 ↦5 or in general ↦ mod 6. For example, + =, which translates in the system as 1 +3 =4. Even though these two groups look different in that the sets contain different elements, they are indeed isomorphic, more generally, the direct product of two cyclic groups Z m and Z n is isomorphic to if and only if m and n are coprime. For example, R is an ordering ≤ and S an ordering ⊑, such an isomorphism is called an order isomorphism or an isotone isomorphism. If X = Y, then this is a relation-preserving automorphism, in a concrete category, such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets. In algebraic categories, an isomorphism is the same as a homomorphism which is bijective on underlying sets, in abstract algebra, two basic isomorphisms are defined, Group isomorphism, an isomorphism between groups Ring isomorphism, an isomorphism between rings. Just as the automorphisms of an algebraic structure form a group, letting a particular isomorphism identify the two structures turns this heap into a group
Canonical isomorphism
–
The group of fifth
roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.
40.
Ancient Greeks
–
Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 12th-9th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and this was followed by the period of Classical Greece, an era that began with the Greco-Persian Wars, lasting from the 5th to 4th centuries BC. Due to the conquests by Alexander the Great of Macedonia, Hellenistic civilization flourished from Central Asia to the end of the Mediterranean Sea. Classical Greek culture, especially philosophy, had a influence on ancient Rome. For this reason Classical Greece is generally considered to be the culture which provided the foundation of modern Western culture and is considered the cradle of Western civilization. Classical Antiquity in the Mediterranean region is considered to have begun in the 8th century BC. Classical Antiquity in Greece is preceded by the Greek Dark Ages and this period is succeeded, around the 8th century BC, by the Orientalizing Period during which a strong influence of Syro-Hittite, Jewish, Assyrian, Phoenician and Egyptian cultures becomes apparent. The end of the Dark Ages is also dated to 776 BC. The Archaic period gives way to the Classical period around 500 BC, Ancient Periods Astronomical year numbering Dates are approximate, consult particular article for details The history of Greece during Classical Antiquity may be subdivided into five major periods. The earliest of these is the Archaic period, in which artists made larger free-standing sculptures in stiff, the Archaic period is often taken to end with the overthrow of the last tyrant of Athens and the start of Athenian Democracy in 508 BC. It was followed by the Classical period, characterized by a style which was considered by observers to be exemplary, i. e. classical, as shown in the Parthenon. This period saw the Greco-Persian Wars and the Rise of Macedon, following the Classical period was the Hellenistic period, during which Greek culture and power expanded into the Near and Middle East. This period begins with the death of Alexander and ends with the Roman conquest, Herodotus is widely known as the father of history, his Histories are eponymous of the entire field. Herodotus was succeeded by authors such as Thucydides, Xenophon, Demosthenes, Plato, most of these authors were either Athenian or pro-Athenian, which is why far more is known about the history and politics of Athens than those of many other cities. Their scope is limited by a focus on political, military and diplomatic history, ignoring economic. In the 8th century BC, Greece began to emerge from the Dark Ages which followed the fall of the Mycenaean civilization, literacy had been lost and Mycenaean script forgotten, but the Greeks adopted the Phoenician alphabet, modifying it to create the Greek alphabet. The Lelantine War is the earliest documented war of the ancient Greek period and it was fought between the important poleis of Chalcis and Eretria over the fertile Lelantine plain of Euboea. Both cities seem to have suffered a decline as result of the long war, a mercantile class arose in the first half of the 7th century BC, shown by the introduction of coinage in about 680 BC
Ancient Greeks
–
The
Parthenon, a temple dedicated to
Athena, located on the
Acropolis in
Athens, is one of the most representative symbols of the culture and sophistication of the ancient Greeks.
Ancient Greeks
–
Dipylon Vase of the late Geometric period, or the beginning of the Archaic period,
c. 750 BC.
Ancient Greeks
–
Political geography of ancient Greece in the Archaic and Classical periods
41.
Set theory
–
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
Set theory
–
Georg Cantor
Set theory
–
A
Venn diagram illustrating the
intersection of two
sets.
42.
Cardinality
–
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = contains 3 elements, there are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted | A |, with a bar on each side, this is the same notation as absolute value. Alternatively, the cardinality of a set A may be denoted by n, A, card, while the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets. Two sets A and B have the same cardinality if there exists a bijection, that is, such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A≈B or A~B, for example, the set E = of non-negative even numbers has the same cardinality as the set N = of natural numbers, since the function f = 2n is a bijection from N to E. A has cardinality less than or equal to the cardinality of B if there exists a function from A into B. A has cardinality less than the cardinality of B if there is an injective function. If | A | ≤ | B | and | B | ≤ | A | then | A | = | B |, the axiom of choice is equivalent to the statement that | A | ≤ | B | or | B | ≤ | A | for every A, B. That is, the cardinality of a set was not defined as an object itself. However, such an object can be defined as follows, the relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all sets which have the same cardinality as A. There are two ways to define the cardinality of a set, The cardinality of a set A is defined as its class under equinumerosity. A representative set is designated for each equivalence class, the most common choice is the initial ordinal in that class. This is usually taken as the definition of number in axiomatic set theory. Assuming AC, the cardinalities of the sets are denoted ℵ0 < ℵ1 < ℵ2 < …. For each ordinal α, ℵ α +1 is the least cardinal number greater than ℵ α
Cardinality
–
Bijective function from N to E. Although E is a proper subset of N, both sets have the same cardinality.
43.
Empty set
–
In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
Empty set
–
The empty set is the set containing no elements.
44.
Cardinal number
–
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a set is a natural number, the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets, cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, in the case of finite sets, this agrees with the intuitive notion of size. In the case of sets, the behavior is more complex. It is also possible for a subset of an infinite set to have the same cardinality as the original set. There is a sequence of cardinal numbers,0,1,2,3, …, n, …, ℵ0, ℵ1, ℵ2, …, ℵ α, …. This sequence starts with the natural numbers including zero, which are followed by the aleph numbers, the aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number, If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory and it is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the numbers form a skeleton of the category of sets. The notion of cardinality, as now understood, was formulated by Georg Cantor, cardinality can be used to compare an aspect of finite sets, e. g. the sets and are not equal, but have the same cardinality, namely three. Cantor applied his concept of bijection to infinite sets, e. g. the set of natural numbers N =, thus, all sets having a bijection with N he called denumerable sets and they all have the same cardinal number. This cardinal number is called ℵ0, aleph-null and he called the cardinal numbers of these infinite sets transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N and he later proved that the set of all real algebraic numbers is also denumerable. His proof used an argument with nested intervals, but in an 1891 paper he proved the result using his ingenious. The new cardinal number of the set of numbers is called the cardinality of the continuum. His continuum hypothesis is the proposition that c is the same as ℵ1 and this hypothesis has been found to be independent of the standard axioms of mathematical set theory, it can neither be proved nor disproved from the standard assumptions
Cardinal number
–
A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.
45.
Base 10
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
Base 10
–
The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the
Warring States period in China.
Base 10
–
Numeral systems
Base 10
–
Ten fingers on two hands, the possible starting point of the decimal counting.
Base 10
–
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period
46.
Numerical digit
–
A digit is a numeric symbol used in combinations to represent numbers in positional numeral systems. The name digit comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 numeral system, i. e. the decimal digits. In a given system, if the base is an integer. For example, the system has ten digits, whereas binary has two digits. In a basic system, a numeral is a sequence of digits. Each position in the sequence has a value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, each digit in a number system represents an integer. For example, in decimal the digit 1 represents the one, and in the hexadecimal system. A positional number system must have a digit representing the integers from zero up to, but not including, thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals 0 to 9 in the rightmost units position. The Hindu–Arabic numeral system uses a decimal separator, commonly a period in English, or a comma in other European languages, to denote the place or units place. Each successive place to the left of this has a value equal to the place value of the previous digit times the base. Similarly, each place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10, the total value of the number is 1 ten,0 ones,3 tenths, and 4 hundredths. Note that the zero, which contributes no value to the number, the place value of any given digit in a numeral can be given by a simple calculation, which in itself is a compliment to the logic behind numeral systems. And to the right, the digit is multiplied by the base raised by a negative n, for example, in the number 10. This system was established by the 7th century in India, but was not yet in its modern form because the use of the digit zero had not yet widely accepted. Instead of a zero, a dot was left in the numeral as a placeholder, the first widely acknowledged use of zero was in 876. The original numerals were very similar to the ones, even down to the glyphs used to represent digits
Numerical digit
47.
Place value
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
Place value
–
Numeral systems
48.
Peano Arithmetic
–
These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic, the Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers, the next four are general statements about equality, in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the underlying logic. The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation, the ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. When Peano formulated his axioms, the language of logic was in its infancy. Peano was unaware of Freges work and independently recreated his logical apparatus based on the work of Boole, the Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or N. The non-logical symbols for the axioms consist of a constant symbol 0, the first axiom states that the constant 0 is a natural number, The next four axioms describe the equality relation. Since they are valid in first-order logic with equality, they are not considered to be part of the Peano axioms in modern treatments. The remaining axioms define the properties of the natural numbers. The naturals are assumed to be closed under a successor function S. Peanos original formulation of the axioms used 1 instead of 0 as the first natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties, however, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1,6,7,8 define a representation of the intuitive notion of natural numbers. However, considering the notion of natural numbers as can be derived from the axioms, axioms 1,6,7,8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number, the intuitive notion that all natural numbers observe a succession relation with one or two other numbers requires an additional axiom, which is sometimes called the axiom of induction. The induction axiom is stated in the following form, In Peanos original formulation. It is now common to replace this second-order principle with a weaker first-order induction scheme, there are important differences between the second-order and first-order formulations, as discussed in the section § Models below. The Peano axioms can be augmented with the operations of addition and multiplication, the respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique using the Peano axioms. Addition is a function that maps two natural numbers to another one and it is defined recursively as, a +0 = a, a + S = S
Peano Arithmetic
49.
Minus sign
–
The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning more and less, respectively, though the signs now seem as familiar as the alphabet or the Hindu-Arabic numerals, they are not of great antiquity. In Europe in the early 15th century the letters P and M were generally used, the symbols appeared for the first time in Luca Pacioli’s mathematics compendium, Summa de arithmetica, geometria, proportioni et proportionalità, first printed and published in Venice in 1494. The + is a simplification of the Latin et, the − may be derived from a tilde written over m when used to indicate subtraction, or it may come from a shorthand version of the letter m itself. In his 1489 treatise Johannes Widmann referred to the symbols − and + as minus and mer, was − ist, das ist minus, und das + ist das mer. They werent used for addition and subtraction here, but to indicate surplus and deficit, the plus sign is a binary operator that indicates addition, as in 2 +3 =5. It can also serve as an operator that leaves its operand unchanged. This notation may be used when it is desired to emphasize the positiveness of a number, the plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or is equivalent to and it is conventional to use the plus sign to only denote commutative operations. Subtraction is the inverse of addition, directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5, a unary operator that acts as an instruction to replace the operand by its additive inverse. For example, if x is 3, then −x is −3, similarly, − is equal to 2. The above is a case of this. All three uses can be referred to as minus in everyday speech, further, some textbooks in the United States encourage −x to be read as the opposite of x or the additive inverse of x to avoid giving the impression that −x is necessarily negative. However, in programming languages and Microsoft Excel in particular, unary operators bind strongest, so in those cases −5^2 is 25. Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers. For example, subtracting −5 from 3 might be read as positive three take away negative 5 and be shown as 3 − −5 becomes 3 +5 =8, in grading systems, the plus sign indicates a grade one level higher and the minus sign a grade lower
Minus sign
–
Plus, minus, and hyphen-minus.
50.
Blackboard bold
–
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol are doubled. The symbols usually denote number sets, one way of producing blackboard bold is to double-strike a character with a small offset on a typewriter. Thus they are referred to as double struck. e. by using the edge rather than point of the chalk. It then made its way back in print form as a style from ordinary bold, possibly starting with the original 1965 edition of Gunning. Some mathematicians do not recognize blackboard bold as a style from bold. Jean-Pierre Serre uses double-struck letters when writing bold on the blackboard, donald Knuth also prefers boldface to blackboard bold, and consequently did not include blackboard bold in the Computer Modern fonts he created for the TeX mathematical typesetting system. The Chicago Manual of Style in 1993 advises, blackboard bold should be confined to the classroom whereas in 2003 it states that open-faced symbols are reserved for systems of numbers. In Unicode, a few of the common blackboard bold characters are encoded in the Basic Multilingual Plane in the Letterlike Symbols area. The rest, however, are encoded outside the BMP, from U+1D538 to U+1D550, U+1D552 to U+1D56B, being outside the BMP, these are relatively new and not widely supported. The following table shows all available Unicode blackboard bold characters, the symbols are nearly universal in their interpretation, unlike their normally-typeset counterparts, which are used for many different purposes. The first column shows the letter as typically rendered by the ubiquitous LaTeX markup system, the second column shows the Unicode codepoint. The third column shows the symbol itself, the fourth column describes known typical usage in mathematical texts. In addition, a blackboard-bold Greek letter mu is used by number theorists. Mathematical alphanumeric symbols Set notation Weisstein, Eric W. Doublestruck, http, //www. w3. org/TR/MathML2/double-struck. html shows blackboard bold symbols together with their Unicode encodings. Encodings in the Basic Multilingual Plane are highlighted in yellow
Blackboard bold
–
An example of blackboard bold letters.