1.
Palindrome
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A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward, such as madam or racecar. Sentence-length palindromes may be written when allowances are made for adjustments to capital letters, punctuation, and word dividers, such as A man, a plan, was it a car or a cat I saw. Composing literature in palindromes is an example of constrained writing, the word palindrome was coined by the English playwright Ben Jonson in the 17th century from the Greek roots palin and dromos. Palindromes date back at least to 79 AD, as a palindrome was found as a graffito at Herculaneum and this palindrome, called the Sator Square, consists of a sentence written in Latin, Sator Arepo Tenet Opera Rotas. It is remarkable for the fact that the first letters of each form the first word, the second letters form the second word. Hence, it can be arranged into a square that reads in four different ways. As such, they can be referred to as palindromatic, the palindromic Latin riddle In girum imus nocte et consumimur igni describes the behavior of moths. It is likely that this palindrome is from medieval rather than ancient times, byzantine Greeks often inscribed the palindrome, Wash sins, not only face ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ ΜΗ ΜΟΝΑΝ ΟΨΙΝ, on baptismal fonts. This practice was continued in many English churches, some well-known English palindromes are, Able was I ere I saw Elba, A man, a plan, a canal - Panama. Madam, Im Adam and Never odd or even, English palindromes of notable length include mathematician Peter Hiltons Doc, note, I dissent. A fast never prevents a fatness, I diet on cod and Scottish poet Alastair Reids T. Eliot, top bard, notes putrid tang emanating, is sad, Id assign it a name, gnat dirt upset on drab pot toilet. The most familiar palindromes in English are character-unit palindromes, the characters read the same backward as forward. Some examples of words are redivider, noon, civic, radar, level, rotor, kayak, reviver, racecar, redder, madam. There are also word-unit palindromes in which the unit of reversal is the word, word-unit palindromes were made popular in the recreational linguistics community by J. A. Lindon in the 1960s. Occasional examples in English were created in the 19th century, several in French and Latin date to the Middle Ages. Palindromes often consist of a sentence or phrase, e. g, mr. Owl ate my metal worm, Was it a cat I saw. Or Go hang a salami, Im a lasagna hog, punctuation, capitalization, and spaces are usually ignored. Some, such as Rats live on no evil star, Live on time, emit no evil, semordnilap is a name coined for words that spell a different word in reverse
2.
Decimal
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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
3.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0
4.
Square number
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In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number
5.
Buckminster Fuller
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Richard Buckminster Bucky Fuller was an American architect, systems theorist, author, designer, and inventor. Fuller published more than 30 books, coining or popularizing terms such as Spaceship Earth, ephemeralization and he also developed numerous inventions, mainly architectural designs, and popularized the widely known geodesic dome. Carbon molecules known as fullerenes were later named by scientists for their structural and mathematical resemblance to geodesic spheres, Fuller was the second World President of Mensa from 1974 to 1983. Fuller was born on July 12,1895, in Milton, Massachusetts, the son of Richard Buckminster Fuller and Caroline Wolcott Andrews and he spent much of his youth on Bear Island, in Penobscot Bay off the coast of Maine. He often made items from materials he found in the woods and he experimented with designing a new apparatus for human propulsion of small boats. Later in life, Fuller took exception to the term invention, Fuller earned a machinists certification, and knew how to use the press brake, stretch press, and other tools and equipment used in the sheet metal trade. Fuller attended Milton Academy in Massachusetts, and after that studying at Harvard College. He was expelled from Harvard twice, first for spending all his money partying with a vaudeville troupe, by his own appraisal, he was a non-conforming misfit in the fraternity environment. Between his sessions at Harvard, Fuller worked in Canada as a mechanic in a textile mill, and later as a laborer in the meat-packing industry. He also served in the U. S. Navy in World War I, as a radio operator, as an editor of a publication. After discharge, he worked again in the packing industry. In 1917, he married Anne Hewlett, Buckminster Fuller recalled 1927 as a pivotal year of his life. His daughter Alexandra had died in 1922 of complications from polio, Fuller dwelled on her death, suspecting that it was connected with the Fullers damp and drafty living conditions. This provided motivation for Fullers involvement in Stockade Building Systems, a business which aimed to provide affordable, in 1927, at age 32, Fuller lost his job as president of Stockade. The Fuller family had no savings, and the birth of their daughter Allegra in 1927 added to the financial challenges, Fuller drank heavily and reflected upon the solution to his familys struggles on long walks around Chicago. During the autumn of 1927, Fuller contemplated suicide, so that his family could benefit from an insurance payment. Fuller said that he had experienced a profound incident which would provide direction and he felt as though he was suspended several feet above the ground enclosed in a white sphere of light. A voice spoke directly to Fuller, and declared, From now on you need never await temporal attestation to your thought and you do not have the right to eliminate yourself
6.
Base (exponentiation)
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In exponentiation, the base is the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b and it is more commonly expressed as the nth power of b, b to the nth power or b to the power n. For example, the power of 10 is 10,000 because 104 =10 ×10 ×10 ×10 =10,000. The term power strictly refers to the expression, but is sometimes used to refer to the exponent. When the nth power of b equals a number a, or a = bn, for example,10 is a fourth root of 10,000. The inverse function to exponentiation with base b is called the logarithm to base b, for example, log1010,000 =4
7.
Sequence
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
8.
Natural number
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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
9.
Numeral system
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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
10.
Radix
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In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for base in the arithmetical sense. In the system with radix 13, for example, a string of such as 398 denotes the number 3 ×132 +9 ×131 +8 ×130. More generally, in a system with radix b, a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, commonly used numeral systems include, For a larger list, see List of numeral systems. The octal and hexadecimal systems are used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, a similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two. However, other systems are possible, e. g. golden ratio base. Base Radix economy Non-standard positional numeral systems Base Convert, a floating-point base calculator MathWorld entry on base
11.
Numerical digit
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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
12.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a 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
13.
Rule of product
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In combinatorics, the rule of product or multiplication principle is a basic counting principle. Stated simply, it is the idea that if there are a ways of doing something and b ways of doing another thing, ⏞ In this example, the rule says, multiply 3 by 2, getting 6. The sets and in this example are disjoint sets, but that is not necessary. The number of ways to choose a member of, and then to do so again, in choosing an ordered pair each of whose components is in, is 3 ×3 =9. As another example, when you decide to order pizza, you must first choose the type of crust, next, you choose one topping, cheese, pepperoni, or sausage. Using the rule of product, you know there are 2 ×3 =6 possible combinations of ordering a pizza. In set theory, this principle is often taken to be the definition of the product of cardinal numbers. We have | S1 | ⋅ | S2 | ⋯ | S n | = | S1 × S2 × ⋯ × S n | where × is the Cartesian product operator. These sets need not be finite, nor is it necessary to have finitely many factors in the product. The rule of sum is another basic counting principle
14.
Parity (mathematics)
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Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations
15.
Cube (algebra)
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In arithmetic and algebra, the cube of a number n is its third power, the result of the number multiplied by itself twice, n3 = n × n × n. It is also the number multiplied by its square, n3 = n × n2 and this is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n and it determines the side of the cube of a given volume. It is also n raised to the one-third power, both cube and cube root are odd functions,3 = −. The cube of a number or any other mathematical expression is denoted by a superscript 3, a cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The perfect cubes up to 603 are, Geometrically speaking, an integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger. For example,27 small cubes can be arranged into one larger one with the appearance of a Rubiks Cube, the difference between the cubes of consecutive integers can be expressed as follows, n3 −3 = 3n +1. There is no minimum perfect cube, since the cube of an integer is negative. For example, −4 × −4 × −4 = −64, unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25,75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6, some cube numbers are also square numbers, for example,64 is a square number and a cube number. This happens if and only if the number is a perfect sixth power, the last digits of each 3rd power are, It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1,8 or 9. That is their values modulo 9 may be only −1,1 and 0, every positive integer can be written as the sum of nine positive cubes. The equation x3 + y3 = z3 has no solutions in integers. In fact, it has none in Eisenstein integers, both of these statements are also true for the equation x3 + y3 = 3z3. The sum of the first n cubes is the nth triangle number squared,13 +23 + ⋯ + n 3 =2 =2. Proofs Charles Wheatstone gives a simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. Indeed, he begins by giving the identity n 3 = + + + ⋯ + ⏟ n consecutive odd numbers, kanim provides a purely visual proof, Benjamin & Orrison provide two additional proofs, and Nelsen gives seven geometric proofs
16.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
17.
Square-free integer
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In mathematics, a square-free, or quadratfrei integer, is an integer which is divisible by no other perfect square than 1. For example,10 is square-free but 18 is not, as 18 is divisible by 9 =32. The smallest positive square-free numbers are 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,39. The radical of an integer is its largest square-free factor, an integer is square-free if and only if it is equal to its radical. Any arbitrary positive integer n can be represented in a way as the product of a powerful number and a square-free integer. The square-free factor is the largest square-free divisor k of n that is coprime with n/k, a positive integer n is square-free if and only if in the prime factorization of n, no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor p of n, also n is square-free if and only if in every factorization n = ab, the factors a and b are coprime. An immediate result of this definition is that all numbers are square-free. A positive integer n is square-free if and only if all abelian groups of n are isomorphic. This follows from the classification of finitely generated abelian groups, a integer n is square-free if and only if the factor ring Z / nZ is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if, for every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice and it is a Boolean algebra if and only if n is square-free. A positive integer n is square-free if and only if μ ≠0, a positive integer n is squarefree if and only if ∑ d 2 ∣ n μ =1. This results from the properties of Möbius function, and the fact that this sum is equal to ∑ d ∣ m μ, where m is the largest divisor of n such that m2 divides n. The Dirichlet generating function for the numbers is ζ ζ = ∑ n =1 ∞ | μ | n s where ζ is the Riemann zeta function. This is easily seen from the Euler product ζ ζ = ∏ p = ∏ p, let Q denote the number of square-free integers between 1 and x. For large n, 3/4 of the positive integers less than n are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Under the Riemann hypothesis, the term can be further reduced to yield Q = x ζ + O =6 x π2 + O
18.
Prime factor
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In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of an integer is a list of the integers prime factors, together with their multiplicities. The fundamental theorem of arithmetic says that every integer has a single unique prime factorization. To shorten prime factorizations, factors are expressed in powers. For example,360 =2 ×2 ×2 ×3 ×3 ×5 =23 ×32 ×5, in which the factors 2,3 and 5 have multiplicities of 3,2 and 1, respectively. For a prime factor p of n, the multiplicity of p is the largest exponent a for which pa divides n exactly. For a positive n, the number of prime factors of n. Perfect square numbers can be recognized by the fact all of their prime factors have even multiplicities. For example, the number 144 has the prime factors 144 =2 ×2 ×2 ×2 ×3 ×3 =24 ×32. These can be rearranged to make the more visible,144 =2 ×2 ×2 ×2 ×3 ×3 = × =2 =2. Because every prime factor appears a number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on, positive integers with no prime factors in common are said to be coprime. Two integers a and b can also be defined as if their greatest common divisor gcd =1. Euclids algorithm can be used to determine whether two integers are coprime without knowing their prime factors, the runs in a time that is polynomial in the number of digits involved. The integer 1 is coprime to every integer, including itself. This is because it has no prime factors, it is the empty product and this implies that gcd =1 for any b ≥1. The function, ω, represents the number of prime factors of n, while the function, Ω. If n = ∏ i =1 ω p i α i, for example,24 =23 ×31, so ω =2 and Ω =3 +1 =4
19.
Divisor function
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In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the function, it counts the number of divisors of an integer. It appears in a number of identities, including relationships on the Riemann zeta function. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities, a related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. The sum of divisors function σx, for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as σ x = ∑ d ∣ n d x, the notations d, ν and τ are also used to denote σ0, or the number-of-divisors function. When x is 1, the function is called the function or sum-of-divisors function. The aliquot sum s of n is the sum of the proper divisors, and equals σ1 − n, the cases x =2 to 5 are listed in A001157 − A001160, x =6 to 24 are listed in A013954 − A013972. For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and σ0 is even, for an integer, one divisor is not paired with a distinct divisor. Similarly, the number σ1 is odd if and only if n is a square or twice a square. For a prime p, σ0 =2 σ0 = n +1 σ1 = p +1 because by definition. Also, where pn# denotes the primorial, σ0 =2 n since n prime factors allow a sequence of binary selection from n terms for each proper divisor formed, clearly,1 < σ0 < n and σ > n for all n >2. The divisor function is multiplicative, but not completely multiplicative and it follows that d is, σ0 = ∏ i =1 r. For example, if n is 24, there are two factors, noting that 24 is the product of 23×31, a1 is 3. Thus we can calculate σ0 as so, σ0 = ∏ i =12 = =4 ⋅2 =8, the eight divisors counted by this formula are 1,2,4,8,3,6,12, and 24. Here s denotes the sum of the divisors of n, that is. This function is the one used to perfect numbers which are the n for which s = n. If s > n then n is an abundant number and if s < n then n is a deficient number
20.
Perfect power
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In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m >1, in this case, n may be called a perfect kth power. If k =2 or k =3, then n is called a square or perfect cube. Sometimes 1 is also considered a perfect power. The sum of the reciprocals of the perfect powers p without duplicates is, ∑ p 1 p = ∑ k =2 ∞ μ ≈0.874464368 … where μ is the Möbius function and this is sometimes known as the Goldbach-Euler theorem. Detecting whether or not a natural number n is a perfect power may be accomplished in many different ways. One of the simplest such methods is to all possible values for k across each of the divisors of n. This method can immediately be simplified by considering only prime values of k. This is because if n = m k for a composite k = a p p is prime. Because of this result, the value of k must necessarily be prime. As an example, consider n = 296·360·724, since gcd =12, n is a perfect 12th power. In 2002 Romanian mathematician Preda Mihăilescu proved that the pair of consecutive perfect powers is 23 =8 and 32 =9. Pillais conjecture states that for any positive integer k there are only a finite number of pairs of perfect powers whose difference is k. As an alternate way to perfect powers, the recursive approach has yet to be found useful. It is based on the observation that the difference between ab and b where a > b may not be constant, but if you take the difference of differences, b times. For example,94 =6561, and 104 is 10000, the difference between 84 and 94 is 2465, meaning the difference of differences is 974. A step further and you have 204, one step further, and you have 24, which is equal to 4. One step further and collating this key row from progressively larger exponents yields a similar to Pascals
21.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
22.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
23.
Base 7
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This is a list of numeral systems, that is, writing systems for expressing numbers. Numeral systems are classified here as to whether they use positional notation, the common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. In this Youtube video, Matt Parker jokingly invented a base-1082 system and this turns out to be 1925. Radix Radix economy Table of bases List of numbers in various languages Numeral prefix
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Base 18
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This is a list of numeral systems, that is, writing systems for expressing numbers. Numeral systems are classified here as to whether they use positional notation, the common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. In this Youtube video, Matt Parker jokingly invented a base-1082 system and this turns out to be 1925. Radix Radix economy Table of bases List of numbers in various languages Numeral prefix
25.
Positional notation
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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
26.
Primorial
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The rest of this article uses the latter interpretation. The name primorial, coined by Harvey Dubner, draws an analogy to primes the same way the name relates to factors. For the nth prime number pn, the primorial pn# is defined as the product of the first n primes, p n # ≡ ∏ k =1 n p k, where pk is the kth prime number. For instance, p5# signifies the product of the first 5 primes, the first six primorials pn# are,1,2,6,30,210,2310. The sequence also includes p0# =1 as empty product, asymptotically, primorials pn# grow according to, p n # = e n log n, where o is little-o notation. This is equivalent to, n # = {1 if n =0,1 # × n if n is prime # if n is composite. For example, 12# represents the product of those primes ≤12,12 # =2 ×3 ×5 ×7 ×11 =2310, since π =5, this can be calculated as,12 # = p π # = p 5 # =2310. Consider the first 12 primorials n#,1,2,6,6,30,30,210,210,210,210,2310,2310. We see that for composite n every term n# simply duplicates the preceding term #, in the above example we have 12# = p5# = 11# since 12 is a composite number. The natural logarithm of n# is the first Chebyshev function, written ϑ or θ, primorials n# grow according to, ln ∼ n. The idea of multiplying all known primes occurs in some proofs of the infinitude of the prime numbers, primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 7009223613394100000♠2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes. Every highly composite number is a product of primorials, primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial n, the fraction φ/n is smaller than it for any lesser integer, any completely multiplicative function is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values. Base systems corresponding to primorials have a proportion of repeating fractions than any smaller base. Every primorial is a sparsely totient number, the n-compositorial of a composite number n is the product of all composite numbers up to and including n. The n-compositorial is equal to the n-factorial divided by the primorial n#, the compositorials are 1,4,24,192,1728, 7004172800000000000♠17280, 7005207360000000000♠207360, 7006290304000000000♠2903040, 7007435456000000000♠43545600, 7008696729600000000♠696729600
27.
Scheherazade
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Scheherazade /ʃəˌhɛrəˈzɑːd, -də/, or Shahrazad, is a character and the storyteller in One Thousand and One Nights. This book includes the tales of Aladdin, Ali Baba and many more, the story goes that Shahryar found out one day that his first wife was unfaithful to him. Therefore, he resolved to marry a new virgin each day as well as behead the previous days wife and he had killed 1,000 such women by the time he was introduced to Scheherazade, the viziers daughter. Against her fathers wishes, Scheherazade volunteered to spend one night with the king, the king lay awake and listened with awe as Scheherazade told her first story. The night passed by, and Scheherazade stopped in the middle of the story, the king asked her to finish, but Scheherazade said there was no time, as dawn was breaking. So, the king spared her life for one day to finish the story the next night, the next night, Scheherazade finished the story and then began a second, even more exciting tale, which she again stopped halfway through at dawn. Again, the king spared her life for one day so she could finish the second story. And so the king kept Scheherazade alive day by day, as he anticipated the finishing of the previous nights story. At the end of 1,001 nights, and 1,000 stories, during these 1,001 nights, the king had fallen in love with Scheherazade. He spared her life, and made her his queen, the earliest forms of Scheherazades name include Shirazad in Masudi, and Shahrazad in Ibn al-Nadim, the latter meaning the person whose realm/dominion is free. It is shortened as Shahrzad in Persian, in explaining his spelling choice for the name, Burton says, Shahrázád = City-freer, in the older version Scheherazade. The Bres Edit corrupts the former to Sháhrzád or Sháhrazád, I have ventured to restore the name as it should be. Having introduced the name, Burton does not continue to use the diacritics on the name, Scheherazade in popular culture The Arabian Nights Entertainments — Project Gutenberg
28.
One Thousand and One Nights
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One Thousand and One Nights is a collection of Middle Eastern and South Asian stories and folk tales compiled in Arabic during the Islamic Golden Age. It is often known in English as the Arabian Nights, from the first English-language edition, the work was collected over many centuries by various authors, translators, and scholars across West, Central, and South Asia and North Africa. The tales themselves trace their roots back to ancient and medieval Arabic, Persian, Mesopotamian, Indian, Jewish, the stories proceed from this original tale, some are framed within other tales, while others begin and end of their own accord. Some editions contain only a few hundred nights, while others include 1,001 or more, the bulk of the text is in prose, although verse is occasionally used for songs and riddles and to express heightened emotion. Most of the poems are single couplets or quatrains, although some are longer, the main frame story concerns Shahryār, whom the narrator calls a Sasanian king ruling in India and China. Shahryār begins to marry a succession of only to execute each one the next morning. Eventually the vizier, whose duty it is to provide them, Scheherazade, the viziers daughter, offers herself as the next bride and her father reluctantly agrees. On the night of their marriage, Scheherazade begins to tell the king a tale, the king, curious about how the story ends, is thus forced to postpone her execution in order to hear the conclusion. The next night, as soon as she finishes the tale, she begins a new one, so it goes on for 1,001 nights. The tales vary widely, they include historical tales, love stories, tragedies, comedies, poems, burlesques, numerous stories depict jinns, ghouls, apes, sorcerers, magicians, and legendary places, which are often intermingled with real people and geography, not always rationally. The different versions have different individually detailed endings but they all end with the giving his wife a pardon. The narrators standards for what constitutes a cliffhanger seem broader than in modern literature, the history of the Nights is extremely complex and modern scholars have made many attempts to untangle the story of how the collection as it currently exists came about. Most scholars agreed that the Nights was a work and that the earliest tales in it came from India and Persia. At some time, probably in the early 8th century, these tales were translated into Arabic under the title Alf Layla and this collection then formed the basis of The Thousand and One Nights. The original core of stories was quite small, then, in Iraq in the 9th or 10th century, this original core had Arab stories added to it – among them some tales about the Caliph Harun al-Rashid. Devices found in Sanskrit literature such as stories and animal fables are seen by some scholars as lying at the root of the conception of the Nights. Indian folklore is represented in the Nights by certain animal stories, the influence of the Panchatantra and Baital Pachisi is particularly notable. The Jataka Tales are a collection of 547 Buddhist stories, which are for the most part moral stories with an ethical purpose
29.
Binomial coefficient
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In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a coefficient is indexed by a pair of integers n ≥ k ≥0 and is written. It is the coefficient of the xk term in the expansion of the binomial power n. The value of the coefficient is given by the expression n. k, arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascals triangle. The properties of binomial coefficients have led to extending the definition to beyond the case of integers n ≥ k ≥0. Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier, the earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingalas Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī, alternative notations include C, nCk, nCk, Ckn, Cnk, and Cn, k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display, in this form the binomial coefficients are easily compared to k-permutations of n, written as P, etc. For natural numbers n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of n, the same coefficient also occurs in the binomial formula, which explains the name binomial coefficient. This shows in particular that is a number for any natural numbers n and k. Most of these interpretations are easily seen to be equivalent to counting k-combinations, several methods exist to compute the value of without actually expanding a binomial power or counting k-combinations. It also follows from tracing the contributions to Xk in n−1, as there is zero Xn+1 or X−1 in n, one might extend the definition beyond the above boundaries to include =0 when either k > n or k <0. This recursive formula then allows the construction of Pascals triangle, surrounded by white spaces where the zeros, or the trivial coefficients, a more efficient method to compute individual binomial coefficients is given by the formula = n k _ k. = n ⋯ k ⋯1 = ∏ i =1 k n +1 − i i and this formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, the denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. Due to the symmetry of the binomial coefficient with regard to k and n−k, calculation may be optimised by setting the limit of the product above to the smaller of k. This formula follows from the formula above by multiplying numerator and denominator by. As a consequence it involves many factors common to numerator and denominator and it is less practical for explicit computation unless common factors are first cancelled
30.
International Standard Book Number
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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
31.
Exponentiation
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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
32.
Achilles number
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An Achilles number is a number that is powerful but not a perfect power. A positive integer n is a number if, for every prime factor p of n. In other words, every prime factor appears at least squared in the factorization, however, not all powerful numbers are Achilles numbers, only those that cannot be represented as mk, where m and k are positive integers greater than 1. Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers. A number n = p1a1 p2a2 … pkak is powerful if min ≥2, if in addition gcd =1 the number is an Achilles number. The smallest pair of consecutive Achilles numbers is,5425069447 =73 ×412 ×9725425069448 =23 ×260412108 is a powerful number and its prime factorization is 22 ·33, and thus its prime factors are 2 and 3. Both 22 =4 and 32 =9 are divisors of 108, however,108 cannot be represented as mk, where m and k are positive integers greater than 1, so 108 is an Achilles number. 360 is not an Achilles number because it is not powerful, one of its prime factors is 5 but 360 is not divisible by 52 =25. Finally,784 is not an Achilles number and it is a powerful number, because not only are 2 and 7 its only prime factors, but also 22 =4 and 72 =49 are divisors of it. Nonetheless, it is a power,784 =24 ⋅72 =2 ⋅72 =2 =282. So it is not an Achilles number
33.
Power of two
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In mathematics, a power of two means a number of the form 2n where n is an integer, i. e. the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to values, so we have 1,2. Because two is the base of the numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100…000 or 0. 00…001, just like a power of ten in the decimal system, a word, interpreted as an unsigned integer, can represent values from 0 to 2n −1 inclusively. Corresponding signed integer values can be positive, negative and zero, either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this show up frequently in computer software. For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees at any time. Powers of two are used to measure computer memory. A byte is now considered eight bits (an octet, resulting in the possibility of 256 values, the prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024. However, in general, the term kilo has been used in the International System of Units to mean 1,000, binary prefixes have been standardized, such as kibi meaning 1,024. Nearly all processor registers have sizes that are powers of two,32 or 64 being most common, powers of two occur in a range of other places as well. For many disk drives, at least one of the size, number of sectors per track. The logical block size is almost always a power of two. Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of two or three powers of two, or powers of two minus one. For example,640 =512 +128 =128 ×5, put another way, they have fairly regular bit patterns. A prime number that is one less than a power of two is called a Mersenne prime, for example, the prime number 31 is a Mersenne prime because it is 1 less than 32. Similarly, a number that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational, the numbers that can be represented as sums of consecutive positive integers are called polite numbers, they are exactly the numbers that are not powers of two
34.
Power of 10
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In mathematics, a power of 10 is any of the integer powers of the number ten, in other words, ten multiplied by itself a certain number of times. By definition, the one is a power of ten. The first few powers of ten are,1,10,100,1000,10000,100000,1000000,10000000. In decimal notation the nth power of ten is written as 1 followed by n zeroes and it can also be written as 10n or as 1En in E notation. See order of magnitude and orders of magnitude for named powers of ten, there are two conventions for naming positive powers of ten, called the long and short scales. Where a power of ten has different names in the two conventions, the long scale namme is shown in brackets, googolplex, a much larger power of ten, was also introduced in that book. Scientific notation is a way of writing numbers of very large, a number written in scientific notation has a significand multiplied by a power of ten. Sometimes written in the form, m × 10n Or more compactly as, where n is positive, this indicates the number zeros after the number, and where the n is negative, this indicates the number of decimal places before the number. As an example,105 =100,000 10−5 =0.00001 The notation of mEn, known as E notation, is used in programming, spreadsheets and databases. Power of two SI prefix Cosmic View, inspiration for the film Powers of Ten Video Powers of Ten, US Public Broadcasting Service, made by Charles and Ray Eames. Starting at a picnic by the lakeside in Chicago, this film transports the viewer to the edges of the universe. Every ten seconds we view the point from ten times farther out until our own galaxy is visible only as a speck of light among many others. Returning to Earth with breathtaking speed, we move inward - into the hand of the sleeping picnicker - with ten times more magnification every ten seconds and our journey ends inside a proton of a carbon atom within a DNA molecule in a white blood cell