The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system; the way of denoting numbers in the decimal system is referred to as decimal notation. A decimal numeral, or just decimal, or casually decimal number, refers to the notation of a number in the decimal numeral system. Decimals may sometimes be identified for containing a decimal separator. "Decimal" may refer to the digits after the decimal separator, such as in "3.14 is the approximation of π to two decimals". The numbers that may be represented in the decimal system are the decimal fractions, the fractions of the form a/10n, where a is an integer, n is a non-negative integer; the decimal system has been extended to infinite decimals, for representing any real number, by using an infinite sequence of digits after the decimal separator. In this context, the decimal numerals with a finite number of non–zero places after the decimal separator are sometimes called terminating decimals.
A repeating decimal is an infinite decimal that after some place repeats indefinitely the same sequence of digits. An infinite decimal represents a rational number if and only if it is a repeating decimal or has a finite number of nonzero digits. Many numeral systems of ancient civilisations use ten and its powers for representing numbers because there are ten fingers on two hands and people started counting by using their fingers. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Chinese numerals. Large numbers were difficult to represent in these old numeral systems, only the best mathematicians were able to multiply or divide large numbers; these difficulties were solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers for forming the decimal numeral system. For writing numbers, the decimal system uses ten decimal digits, a decimal mark, for negative numbers, a minus sign "−".
The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. For representing a non-negative number, a decimal consists of either a sequence of digits such as 2017, or in full generality, a m a m − 1 … a 0 or two sequence of digits separated by a decimal mark such as 3.14159, 15.00, or in full generality a m a m − 1 … a 0. B 1 b 2 … b n It is assumed that, if m > 0, the first digit am is not zero, but, in some circumstances, it may be useful to have one or more 0's on the left. This does not change the value represented by the decimal. For example, 3.14 = 03.14 = 003.14. If bn =0, it may be removed, conversely, trailing zeros may be added without changing the represented number: for example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200. Sometimes the extra zeros are used for indicating the accuracy of a measurement. For example, 15.00 m may indicate that the measurement error is less than one centimeter, while 15 m may mean that the length is fifteen meters, that the error may exceed 10 cm. For representing a negative number, a minus sign is placed before am.
The numeral a m a m − 1 … a 0. B 1 b 2 … b n represents the number a m 10 m + a m − 1 10 m − 1 + ⋯ + a 0 10 0 + b 1 10 1 + b 2 10 2 + ⋯ + b n 10 n Therefore, the contribution of each digit to the value of a number depends on its position in the numeral; that is, the decimal system is a positional numeral system The numbers that are represented by decimal numerals are the decimal fractions, that is, the rational numbers that may be expressed as a fraction, the denominator of, a power of ten. For example, the numerals 0.8, 14.89, 0.00024 represent the fractions 8/10, 1489/100, 24/100000. More a decimal with n digits after the separator represents the fraction with denominator 10n, whose numerator is the integer obtained by removing the separator. Expressed as a reduced fraction, the decimal numbers are those whose denominator is a product of a powe
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.
Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.
The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide
In elementary arithmetic, a carry is a digit, transferred from one column of digits to another column of more significant digits. It is part of the standard algorithm to add numbers together by starting with the rightmost digits and working to the left. For example, when 6 and 7 are added to make 13, the "3" is written to the same column and the "1" is carried to the left; when used in subtraction the operation is called a borrow. Carrying is emphasized in traditional mathematics, while curricula based on reform mathematics do not emphasize any specific method to find a correct answer. Carrying makes a few appearances in higher mathematics as well. In computing, carrying is an important function of adder circuits. A typical example of carry is in the following pencil-and-paper addition: ¹ 27 + 59 ---- 86 7 + 9 = 16, the digit 1 is the carry; the opposite is a borrow, as in −1 47 − 19 ---- 28 Here, 7 − 9 = −2, so try + 7 = 8, the 10 is got by taking 1 from the next digit to the left. There are two ways in which this is taught: The ten is moved from the next digit left, leaving in this example 3 − 1 in the tens column.
According to this method, the term "borrow" is a misnomer. The ten is copied from the next digit left, then'paid back' by adding it to the subtrahend in the column from which it was'borrowed', giving in this example 4 − in the tens column. Traditionally, carry is taught in the addition of multi-digit numbers in the 2nd or late first year of elementary school. However, since the late 20th century, many adopted curricula developed in the United States such as TERC omitted instruction of the traditional carry method in favor of invented arithmetic methods, methods using coloring and charts; such omissions were criticized by such groups as Mathematically Correct, some states and districts have since abandoned this experiment, though it remains used. Kummer's theorem states that the number of carries involved in adding two numbers in base p is equal to the exponent of the highest power of p dividing a certain binomial coefficient; when several random numbers of many digits are added, the statistics of the carry digits bears an unexpected connection with Eulerian numbers and the statistics of riffle shuffle permutations.
In abstract algebra, the carry operation for two-digit numbers can be formalized using the language of group cohomology. This viewpoint can be applied to alternative characterizations of the real numbers; when speaking of a digital circuit like an adder, the word carry is used in a similar sense. In most computers, the carry from the most significant bit of an arithmetic operation is placed in a special carry bit which can be used as a carry-in for multiple precision arithmetic or tested and used to control execution of a computer program; the same carry bit is generally used to indicate borrows in subtraction, though the bit's meaning is inverted due to the effects of two's complement arithmetic. A carry bit value of "1" signifies that an addition overflowed the ALU, must be accounted for when adding data words of lengths greater than that of the CPU. For subtractive operations, two conventions are employed as most machines set the carry flag on borrow while some machines instead reset the carry flag on borrow.
Weisstein, Eric W. "Carry". MathWorld. Weisstein, Eric W. "Borrow". MathWorld. Carrying - nLab
In mathematics, a perfect power is a positive integer that can be resolved into equal factors, whose root can be extracted. I.e. 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, k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3 n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are considered perfect powers. A sequence of perfect powers can be generated by iterating through the possible values for k; the first few ascending perfect powers in numerical order are: 2 2 = 4, 2 3 = 8, 3 2 = 9, 2 4 = 16, 4 2 = 16, 5 2 = 25, 3 3 = 27, 2 5 = 32, 6 2 = 36, 7 2 = 49, 2 6 = 64, 4 3 = 64, 8 2 = 64, … The sum of the reciprocals of the perfect powers is 1: ∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = 1. Which can be proved as follows: ∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 ∑ k = 0 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 = ∑ m = 2 ∞ 1 m = ∑ m = 2 ∞ = 1.
The first perfect powers without duplicates are:, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024... 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 ζ is the Riemann zeta function. According to Euler, Goldbach showed that the sum of 1/p − 1 over the set of perfect powers p, excluding 1 and excluding duplicates, is 1: ∑ p 1 p − 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + ⋯ = 1; this is sometimes known as the Goldbach–Euler theorem. Detecting whether or not a given natural number n is a perfect power may be accomplished in many different ways, with varying levels of complexity. One of the simplest such methods is to consider all possible values for k across each of the divisors of n, up to k ≤ log 2 n. So if the divisors of n are n 1, n 2, …, n j one of the values n 1 2, n 2 2
Richard Buckminster Fuller was an American architect, systems theorist, designer and futurist. Fuller published more than 30 books, coining or popularizing terms such as "Spaceship Earth", "Dymaxion" house/car, synergetic, "tensegrity", he developed numerous inventions architectural designs, popularized the known geodesic dome. Carbon molecules known as fullerenes were 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, the son of Richard Buckminster Fuller and Caroline Wolcott Andrews, grand-nephew of Margaret Fuller, an American journalist and women's rights advocate associated with the American transcendentalism movement; the unusual middle name, was an ancestral family name. As a child, Richard Buckminster Fuller tried numerous variations of his name, he used to sign his name differently each year in the guest register of his family summer vacation home at Bear Island, Maine.
He settled on R. Buckminster Fuller. Fuller spent much of his youth in Penobscot Bay off the coast of Maine, he attended Froebelian Kindergarten. He disagreed with the way geometry was taught in school, being unable to experience for himself that a chalk dot on the blackboard represented an "empty" mathematical point, or that a line could stretch off to infinity. To him these were illogical, led to his work on synergetics, he made items from materials he found in the woods, sometimes made his own tools. He experimented with designing a new apparatus for human propulsion of small boats. By age 12, he had invented a'push pull' system for propelling a rowboat by use of an inverted umbrella connected to the transom with a simple oar lock which allowed the user to face forward to point the boat toward its destination. In life, Fuller took exception to the term "invention". Years he decided that this sort of experience had provided him with not only an interest in design, but a habit of being familiar with and knowledgeable about the materials that his projects would require.
Fuller earned a machinist's certification, knew how to use the press brake, stretch press, other tools and equipment used in the sheet metal trade. Fuller attended Milton Academy in Massachusetts, after that began studying at Harvard College, where he was affiliated with Adams House, he was expelled from Harvard twice: first for spending all his money partying with a vaudeville troupe, after having been readmitted, for his "irresponsibility and lack of interest". 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, as a laborer in the meat-packing industry, he served in the U. S. Navy in World War I, as a shipboard radio operator, as an editor of a publication, as a crash rescue boat commander. After discharge, he worked again in the meat packing industry. In 1917, he married Anne Hewlett. During the early 1920s, he and his father-in-law developed the Stockade Building System for producing light-weight and fireproof housing—although the company would fail in 1927.
Buckminster Fuller recalled 1927 as a pivotal year of his life. His daughter Alexandra had died in 1922 of complications from polio and spinal meningitis just before her fourth birthday. Stanford historian, Barry Katz, found signs that around this time in his life Fuller was suffering from depression and anxiety. Fuller dwelled on his daughter's death, suspecting that it was connected with the Fullers' damp and drafty living conditions; this provided motivation for Fuller's involvement in Stockade Building Systems, a business which aimed to provide affordable, efficient housing. In 1927, at age 32, Fuller lost his job as president of Stockade; the Fuller family had no savings, the birth of their daughter Allegra in 1927 added to the financial challenges. Fuller drank and reflected upon the solution to his family's struggles on long walks around Chicago. During the autumn of 1927, Fuller contemplated suicide by drowning in Lake Michigan, so that his family could benefit from a life insurance payment.
Fuller said that he had experienced a profound incident which would provide direction and purpose for his life. 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, declared: From now on you need never await temporal attestation to your thought. You think the truth. You do not have the right to eliminate yourself. You do not belong to you. You belong to Universe. Your significance will remain forever obscure to you, but you may assume that you are fulfilling your role if you apply yourself to converting your experiences to the highest advantage of others. Fuller stated, he chose to embark on "an experiment, to find what a single individual could contribute to changing the world and benefiting all humanity". Speaking to audiences in life, Fuller would recount the story of his Lake Michigan experience, its transformative impact on his life. Historians have been unable to identify direct evidence for this experience within the 1927 papers of Fuller's Chronofile archives, housed at Stanford University.
Stanford historian Barry Katz suggests that the suicide story may be a myth which Fuller constructed in life, to summarize this formative period of his career. In 1927 Fuller resolved to think independently which included a commitment
A palindrome is a word, phrase, or other sequence of characters which reads the same backward as forward, such as madam or racecar or the number 10801. Sentence-length palindromes may be written when allowances are made for adjustments to capital letters and word dividers, such as "A man, a plan, a canal, Panama!", "Was it a car or a cat I saw?" or "No'x' in Nixon". 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, a city buried by ash in that year; 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 word form the first word, the second letters form the second word, so forth. Hence, it can be arranged into a word square that reads in four different ways: horizontally or vertically from either top left to bottom right or bottom right to top left.
As such, they can be referred to as palindromatic. A palindrome with the same square property is the Hebrew palindrome, "We explained the glutton, in the honey was burned and incinerated", credited to Abraham ibn Ezra in 1924, referring to the halachic question as to whether a fly landing in honey makes the honey treif; the palindromic Latin riddle "In girum imus nocte et consumimur igni" describes the behavior of moths. It is that this palindrome is from medieval rather than ancient times; the second word, borrowed from Greek, should properly be spelled gyrum. Byzantine Greeks inscribed the palindrome, "Wash sins, not only face" ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ ΜΗ ΜΟΝΑΝ ΟΨΙΝ, on baptismal fonts; this practice was continued in many English churches. Examples include the font at St. Mary's Church and the font in the basilica of St. Sophia, the font of St. Stephen d'Egres, Paris; some well-known English palindromes are, "Able was I ere I saw Elba", "A man, a plan, a canal – Panama", "Madam, I'm Adam" and "Never odd or even".
English palindromes of notable length include mathematician Peter Hilton's "Doc, note: I dissent. A fast never prevents a fatness. I diet on cod" and Scottish poet Alastair Reid's "T. Eliot, top bard, notes putrid tang emanating, is sad; the most familiar palindromes in English are character-unit palindromes. The characters read the same backward as forward; some examples of palindromic words are redivider, civic, level, kayak, racecar, redder and refer. There are 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. There are line-unit palindromes. Palindromes consist of a sentence or phrase, e.g. "Mr. Owl ate my metal worm", "Was it a car or a cat I saw?", "Murder for a jar of red rum" or "Go hang a salami, I'm a lasagna hog". Punctuation and spaces are ignored.
Some, such as "Rats live on no evil star", "Live on time, emit no evil", "Step on no pets", include the spaces. Semordnilap is a name coined for words; the word was coined by Martin Gardner in his notes to C. C. Bombaugh's book Oddities and Curiosities of Words and Literature in 1961. An example of this is the word stressed, desserts spelled backward; some semordnilaps are deliberate. An example in electronics is the mho, a unit of electrical conductance, ohm spelled backwards, the unit of electrical resistance and the reciprocal of conductance; the daraf, a unit of elastance, is farad spelled backwards, the unit of capacitance and the reciprocal of elastance. In fiction, many characters have names deliberately made to be semordnilaps of other names or words, the most used of, Alucard. Semordnilaps are known as emordnilaps, word reversals, reversible anagrams, heteropalindromes, semi-palindromes, half-palindromes, mynoretehs, volvograms, or anadromes, they have sometimes been called antigrams, though this term refers to anagrams which have opposite meanings.
In 2017, a six-year-old Canadian named Levi Budd called this a levidrome, which garnered support into making it a word from celebrities William Shatner and Patricia Arquette As of October 2018, none of these terms have been accepted as official entries in the Oxford English Dictionary. Some names are palindromes, such as the given names Hannah, Anna, Bob and Otto, or the surnames Harrah, Renner and Nenonen. Lon Nol was Prime Minister of Cambodia. Nisio Isin is a Japanese novelist and manga writer, whose pseudonym is a pal
One Thousand and One Nights
One Thousand and One Nights is a collection of Middle Eastern folk tales compiled in Arabic during the Islamic Golden Age. It is known in English as the Arabian Nights, from the first English-language edition, which rendered the title as The Arabian Nights' Entertainment; the work was collected over many centuries by various authors and scholars across West and South Asia and North Africa. Some tales themselves trace their roots back to ancient and medieval Arabic, Indian, Greek and Turkish folklore and literature. In particular, many tales were folk stories from the Abbasid and Mamluk eras, while others the frame story, are most drawn from the Pahlavi Persian work Hezār Afsān, which in turn relied on Indian elements. What is common throughout all the editions of the Nights is the initial frame story of the ruler Shahryār and his wife Scheherazade and the framing device incorporated throughout the tales themselves; the stories proceed from this original tale. Some editions contain only a few hundred nights.
The bulk of the text is in prose, although verse is used for songs and riddles and to express heightened emotion. Most of the poems are single quatrains, although some are longer; some of the stories associated with The Nights, in particular "Aladdin's Wonderful Lamp", "Ali Baba and the Forty Thieves", "The Seven Voyages of Sinbad the Sailor", were not part of The Nights in its original Arabic versions but were added to the collection by Antoine Galland and other European translators. The main frame story concerns Shahryār, whom the narrator calls a "Sasanian king" ruling in "India and China". Shahryār is shocked to learn. In his bitterness and grief, he decides. Shahryār begins to marry a succession of virgins only to execute each one the next morning, before she has a chance to dishonor him; the vizier, whose duty it is to provide them, cannot find any more virgins. Scheherazade, the vizier's daughter, offers herself as the next bride and her father reluctantly agrees. On the night of their marriage, Scheherazade does not end it.
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 another one, the king, eager to hear the conclusion of that tale as well, postpones her execution once again; this goes on for one one nights, hence the name. The tales vary widely: they include historical tales, love stories, comedies, poems and various forms of erotica. Numerous stories depict jinns, apes, sorcerers and legendary places, which are intermingled with real people and geography, not always rationally. Common protagonists include the historical Abbasid caliph Harun al-Rashid, his Grand Vizier, Jafar al-Barmaki, the famous poet Abu Nuwas, despite the fact that these figures lived some 200 years after the fall of the Sassanid Empire, in which the frame tale of Scheherazade is set. Sometimes a character in Scheherazade's tale will begin telling other characters a story of his own, that story may have another one told within it, resulting in a richly layered narrative texture.
The different versions have different individually detailed endings but they all end with the king giving his wife a pardon and sparing her life. The narrator's standards for what constitutes a cliffhanger seem broader than in modern literature. While in many cases a story is cut off with the hero in danger of losing his life or another kind of deep trouble, in some parts of the full text Scheherazade stops her narration in the middle of an exposition of abstract philosophical principles or complex points of Islamic philosophy, in one case during a detailed description of human anatomy according to Galen—and in all these cases turns out to be justified in her belief that the king's curiosity about the sequel would buy her another day of life; the history of the Nights is complex and modern scholars have made many attempts to untangle the story of how the collection as it exists came about. Robert Irwin summarises their findings: In the 1880s and 1890s a lot of work was done on the Nights by Zotenberg and others, in the course of which a consensus view of the history of the text emerged.
Most scholars agreed that the Nights was a composite work and that the earliest tales in it came from India and Persia. At some time in the early 8th century, these tales were translated into Arabic under the title Alf Layla, or'The Thousand Nights'; this collection formed the basis of The Thousand and One Nights. The original core of stories was quite small. 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. From the 10th century onwards independent sagas and story cycles were added to the compilation Then, from the 13th century onwards, a further layer of stories was add