1.
Geheel getal
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

2.
Factorisatie
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors

3.
Decimaal talstelsel
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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

4.
Kwadraatvrij geheel getal
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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

5.
Priemgetal
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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

6.
121 (getal)
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121 is the natural number following 120 and preceding 122. One hundred twenty-one is a square and is the sum of three consecutive primes, there are no squares besides 121 known to be of the form 1 + p + p 2 + p 3 + p 4, where p is prime. Other such squares must have at least 35 digits, there are only two other squares known to be of the form n. Another example of 121 being of the few examples supporting a conjecture is that Fermat conjectured that 4 and 121 are the perfect squares of the form x3 -4. It is also a number and a centered octagonal number. In base 10, it is a Smith number since its digits add up to the value as its factorization. But it can not be expressed as the sum of any other number plus that numbers digits, making 121 a self number

7.
666 (getal)
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666 is the natural number following 665 and preceding 667. Six hundred and sixty-six is called the number of the Beast in chapter 13 of the Book of Revelation, of the New Testament,666 is the sum of the first 36 natural numbers, and thus it is a triangular number. Notice that 36 =15 +21,15 and 21 are also triangular numbers, in base 10,666 is a repdigit and a Smith number. A prime reciprocal magic square based on 1/149 in base 10 has a total of 666. The prime factorization of 666 is 2 •32 •37, some manuscripts of the original Greek use the symbols χξϛ chi xi stigma, while other manuscripts spell out the number in words. In modern popular culture,666 has become one of the most widely recognized symbols for the Antichrist or, alternatively, the number 666 is purportedly used to invoke Satan. Earnest references to the number occur both among apocalypticist Christian groups and in explicitly anti-Christian subcultures, references in contemporary Western art or literature are, more likely than not, intentional references to the Beast symbolism. Such popular references are therefore too numerous to list and it is common to see the symbolic role of the integer 666 transferred to the digit sequence 6-6-6. Some people take the Satanic associations of 666 so seriously that they actively avoid things related to 666 or the digits 6-6-6, in some early biblical manuscripts, including Papyrus 115, the number is cited as 616. In the Bible,666 is the number of talents of gold Solomon collected each year, in the Bible,666 is the number of Adonikams descendants who return to Jerusalem and Judah from the Babylonian exile. In the Bible, there may be a latent reference to 666 in the name of the great sixth-century BC king of Babylon, commonly spelled Nebuchadnezzar, transliterating from the Book of Daniel, the name is Nebuchadrezzar or Nebuchadrezzur in the Book of Jeremiah. The number of name can be calculated, since Hebrew letters double as numbers. Nebuchadrezzar is 663, and Nebuchadrezzur,669, midway between the two variants is 666. If the mysteries of Jeremiah are to be related to those of Revelation, Nebuchadrezzar, using gematria, Neron Caesar transliterated from Greek into Hebrew produces the number 666. The Latin spelling of Nero Caesar transliterated into Hebrew produces the number 616, thus, in the Bible,666 may have been a coded reference to Nero the Roman Emperor from 55 to 68 AD. Is the magic sum, or sum of the constants of a six by six magic square. Is the sum of all the numbers on a roulette wheel, was a winning lottery number in the 1980 Pennsylvania Lottery scandal, in which equipment was tampered to favor a 4 or 6 as each of the three individual random digits. Was the original name of the Macintosh SevenDust computer virus that was discovered in 1998, the number is a frequent visual element of Aryan Brotherhood tattoos

8.
Palindroom
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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

9.
Martin Gardner
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He was considered a leading authority on Lewis Carroll. The Annotated Alice, which incorporated the text of Carrolls two Alice books, was his most successful work and sold over a million copies and he had a lifelong interest in magic and illusion and was regarded as one of the most important magicians of the twentieth century. He was considered the dean of American puzzlers and he was a prolific and versatile author, publishing more than 100 books. Gardner was one of the foremost anti-pseudoscience polemicists of the 20th century and his book Fads and Fallacies in the Name of Science, published in 1957, became a classic and seminal work of the skeptical movement. In 1976 he joined with fellow skeptics to found CSICOP, an organization promoting scientific inquiry, Gardner, son of a petroleum geologist, grew up in and around Tulsa, Oklahoma. His lifelong interest in puzzles started in his boyhood when his father gave him a copy of Sam Loyds Cyclopedia of 5000 Puzzles, Tricks and he attended the University of Chicago, where he earned his bachelors degree in philosophy in 1936. Early jobs included reporter on the Tulsa Tribune, writer at the University of Chicago Office of Press Relations, during World War II, he served for four years in the U. S. Navy as a yeoman on board the destroyer escort USS Pope in the Atlantic. His ship was still in the Atlantic when the war came to an end with the surrender of Japan in August 1945, after the war, Gardner returned to the University of Chicago. He attended graduate school for a year there, but he did not earn an advanced degree, in 1950 he wrote an article in the Antioch Review entitled The Hermit Scientist. His paper-folding puzzles at that magazine led to his first work at Scientific American, appropriately enough—given his interest in logic and mathematics—they lived on Euclid Avenue. The year 1960 saw the edition of his best-selling book ever. In 1979, Gardner retired from Scientific American and he and his wife Charlotte moved to Hendersonville and he also revised some of his older books such as Origami, Eleusis, and the Soma Cube. Charlotte died in 2000 and two years later Gardner returned to Norman, Oklahoma, where his son, James Gardner, was a professor of education at the University of Oklahoma and he died there on May 22,2010. An autobiography — Undiluted Hocus-Pocus, The Autobiography of Martin Gardner — was published posthumously, the main-belt asteroid 2587 Gardner discovered by Edward L. G. Bowell at Anderson Mesa Station in 1980 is named after Martin Gardner. Martin Gardner had a impact on mathematics in the second half of the 20th century. His column was called Mathematical Games but it was more than that. His writing introduced many readers to real mathematics for the first time in their lives, the column lasted for 25 years and was read avidly by the generation of mathematicians and physicists who grew up in the years 1956 to 1981. It was the inspiration for many of them to become mathematicians or scientists themselves