1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Integer
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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
3.
Square (algebra)
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In mathematics, a square is the result of multiplying a number by itself. The verb to square is used to denote this operation, squaring is the same as raising to the power 2, and is denoted by a superscript 2, for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the adjective which corresponds to squaring is quadratic. The square of an integer may also be called a number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, for instance, the square of the linear polynomial x +1 is the quadratic polynomial x2 + 2x +1. One of the important properties of squaring, for numbers as well as in other mathematical systems, is that. That is, the function satisfies the identity x2 =2. This can also be expressed by saying that the function is an even function. The squaring function preserves the order of numbers, larger numbers have larger squares. In other words, squaring is a function on the interval. Hence, zero is its global minimum, the only cases where the square x2 of a number is less than x occur when 0 < x <1, that is, when x belongs to an open interval. This implies that the square of an integer is never less than the original number, every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of one number, itself. For this reason, it is possible to define the square root function, no square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. There are several uses of the squaring function in geometry. The name of the squaring function shows its importance in the definition of the area, the area depends quadratically on the size, the area of a shape n times larger is n2 times greater. The squaring function is related to distance through the Pythagorean theorem and its generalization, Euclidean distance is not a smooth function, the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance, which has a paraboloid as its graph, is a smooth, the dot product of a Euclidean vector with itself is equal to the square of its length, v⋅v = v2
4.
Multiplication
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Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
5.
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
6.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
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Unit square
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In mathematics, a unit square is a square whose sides have length 1. Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points, and. In a Cartesian coordinate system with coordinates the unit square is defined as the square consisting of the points where x and y lie in a closed unit interval from 0 to 1. That is, the square is the Cartesian product I × I. The unit square can also be thought of as a subset of the complex plane, in this view, the four corners of the unit square are at the four complex numbers 0,1, i, and 1 + i. It is not known whether any point in the plane is a distance from all four vertices of the unit square. However, no point is on an edge of the square. Unit circle Unit sphere Unit cube Weisstein, Eric W. Unit square
8.
Sign (mathematics)
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In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero, along with its application to real numbers, change of sign is used throughout mathematics and physics to denote the additive inverse, even for quantities which are not real numbers. Also, the sign can indicate aspects of mathematical objects that resemble positivity and negativity. A real number is said to be if its value is greater than zero. The attribute of being positive or negative is called the sign of the number, zero itself is not considered to have a sign. Also, signs are not defined for complex numbers, although the argument generalizes it in some sense, in common numeral notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes positive three, and −3 denotes negative three, when no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of numbers through subtraction. In this context, it makes sense to write − = +3, any non-zero number can be changed to a positive one using the absolute value function. For example, the value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| =3 and |3| =3, the number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the inverse of itself. Note that this definition is culturally determined, in France and Belgium,0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be strictly positive resp, in some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers. One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits and this notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively, these behaviours are not necessarily the same. Because zero is positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number. A number is negative if it is less than zero, a number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero, thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero
9.
Square root
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In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss
10.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
11.
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
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.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
14.
Floor function
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In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. More precisely, floor = ⌊ x ⌋ is the greatest integer less than or equal to x, carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. This remained the standard in mathematics until Kenneth E. Iverson introduced the names floor and ceiling, both notations are now used in mathematics, this article follows Iverson. e. The value of x rounded to an integer towards 0, the language APL uses ⌊x, other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with boldface or double brackets, the ceiling function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL that is designed to use standard symbols, uses >. for ceiling. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\. x[, the fractional part is the sawtooth function, denoted by for real x and defined by the formula = x − ⌊ x ⌋. HTML4.0 uses the names, &lfloor, &rfloor, &lceil. Unicode contains codepoints for these symbols at U+2308–U+230B, ⌈x⌉, ⌊x⌋, in the following formulas, x and y are real numbers, k, m, and n are integers, and Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Since there is exactly one integer in an interval of length one. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and these formulas can be used to simplify expressions involving floors and ceilings. In the language of order theory, the function is a residuated mapping. These formulas show how adding integers to the arguments affect the functions, negating the argument complements the fractional part, + = {0 if x ∈ Z1 if x ∉ Z. The floor, ceiling, and fractional part functions are idempotent, the result of nested floor or ceiling functions is the innermost function, ⌊ ⌈ x ⌉ ⌋ = ⌈ x ⌉, ⌈ ⌊ x ⌋ ⌉ = ⌊ x ⌋. If m and n are integers and n ≠0,0 ≤ ≤1 −1 | n |. If n is a positive integer ⌊ x + m n ⌋ = ⌊ ⌊ x ⌋ + m n ⌋, ⌈ x + m n ⌉ = ⌈ ⌈ x ⌉ + m n ⌉. For m =2 these imply n = ⌊ n 2 ⌋ + ⌈ n 2 ⌉
15.
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
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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
17.
Recursion
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Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic, the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines a number of instances, it is often done in such a way that no loop or infinite chain of references can occur. The ancestors of ones ancestors are also ones ancestors, the Fibonacci sequence is a classic example of recursion, Fib =0 as base case 1, Fib =1 as base case 2, For all integers n >1, Fib, = Fib + Fib. Many mathematical axioms are based upon recursive rules, for example, the formal definition of the natural numbers by the Peano axioms can be described as,0 is a natural number, and each natural number has a successor, which is also a natural number. By this base case and recursive rule, one can generate the set of all natural numbers, recursively defined mathematical objects include functions, sets, and especially fractals. There are various more tongue-in-cheek definitions of recursion, see recursive humor, Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be recursive, to understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, the running of a procedure involves actually following the rules and performing the steps. An analogy, a procedure is like a recipe, running a procedure is like actually preparing the meal. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure. For instance, a recipe might refer to cooking vegetables, which is another procedure that in turn requires heating water, for this reason recursive definitions are very rare in everyday situations. An example could be the procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point, If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively, if every trial fails by reaching only dead ends, return on the path led to this branching point. Whether this actually defines a terminating procedure depends on the nature of the maze, in any case, executing the procedure requires carefully recording all currently explored branching points, and which of their branches have already been exhaustively tried. This can be understood in terms of a definition of a syntactic category. A sentence can have a structure in which what follows the verb is another sentence, Dorothy thinks witches are dangerous, so a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, and optionally another sentence
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Triangular number
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime
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Centered octagonal number
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A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. The centered octagonal numbers are the same as the odd square numbers, thus, the nth centered octagonal number is given by the formula 2 =4 n 2 −4 n +1. The first few centered octagonal numbers are 1,9,25,49,81,121,169,225,289,361,441,529,625,729,841,961,1089. Calculating Ramanujans tau function on an octagonal number yields an odd number
20.
Integer factorization
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In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these integers are further restricted to numbers, the process is called prime factorization. When the numbers are large, no efficient, non-quantum integer factorization algorithm is known. However, it has not been proven that no efficient algorithm exists, the presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such as RSA. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, not all numbers of a given length are equally hard to factor. The hardest instances of these problems are semiprimes, the product of two prime numbers, many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure, by the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. If the integer is then it can be recognized as such in polynomial time. If composite however, the theorem gives no insight into how to obtain the factors, given a general algorithm for integer factorization, any integer can be factored down to its constituent prime factors simply by repeated application of this algorithm. The situation is complicated with special-purpose factorization algorithms, whose benefits may not be realized as well or even at all with the factors produced during decomposition. For example, if N =10 × p × q where p < q are very large primes, trial division will quickly produce the factors 2 and 5 but will take p divisions to find the next factor. Among the b-bit numbers, the most difficult to factor in practice using existing algorithms are those that are products of two primes of similar size, for this reason, these are the integers used in cryptographic applications. The largest such semiprime yet factored was RSA-768, a 768-bit number with 232 decimal digits and this factorization was a collaboration of several research institutions, spanning two years and taking the equivalent of almost 2000 years of computing on a single-core 2.2 GHz AMD Opteron. Like all recent factorization records, this factorization was completed with an optimized implementation of the general number field sieve run on hundreds of machines. No algorithm has been published that can factor all integers in polynomial time, neither the existence nor non-existence of such algorithms has been proved, but it is generally suspected that they do not exist and hence that the problem is not in class P. The problem is clearly in class NP but has not been proved to be in, or not in and it is generally suspected not to be in NP-complete. There are published algorithms that are faster than O for all positive ε, i. e. sub-exponential, the best published asymptotic running time is for the general number field sieve algorithm, which, for a b-bit number n, is, O. For current computers, GNFS is the best published algorithm for large n, for a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time
21.
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
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Base 12
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
23.
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
24.
Modular arithmetic
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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7,00 now, then 8 hours later it will be 3,00. Usual addition would suggest that the time should be 7 +8 =15. Likewise, if the clock starts at 12,00 and 21 hours elapse, then the time will be 9,00 the next day, because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below,12 is congruent not only to 12 itself, Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers, addition, subtraction, and multiplication. For a positive n, two integers a and b are said to be congruent modulo n, written, a ≡ b. The number n is called the modulus of the congruence, for example,38 ≡14 because 38 −14 =24, which is a multiple of 12. The same rule holds for negative values, −8 ≡72 ≡ −3 −3 ≡ −8. Equivalently, a ≡ b mod n can also be thought of as asserting that the remainders of the division of both a and b by n are the same, for instance,38 ≡14 because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that 38 −14 =24 is a multiple of 12. A remark on the notation, Because it is common to consider several congruence relations for different moduli at the same time, in spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been if the notation a ≡n b had been used. The properties that make this relation a congruence relation are the following, if a 1 ≡ b 1 and a 2 ≡ b 2, then, a 1 + a 2 ≡ b 1 + b 2 a 1 − a 2 ≡ b 1 − b 2. The above two properties would still hold if the theory were expanded to all real numbers, that is if a1, a2, b1, b2. The next property, however, would fail if these variables were not all integers, the notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is referred to as the modulo operation. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12, as this remainder is 2, we have 14 mod 12 =2
25.
Canonical representation of a positive integer
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For example,1200 =24 ×31 ×52 =3 ×2 ×2 ×2 ×2 ×5 ×5 =5 ×2 ×3 ×2 ×5 ×2 ×2 = etc. The requirement that the factors be prime is necessary, factorizations containing composite numbers may not be unique. This theorem is one of the reasons why 1 is not considered a prime number, if 1 were prime. Book VII, propositions 30,31 and 32, and Book IX, proposition 14 of Euclids Elements are essentially the statement, proposition 30 is referred to as Euclids lemma. And it is the key in the proof of the theorem of arithmetic. Proposition 31 is proved directly by infinite descent, proposition 32 is derived from proposition 31, and prove that the decomposition is possible. Book IX, proposition 14 is derived from Book VII, proposition 30, indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss Disquisitiones Arithmeticae is a modern statement. < pk are primes and the αi are positive integers and this representation is commonly extended to all positive integers, including one, by the convention that the empty product is equal to 1. This representation is called the representation of n, or the standard form of n. For example 999 = 33×37,1000 = 23×53,1001 = 7×11×13 Note that factors p0 =1 may be inserted without changing the value of n, allowing negative exponents provides a canonical form for positive rational numbers. However, as Integer factorization of large integers is much harder than computing their product, gcd or lcm, these formulas have, in practice, many arithmetical functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers, the proof uses Euclids lemma, if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b. We need to show that every integer greater than 1 is either prime or a product of primes, for the base case, note that 2 is prime. By induction, assume true for all numbers between 1 and n, if n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n, by the induction hypothesis, a = p1p2. pj and b = q1q2. qk are products of primes. But then n = ab = p1p2. pjq1q2. qk is a product of primes, assume that s >1 is the product of prime numbers in two different ways, s = p 1 p 2 ⋯ p m = q 1 q 2 ⋯ q n. We must show m = n and that the qj are a rearrangement of the pi, by Euclids lemma, p1 must divide one of the qj, relabeling the qj if necessary, say that p1 divides q1
26.
Factorization
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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
27.
Difference of two squares
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In mathematics, the difference of two squares is a squared number subtracted from another squared number. Every difference of squares may be factored according to the identity a 2 − b 2 = in elementary algebra, the proof of the factorization identity is straightforward. Among many uses, it gives a proof of the AM–GM inequality in two variables. The difference of two squares can also be illustrated geometrically as the difference of two areas in a plane. In the diagram, the shaded part represents the difference between the areas of the two squares, i. e. a 2 − b 2. The area of the part can be found by adding the areas of the two rectangles, a + b, which can be factorized to. Therefore a 2 − b 2 = Another geometric proof proceeds as follows, We start with the shown in the first diagram below. The side of the square is a, and the side of the small removed square is b. The area of the region is a 2 − b 2. A cut is made, splitting the region into two pieces, as shown in the second diagram. The larger piece, at the top, has width a, the smaller piece, at the bottom, has width a-b and height b. Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, since this rectangle came from rearranging the original figure, it must have the same area as the original figure. The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. Since the two factors found by this method are complex conjugates, we can use this in reverse as a method of multiplying a number to get a real number. This is used to get real denominators in complex fractions, the difference of two squares can also be used in the rationalising of irrational denominators. This is a method for removing surds from expressions, applying to division by some combinations involving square roots, here, the irrational denominator 3 +4 has been rationalised to 13. The difference of two squares can also be used as a short cut
28.
Perfect number
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In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Equivalently, a number is a number that is half the sum of all of its positive divisors i. e. σ1 = 2n. This definition is ancient, appearing as early as Euclids Elements where it is called τέλειος ἀριθμός. Euclid also proved a formation rule whereby q /2 is a perfect number whenever q is a prime of the form 2 p −1 for prime p —what is now called a Mersenne prime. Much later, Euler proved that all even numbers are of this form. This is known as the Euclid–Euler theorem and it is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first perfect number is 6 and its proper divisors are 1,2, and 3, and 1 +2 +3 =6. Equivalently, the number 6 is equal to half the sum of all its positive divisors, the next perfect number is 28 =1 +2 +4 +7 +14. This is followed by the perfect numbers 496 and 8128, in about 300 BC Euclid showed that if 2p−1 is prime then 2p−1 is perfect. The first four numbers were the only ones known to early Greek mathematics. Philo of Alexandria in his first-century book On the creation mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen, and by Didymus the Blind, st Augustine defines perfect numbers in City of God in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician Ismail ibn Fallūs mentioned the next three numbers and listed a few more which are now known to be incorrect. Euclid proved that 2p−1 is a perfect number whenever 2p −1 is prime. Prime numbers of the form 2p −1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, for 2p −1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p −1 with a prime p are prime, in fact, Mersenne primes are very rare—of the 9,592 prime numbers p less than 100,000, 2p −1 is prime for only 28 of them. Nicomachus conjectured that every number is of the form 2p−1 where 2p −1 is prime. Ibn al-Haytham circa 1000 AD conjectured that every perfect number is of that form
29.
Square pyramidal number
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In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid. The first few square pyramidal numbers are,1,5,14,30,55,91,140,204,285,385,506,650,819 and this is a special case of Faulhabers formula, and may be proved by a mathematical induction. An equivalent formula is given in Fibonaccis Liber Abaci, in modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial L of a polyhedron P is a polynomial that counts the number of points in a copy of P that is expanded by multiplying all its coordinates by the number t. The Ehrhart polynomial of a pyramid base is a unit square with integer coordinates. The square pyramidal numbers can also be expressed as sums of binomial coefficients, the smaller tetrahedral number represents 1 +3 +6 + ⋯ + T and the larger 1 +3 +6 + ⋯ + T. Offsetting the larger and adding, we arrive at 1,1 +3,3 +6 …, Square pyramidal numbers are also related to tetrahedral numbers in a different way, P n =14. The sum of two square pyramidal numbers is an octahedral number. Augmenting a pyramid whose base edge has n balls by adding to one of its faces a tetrahedron whose base edge has n −1 balls produces a triangular prism. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism and this geometric dissection leads to another relation, P n = n −. Besides 1, there is one other number that is both a square and a pyramid number,4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918, in the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes. Also, P n = − which is the difference of two pentatope numbers and this can be seen by expanding, n − n = n = n and dividing through by 24. A common mathematical puzzle involves finding the number of squares in a n by n square grid. This number can be derived as follows, The number of 1 ×1 boxes found in the grid is n2, the number of 2 ×2 boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of 2 ×2 boxes, the number of k × k boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of k × k boxes and it follows that the number of squares in an n × n square grid is, n 2 +2 +2 +2 + … +12 = n 6
30.
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
31.
Methods of computing square roots
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In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a non-negative real number. For the square roots of a negative or complex number, see below, finding S is the same as solving the equation f = x 2 − S =0 for a positive x. Therefore, any general numerical root-finding algorithm can be used, many square root algorithms require an initial seed value. If the initial seed value is far away from the square root. It is therefore useful to have an estimate, which may be very inaccurate. For S =125348 =12.5348 ×104, for S =125348 =111101001101001002 =1.11101001101001002 ×216 the binary approximation gives S ≈28 =1000000002 =256. These approximations are useful to find better seeds for iterative algorithms and it can be derived from Newtons method. The process of updating is iterated until desired accuracy is obtained and this is a quadratically convergent algorithm, which means that the number of correct digits of the approximation roughly doubles with each iteration. It proceeds as follows, Begin with a positive starting value x0. Let xn +1 be the average of xn and S/xn, repeat step 2 until the desired accuracy is achieved. It can also be represented as, x 0 ≈ S, x n +1 =12, S = lim n → ∞ x n. Let the relative error in xn be defined by ε n = x n S −1 and thus x n = S ⋅. Then it can be shown that ε n +1 = ε n 22 and thus that 0 ≤ ε n +2 ≤ min and consequently that convergence is assured provided that x0 and S are both positive. If using the rough estimate above with the Babylonian method, then the least accurate cases in ascending order are as follows, S =1, x 0 =2, x 1 =1.250, ε1 =0.250. S =10, x 0 =2, x 1 =3.500, S =10, x 0 =6, x 1 =3.833, ε1 <0.213. S =100, x 0 =6, x 1 =11.333, thus in any case, ε1 ≤2 −2. ε8 <2 −383 <10 −115, rounding errors will slow the convergence. It is recommended to keep at least one extra digit beyond the desired accuracy of the xn being calculated to minimize round off error and this is a method to find each digit of the square root in a sequence
32.
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
33.
Pythagorean triple
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A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written, and an example is. If is a Pythagorean triple, then so is for any integer k. A primitive Pythagorean triple is one in which a, b and c are coprime, a right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. However, right triangles with non-integer sides do not form Pythagorean triples, for instance, the triangle with sides a = b =1 and c = √2 is right, but is not a Pythagorean triple because √2 is not an integer. Moreover,1 and √2 do not have a common multiple because √2 is irrational. There are 16 primitive Pythagorean triples with c ≤100, Note, for example, each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot. The formula states that the integers a = m 2 − n 2, b =2 m n, c = m 2 + n 2 form a Pythagorean triple. The triple generated by Euclids formula is primitive if and only if m and n are coprime, every primitive triple arises from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples and this relationship of a, b and c to m and n from Euclids formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclids formula does not produce all triples—for example and this can be remedied by inserting an additional parameter k to the formula. That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra, many formulas for generating triples with particular properties have been developed since the time of Euclid. A proof of the necessity that a, b, c be expressed by Euclids formula for any primitive Pythagorean triple is as follows, all such triples can be written as where a2 + b2 = c2 and a, b, c are coprime. Thus a, b, c are pairwise coprime, as a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd, from a 2 + b 2 = c 2 we obtain c 2 − a 2 = b 2 and hence = b 2. Since b is rational, we set it equal to m n in lowest terms, thus b = n m, as being the reciprocal of b. As m n is fully reduced, m and n are coprime, and they cannot be both even. If they were odd, the numerator of m 2 − n 22 m n would be a multiple of 4
34.
Quadratic function
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A univariate quadratic function has the form f = a x 2 + b x + c, a ≠0 in the single variable x. The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis. If the quadratic function is set equal to zero, then the result is a quadratic equation, the solutions to the univariate equation are called the roots of the univariate function. In general there can be a large number of variables, in which case the resulting surface is called a quadric. The adjective quadratic comes from the Latin word quadrātum, a term like x2 is called a square in algebra because it is the area of a square with side x. In general, a prefix indicates the number 4. Quadratum is the Latin word for square, the coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring. When using the quadratic polynomial, authors sometimes mean having degree exactly 2. If the degree is less than 2, this may be called a degenerate case, usually the context will establish which of the two is meant. Sometimes the word order is used with the meaning of degree, a quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z. Any single-variable quadratic polynomial may be written as a x 2 + b x + c, where x is the variable, and a, b, and c represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation a x 2 + b x + c =0, each quadratic polynomial has an associated quadratic function, whose graph is a parabola. Such polynomials are fundamental to the study of sections, which are characterized by equating the expression for f to zero. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces, in linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space. F = a 2 + k is called the vertex form, the coefficient a is the same value in all three forms. To convert the standard form to factored form, one only the quadratic formula to determine the two roots r1 and r2. To convert the standard form to form, one needs a process called completing the square. To convert the factored form to form, one needs to multiply
35.
Square triangular number
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In mathematics, a square triangular number is a number which is both a triangular number and a perfect square. There are infinitely many triangular numbers, the first few are 0,1,36,1225,41616,1413721,48024900,1631432881,55420693056,1882672131025. Write Nk for the kth square triangular number, and write sk and tk for the sides of the square and triangle. Define the triangular root of a triangular number N = n 2 to be n, from this definition and the quadratic formula, n =8 N +1 −12. Therefore, N is triangular if and only if 8 N +1 is square, consequently, a number M2 is square and triangular if and only if 8 M2 +1 is square, i. e. There are numbers x and y such that x 2 −8 y 2 =1 and this is an instance of the Pell equation, with n =8. All Pell equations have the solution, for any n, this solution is called the zeroth. Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, the first non-trivial solution when n=8 is easy to find, it is. A solution to the Pell equation for n=8 yields a square number and its square and triangular roots as follows, s k = y k, t k = x k −12. Hence, the first square triangular number, derived from, is 1, the sequences Nk, sk and tk are the OEIS sequences A001110, A001109, and A001108 respectively. In 1778 Leonhard Euler determined the explicit formula N k =2, other equivalent formulas that may be convenient include N k =1322 =132 =132. The corresponding explicit formulas for sk and tk are s k = k − k 42 and t k = k + k −24, the problem of finding square triangular numbers reduces to Pells equation in the following way. Every triangular number is of the form t/2, therefore we seek integers t, s such that t 2 = s 2. There are many identities about the Pell numbers, and these translate into identities about the triangular numbers. There are recurrence relations for the triangular numbers, as well as for the sides of the square. We have N k =34 N k −1 − N k −2 +2, N k =2, with N0 =0 and N1 =1. All square triangular numbers have the form b2c2, where b / c is a convergent to the continued fraction for the root of 2. A. V. Sylwester gave a proof that there are an infinity of square triangular numbers, to wit, If the triangular number n/2 is square
36.
Neil Sloane
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Neil James Alexander Sloane is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, Sloane is best known for being the creator and maintainer of the On-Line Encyclopedia of Integer Sequences. Sloane was born in Wales and brought up in Australia and he studied at Cornell University, New York state, under Nick DeClaris, Frank Rosenblatt, Frederick Jelinek and Wolfgang Heinrich Johannes Fuchs, receiving his Ph. D. in 1967. His doctoral dissertation was titled Lengths of Cycle Times in Random Neural Networks, Sloane joined AT&T Bell Labs in 1968 and retired from AT&T Labs in 2012. He became an AT&T Fellow in 1998 and he is also a Fellow of the Learned Society of Wales, an IEEE Fellow, a Fellow of the American Mathematical Society, and a member of the National Academy of Engineering. He is a winner of a Lester R. Ford Award in 1978, in 2005 Sloane received the IEEE Richard W. Hamming Medal. In 2008 he received the Mathematical Association of America David P. Robbins award, in 2014, to celebrate his 75th birthday, Neil Sloane shared some of his favorite integer sequences. Besides mathematics, he loves rock climbing and has authored two rock-climbing guides to New Jersey, N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, NY,1973. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North-Holland, M. Harwit and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, San Diego CA,1979. N. J. A. Sloane and A. D. Wyner, editors, Claude Elwood Shannon, Collected Papers, IEEE Press, N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego,1995. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 1st edn. A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Theory and Applications, Springer-Verlag, NY,1999. G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer-Verlag,2006
37.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
38.
Richard K. Guy
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Richard Kenneth Guy is a British mathematician, professor emeritus in the Department of Mathematics at the University of Calgary. He is known for his work in theory, geometry, recreational mathematics, combinatorics. He is best known for co-authorship of Winning Ways for your Mathematical Plays and he has also published over 300 papers. For this paper he received the MAA Lester R. Ford Award, Guy was born 30 Sept 1916 in Nuneaton, Warwickshire, England, to Adeline Augusta Tanner and William Alexander Charles Guy. Both of his parents were teachers, rising to the rank of headmistress and headmaster and he attended Warwick School for Boys, the third oldest school in Britain, but was not enthusiastic about most of the curriculum. He was good at sports, however, and excelled in mathematics, at the age of 17 he read Dicksons History of the Theory of Numbers. He said it was better than the works of Shakespeare. By then he had developed a passion for mountain climbing. In 1935 Guy entered Gonville and Caius College, at the University of Cambridge as a result of winning several scholarships, to win the most important of these he had to travel to Cambridge and write exams for two days. His interest in games began while at Cambridge where he became a composer of chess problems. In 1938, he graduated with an honours degree, he himself thinks that his failure to get a first may have been related to his obsession with chess. Although his parents advised against it, Guy decided to become a teacher. He met his future wife Nancy Louise Thirian through her brother Michael who was a fellow scholarship winner at Gonville and he and Louise shared loves of mountains and dancing. He wooed her through correspondence, and they married in December 1940, in November 1942, Guy received an emergency commission in the Meteorological Branch of the Royal Air Force, with the rank of flight lieutenant. He was posted to Reykjavik, and later to Bermuda, as a meteorologist and he tried to get permission for Louise to join him but was refused. While in Iceland, he did some glacier travel, skiing and mountain climbing, marking the beginning of another love affair. When Guy returned to England after the war, he went back to teaching, this time at Stockport Grammar School, in 1947 the family moved to London, where he got a job teaching math at Goldsmiths College. In 1951 he moved to Singapore, where he taught at the University of Malaya for the next decade and he then spent a few years at the Indian Institute of Technology in Delhi, India