1.
Integer factorization
–
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
Integer factorization
–
This image demonstrates the prime decomposition of 864. A shorthand way of writing the resulting prime factors is 2 5 × 3 3
2.
Greatest common divisor
–
In mathematics, the greatest common divisor of two or more integers, when at least one of them is not zero, is the largest positive integer that is a divisor of both numbers. For example, the GCD of 8 and 12 is 4, the greatest common divisor is also known as the greatest common factor, highest common factor, greatest common measure, or highest common divisor. This notion can be extended to polynomials and other commutative rings, in this article we will denote the greatest common divisor of two integers a and b as gcd. What is the greatest common divisor of 54 and 24, the number 54 can be expressed as a product of two integers in several different ways,54 ×1 =27 ×2 =18 ×3 =9 ×6. Thus the divisors of 54 are,1,2,3,6,9,18,27,54, similarly, the divisors of 24 are,1,2,3,4,6,8,12,24. The numbers that these two share in common are the common divisors of 54 and 24,1,2,3,6. The greatest of these is 6 and that is, the greatest common divisor of 54 and 24. The greatest common divisor is useful for reducing fractions to be in lowest terms, for example, gcd =14, therefore,4256 =3 ⋅144 ⋅14 =34. Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1, for example,9 and 28 are relatively prime. For example, a 24-by-60 rectangular area can be divided into a grid of, 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, therefore,12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, in practice, this method is only feasible for small numbers, computing prime factorizations in general takes far too long. Here is another example, illustrated by a Venn diagram. Suppose it is desired to find the greatest common divisor of 48 and 180, first, find the prime factorizations of the two numbers,48 =2 ×2 ×2 ×2 ×3,180 =2 ×2 ×3 ×3 ×5. What they share in common is two 2s and a 3, Least common multiple =2 ×2 × ×3 ×5 =720 Greatest common divisor =2 ×2 ×3 =12. To compute gcd, divide 48 by 18 to get a quotient of 2, then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as, gcd = a gcd = gcd, in this sense the GCD problem is analogous to e. g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete. Shallcross et al. showed that a problem is NC-equivalent to the problem of integer linear programming with two variables, if either problem is in NC or is P-complete, the other is as well
Greatest common divisor
–
Contents
3.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
4.
American and British English spelling differences
–
Many of the differences between American and British English date back to a time when spelling standards had not yet developed. For instance, some spellings seen as American today were once used in Britain. But English-language spelling reform has rarely been adopted otherwise, and thus modern English orthography varies somewhat between countries and is far from phonemic in any country, in the early 18th century, English spelling was inconsistent. These differences became noticeable after the publishing of influential dictionaries, todays British English spellings mostly follow Johnsons A Dictionary of the English Language, while many American English spellings follow Websters An American Dictionary of the English Language. Webster was a proponent of English spelling reform for reasons both philological and nationalistic, in A Companion to the American Revolution, John Algeo notes, it is often assumed that characteristically American spellings were invented by Noah Webster. He was very influential in popularizing certain spellings in America, rather he chose already existing options such as center, color and check for the simplicity, analogy or etymology. William Shakespeares first folios, for example, used spellings like center and color as much as centre, Webster did attempt to introduce some reformed spellings, as did the Simplified Spelling Board in the early 20th century, but most were not adopted. In Britain, the influence of those who preferred the Norman spellings of words proved to be decisive, later spelling adjustments in the United Kingdom had little effect on todays American spellings and vice versa. For the most part, the systems of most Commonwealth countries. Australian spelling has also strayed slightly from British spelling, with some American spellings incorporated as standard, New Zealand spelling is almost identical to British spelling, except in the word fiord. There is also an increasing use of macrons in words that originated in Māori, most words ending in an unstressed -our in British English end in -or in American English. Wherever the vowel is unreduced in pronunciation, e. g. contour, velour, paramour and troubadour the spelling is the same everywhere, most words of this kind came from Latin, where the ending was spelled -or. They were first adopted into English from early Old French, after the Norman conquest of England, the ending became -our to match the Old French spelling. The -our ending was not only used in new English borrowings, however, -or was still sometimes found, and the first three folios of Shakespeares plays used both spellings before they were standardised to -our in the Fourth Folio of 1685. After the Renaissance, new borrowings from Latin were taken up with their original -or ending, Websters 1828 dictionary had only -or and is given much of the credit for the adoption of this form in the United States. Johnson, unlike Webster, was not an advocate of spelling reform and he preferred French over Latin spellings because, as he put it, the French generally supplied us. In Jeffersons original draft it is spelled honour, Honor and honour were equally frequent in Britain until the 17th century, honor still is, in the UK, the usual spelling as a persons name and appears in Honor Oak, a district of London. In derivatives and inflected forms of the words, British usage depends on the nature of the suffix used
American and British English spelling differences
–
Extract from the Orthography section of the first edition (1828) of
Webster 's "
ADEL ", which popularized the "American standard" spellings of -er (6); -or (7); the dropped -e (8); -or (10); -se (11); and the doubling of consonants with a suffix (15).
American and British English spelling differences
–
An 1814 American medical text showing British English spellings that were still in use ("tumours", "colour", "centres", etc.).
5.
Number
–
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not
Number
–
The number 605 in
Khmer numerals, from an inscription from 683 AD. An early use of zero as a decimal figure.
Number
–
Subsets of the
complex numbers.
6.
Polynomial
–
In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
Polynomial
–
The
graph of a polynomial function of degree 3
7.
Matrix (mathematics)
–
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 ×3, the individual items in an m × n matrix A, often denoted by ai, j, where max i = m and max j = n, are called its elements or entries. Provided that they have the size, two matrices can be added or subtracted element by element. The rule for multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field, a major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f = 4x. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations, if the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a transformation is obtainable from the matrixs eigenvalues. Applications of matrices are found in most scientific fields, in computer graphics, they are used to manipulate 3D models and project them onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions, Matrices are used in economics to describe systems of economic relationships. A major branch of analysis is devoted to the development of efficient algorithms for matrix computations. Matrix decomposition methods simplify computations, both theoretically and practically, algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory, a simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. A matrix is an array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is an array of scalars each of which is a member of F. Most of this focuses on real and complex matrices, that is, matrices whose elements are real numbers or complex numbers. More general types of entries are discussed below, for instance, this is a real matrix, A =
Matrix (mathematics)
–
Each element of a matrix is often denoted by a variable with two
subscripts. For instance, a 2,1 represents the element at the second row and first column of a matrix A.
8.
Product (mathematics)
–
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance,6 is the product of 2 and 3, the order in which real or complex numbers are multiplied has no bearing on the product, this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, matrix multiplication, for example, and multiplication in other algebras is in general non-commutative. There are many different kinds of products in mathematics, besides being able to multiply just numbers, polynomials or matricies, an overview of these different kinds of products is given here. Placing several stones into a pattern with r rows and s columns gives r ⋅ s = ∑ i =1 s r = ∑ j =1 r s stones. Integers allow positive and negative numbers, the product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, the product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏. The product of a sequence consisting of one number is just that number itself. The product of no factors at all is known as the empty product, commutative rings have a product operation. Under the Fourier transform, convolution becomes point-wise function multiplication, others have very different names but convey essentially the same idea. A brief overview of these is given here, by the very definition of a vector space, one can form the product of any scalar with any vector, giving a map R × V → V. A scalar product is a map, ⋅, V × V → R with the following conditions. From the scalar product, one can define a norm by letting ∥ v ∥, = v ⋅ v, now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping f map V to W, and let the linear mapping g map W to U, then one can get g ∘ f = g = g j k f i j v i b U k. Or in matrix form, g ∘ f = G F v, in which the i-row, j-column element of F, denoted by Fij, is fji, the composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication. To see this, let r = dim, s = dim, let U = be a basis of U, V = be a basis of V and W = be a basis of W. Then B ⋅ A = M W U ∈ R s × t is the matrix representing g ∘ f, U → W, in other words, the matrix product is the description in coordinates of the composition of linear functions. For inifinite-dimensional vector spaces, one also has the, Tensor product of Hilbert spaces Topological tensor product, the tensor product, outer product and Kronecker product all convey the same general idea
Product (mathematics)
–
3 by 4 is 12
9.
Multiplication
–
Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
Multiplication
–
4 × 5 = 20, the rectangle is composed of 20 squares, having dimensions of 4 by 5.
Multiplication
–
Four bags of three
marbles gives twelve marbles (4 × 3 = 12).
10.
Prime number
–
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
Prime number
–
The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
11.
Fundamental theorem of arithmetic
–
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
Fundamental theorem of arithmetic
–
The unique factorization theorem was proved by
Gauss with his 1801 book
Disquisitiones Arithmeticae. In this book, Gauss used the fundamental theorem for proving the
law of quadratic reciprocity.
Fundamental theorem of arithmetic
–
Overview
12.
Zero of a function
–
In other words, a zero of a function is an input value that produces an output of zero. A root of a polynomial is a zero of the polynomial function. If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis, an alternative name for such a point in this context is an x-intercept. Every equation in the unknown x may be rewritten as f =0 by regrouping all terms in the left-hand side and it follows that the solutions of such an equation are exactly the zeros of the function f. Every real polynomial of odd degree has an odd number of roots, likewise. Consequently, real odd polynomials must have at least one real root, the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs, vietas formulas relate the coefficients of a polynomial to sums and products of its roots. Computing roots of functions, for polynomial functions, frequently requires the use of specialised or approximation techniques. However, some functions, including all those of degree no greater than 4. In topology and other areas of mathematics, the set of a real-valued function f, X → R is the subset f −1 of X. Zero sets are important in many areas of mathematics. One area of importance is algebraic geometry, where the first definition of an algebraic variety is through zero-sets. For instance, for each set S of polynomials in k, one defines the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. Then a subset V of An is called an algebraic set if V = Z for some S. These affine algebraic sets are the building blocks of algebraic geometry. Zero Pole Fundamental theorem of algebra Newtons method Sendovs conjecture Mardens theorem Vanish at infinity Zero crossing Weisstein, Eric W. Root
Zero of a function
–
A graph of the function cos(x) on the domain, with x -intercepts indicated in red. The function has zeroes where x is,, and.
13.
Polynomial expansion
–
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. During the expansion, simplifications such as grouping of like terms or cancellations of terms may also be applied and it is customary to reintroduce powers in the final result when terms involve products of identical symbols. Simple examples of polynomial expansions are the known rules 2 = x 2 +2 x y + y 2 = x 2 − y 2 when used from left to right. The opposite process of trying to write an expanded polynomial as a product is called polynomial factorization, to multiply two factors, each term of the first factor must be multiplied by each term of the other factor. If both factors are binomials, the FOIL rule can be used, which stands for First Outer Inner Last, for example, expanding yields 2 x 2 −5 x +4 x −10 =2 x 2 − x −10. When expanding n, a relationship exists between the coefficients of the terms when written in order of descending powers of x and ascending powers of y. The coefficients will be the numbers in the th row of Pascals triangle. com Online Calculator with Symbolic Calculations, livephysics. com
Polynomial expansion
–
Two expressions can be multiplied by using the commutative law, associative law and distributive law. (To multiply more than 2 expressions, just multiply 2 at a time)
14.
Public key cryptography
–
In a public key encryption system, any person can encrypt a message using the public key of the receiver, but such a message can be decrypted only with the receivers private key. For this to work it must be easy for a user to generate a public. The strength of a public key cryptography system relies on the degree of difficulty for a properly generated private key to be determined from its public key. Security then depends only on keeping the key private. Public key algorithms, unlike symmetric key algorithms, do not require a secure channel for the exchange of one secret keys between the parties. Because of the complexity of asymmetric encryption, it is usually used only for small blocks of data. This symmetric key is used to encrypt the rest of the potentially long message sequence. The symmetric encryption/decryption is based on algorithms and is much faster. In a public key system, a person can combine a message with a private key to create a short digital signature on the message. Thus the authenticity of a message can be demonstrated by the signature, Public key algorithms are fundamental security ingredients in cryptosystems, applications and protocols. They underpin various Internet standards, such as Transport Layer Security, S/MIME, PGP, some public key algorithms provide key distribution and secrecy, some provide digital signatures, and some provide both. Public key cryptography finds application in, among others, the information technology security discipline, information security is concerned with all aspects of protecting electronic information assets against security threats. Public key cryptography is used as a method of assuring the confidentiality, authenticity and non-repudiability of electronic communications, two of the best-known uses of public key cryptography are, Public key encryption, in which a message is encrypted with a recipients public key. The message cannot be decrypted by anyone who does not possess the matching private key, who is presumed to be the owner of that key. This is used in an attempt to ensure confidentiality, digital signatures, in which a message is signed with the senders private key and can be verified by anyone who has access to the senders public key. This verification proves that the sender had access to the private key, an analogy to public key encryption is that of a locked mail box with a mail slot. The mail slot is exposed and accessible to the public – its location is, in essence, anyone knowing the street address can go to the door and drop a written message through the slot. However, only the person who possesses the key can open the mailbox, an analogy for digital signatures is the sealing of an envelope with a personal wax seal
Public key cryptography
–
An unpredictable (typically large and
random) number is used to begin generation of an acceptable pair of
keys suitable for use by an asymmetric key algorithm.
15.
RSA (algorithm)
–
RSA is one of the first practical public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the key is public and differs from the decryption key which is kept secret. In RSA, this asymmetry is based on the difficulty of factoring the product of two large prime numbers, the factoring problem. RSA is made of the letters of the surnames of Ron Rivest, Adi Shamir, and Leonard Adleman. Clifford Cocks, an English mathematician working for the UK intelligence agency GCHQ, had developed an equivalent system in 1973, a user of RSA creates and then publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers must be kept secret, breaking RSA encryption is known as the RSA problem, whether it is as hard as the factoring problem remains an open question. RSA is a relatively slow algorithm, and because of this it is commonly used to directly encrypt user data. More often, RSA passes encrypted shared keys for symmetric key cryptography which in turn can perform bulk encryption-decryption operations at higher speed. The idea of an asymmetric public-private key cryptosystem is attributed to Whitfield Diffie and Martin Hellman and they also introduced digital signatures and attempted to apply number theory, their formulation used a shared secret key created from exponentiation of some number, modulo a prime numbers. However, they open the problem of realizing a one-way function. Ron Rivest, Adi Shamir, and Leonard Adleman at MIT made several attempts over the course of a year to create a function that is hard to invert. Rivest and Shamir, as scientists, proposed many potential functions while Adleman. They tried many approaches including knapsack-based and permutation polynomials, for a time they thought it was impossible for what they wanted to achieve due to contradictory requirements. In April 1977, they spent Passover at the house of a student, Rivest, unable to sleep, lay on the couch with a math textbook and started thinking about their one-way function. He spent the rest of the night formalizing his idea and had much of the paper ready by daybreak, the algorithm is now known as RSA – the initials of their surnames in same order as their paper. Clifford Cocks, an English mathematician working for the UK intelligence agency GCHQ, however, given the relatively expensive computers needed to implement it at the time, it was mostly considered a curiosity and, as far as is publicly known, was never deployed. His discovery, however, was not revealed until 1997 due to its secret classification, Kid-RSA is a simplified public-key cipher published in 1997, designed for educational purposes. Some people feel that learning Kid-RSA gives insight into RSA and other public-key ciphers, Patent 4,405,829 for a Cryptographic communications system and method that used the algorithm, on September 20,1983
RSA (algorithm)
–
Adi Shamir, one of the authors of RSA:
Rivest,
Shamir and
Adleman
16.
Matrix (math)
–
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 ×3, the individual items in an m × n matrix A, often denoted by ai, j, where max i = m and max j = n, are called its elements or entries. Provided that they have the size, two matrices can be added or subtracted element by element. The rule for multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field, a major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f = 4x. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations, if the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a transformation is obtainable from the matrixs eigenvalues. Applications of matrices are found in most scientific fields, in computer graphics, they are used to manipulate 3D models and project them onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions, Matrices are used in economics to describe systems of economic relationships. A major branch of analysis is devoted to the development of efficient algorithms for matrix computations. Matrix decomposition methods simplify computations, both theoretically and practically, algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory, a simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. A matrix is an array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is an array of scalars each of which is a member of F. Most of this focuses on real and complex matrices, that is, matrices whose elements are real numbers or complex numbers. More general types of entries are discussed below, for instance, this is a real matrix, A =
Matrix (math)
–
Each element of a matrix is often denoted by a variable with two
subscripts. For instance, a 2,1 represents the element at the second row and first column of a matrix A.
17.
Triangular matrix
–
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the diagonal are zero. Similarly, a matrix is called upper triangular if all the entries below the main diagonal are zero. A triangular matrix is one that is lower triangular or upper triangular. A matrix that is both upper and lower triangular is called a diagonal matrix, because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. A matrix of the form L = is called a triangular matrix or left triangular matrix. The variable L is commonly used to represent a lower triangular matrix, a matrix that is both upper and lower triangular is diagonal. Matrices that are similar to triangular matrices are called triangularisable, many operations on upper triangular matrices preserve the shape, The sum of two upper triangular matrices is upper triangular. The product of two upper triangular matrices is upper triangular, the inverse of an invertible upper triangular matrix is upper triangular. The product of a triangular matrix by a constant is an upper triangular matrix. Together these facts mean that the triangular matrices form a subalgebra of the associative algebra of square matrices for a given size. The Lie algebra of all upper triangular matrices is referred to as a Borel subalgebra of the Lie algebra of all square matrices. All these results hold if upper triangular is replaced by lower triangular throughout, however, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a triangular matrix can be any matrix. This matrix is upper triangular and this matrix is lower triangular, if the entries on the main diagonal of a triangular matrix are all 1, the matrix is called unitriangular. Other names used for these matrices are unit triangular, or very rarely normed triangular, however a unit triangular matrix is not the same as the unit matrix, and a normed triangular matrix has nothing to do with the notion of matrix norm. The identity matrix is the matrix which is both upper and lower unitriangular. The set of unitriangular matrices forms a Lie group, if the entries on the main diagonal of a triangular matrix are all 0, the matrix is called strictly triangular
Triangular matrix
–
Binary lower unitriangular
Toeplitz matrices, multiplied using
F_{2} operations They form the
Cayley table of
Z_{4} and correspond to powers of the 4-bit Gray code permutation.
18.
QR decomposition
–
Any real square matrix A may be decomposed as A = Q R, where Q is an orthogonal matrix and R is an upper triangular matrix. If A is invertible, then the factorization is unique if we require that the elements of R be positive. If instead A is a square matrix, then there is a decomposition A = QR where Q is a unitary matrix. If A has n linearly independent columns, then the first n columns of Q form a basis for the column space of A. More generally, the first k columns of Q form a basis for the span of the first k columns of A for any 1 ≤ k ≤ n. The fact that any column k of A only depends on the first k columns of Q is responsible for the form of R. If A is of rank n and we require that the diagonal elements of R1 are positive then R1 and Q1 are unique. R1 is then equal to the upper triangular factor of the Cholesky decomposition of A* A. Analogously, we can define QL, RQ, and LQ decompositions, with L being a lower triangular matrix. There are several methods for computing the QR decomposition, such as by means of the Gram–Schmidt process, Householder transformations. Each has a number of advantages and disadvantages, consider the Gram–Schmidt process applied to the columns of the full column rank matrix A =, with inner product ⟨ v, w ⟩ = v ⊤ w. This can be written in form, A = Q R where. Consider the decomposition of A =, recall that an orthonormal matrix Q has the property Q T Q = I. Then, we can calculate Q by means of Gram–Schmidt as follows, thus, we have Q T A = Q T Q R = R, R = Q T A =. The RQ decomposition transforms a matrix A into the product of a triangular matrix R. The only difference from QR decomposition is the order of these matrices, QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column. RQ decomposition is Gram–Schmidt orthogonalization of rows of A, started from the last row, the Gram-Schmidt process is inherently numerically unstable. While the application of the projections has an appealing geometric analogy to orthogonalisation, a significant advantage however is the ease of implementation, which makes this a useful algorithm to use for prototyping if a pre-built linear algebra library is unavailable. A Householder reflection is a transformation takes a vector and reflects it about some plane or hyperplane
QR decomposition
19.
Function (mathematics)
–
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
Function (mathematics)
–
A function f takes an input x, and returns a single output f (x). One metaphor describes the function as a "machine" or "
black box " that for each input returns a corresponding output.
20.
Function composition
–
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function. The resulting composite function is denoted g ∘ f, X → Z, the notation g ∘ f is read as g circle f, or g round f, or g composed with f, g after f, g following f, or g of f, or g on f. Intuitively, composing two functions is a process in which the output of the inner function becomes the input of the outer function. The composition of functions is a case of the composition of relations. The composition of functions has some additional properties, Composition of functions on a finite set, If f =, and g =, then g ∘ f =. The composition of functions is always associative—a property inherited from the composition of relations, since there is no distinction between the choices of placement of parentheses, they may be left off without causing any ambiguity. In a strict sense, the composition g ∘ f can be only if fs codomain equals gs domain, in a wider sense it is sufficient that the former is a subset of the latter. The functions g and f are said to commute with each other if g ∘ f = f ∘ g, commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, | x | +3 = | x + 3 | only when x ≥0, the composition of one-to-one functions is always one-to-one. Similarly, the composition of two functions is always onto. It follows that composition of two bijections is also a bijection, the inverse function of a composition has the property that −1 =. Derivatives of compositions involving differentiable functions can be using the chain rule. Higher derivatives of functions are given by Faà di Brunos formula. Suppose one has two functions f, X → X, g, X → X having the domain and codomain. Then one can form chains of transformations composed together, such as f ∘ f ∘ g ∘ f, such chains have the algebraic structure of a monoid, called a transformation monoid or composition monoid. In general, transformation monoids can have remarkably complicated structure, one particular notable example is the de Rham curve. The set of all functions f, X → X is called the transformation semigroup or symmetric semigroup on X. If the transformation are bijective, then the set of all combinations of these functions forms a transformation group
Function composition
–
g ∘ f, the composition of f and g. For example, (g ∘ f)(c) = #.
21.
Surjective function
–
It is not required that x is unique, the function f may map one or more elements of X to the same element of Y. The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the functions codomain, any function induces a surjection by restricting its codomain to its range. Every surjective function has an inverse, and every function with a right inverse is necessarily a surjection. The composite of surjective functions is always surjective, any function can be decomposed into a surjection and an injection. A surjective function is a function whose image is equal to its codomain, equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with f = y. Surjections are sometimes denoted by a two-headed rightwards arrow, as in f, X ↠ Y, symbolically, If f, X → Y, then f is said to be surjective if ∀ y ∈ Y, ∃ x ∈ X, f = y. For any set X, the identity function idX on X is surjective, the function f, Z → defined by f = n mod 2 is surjective. The function f, R → R defined by f = 2x +1 is surjective, because for every real number y we have an x such that f = y, an appropriate x is /2. However, this function is not injective since e. g. the pre-image of y =2 is, the function g, R → R defined by g = x2 is not surjective, because there is no real number x such that x2 = −1. However, the g, R → R0+ defined by g = x2 is surjective because for every y in the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y. The natural logarithm ln, → R is a surjective. Its inverse, the function, is not surjective as its range is the set of positive real numbers. The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the linear group of degree n, i. e. the group of all n×n invertible matrices. Under this definition the matrix exponential is surjective for complex matrices, the projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty. In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function, a function is bijective if and only if it is both surjective and injective. If a function is identified with its graph, then surjectivity is not a property of the function itself, unlike injectivity, surjectivity cannot be read off of the graph of the function alone. The function g, Y → X is said to be an inverse of the function f, X → Y if f = y for every y in Y
Surjective function
–
A surjective function from
domain X to
codomain Y. The function is surjective because every point in the codomain is the value of f (x) for at least one point x in the domain.
22.
Injective function
–
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness, it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence, occasionally, an injective function from X to Y is denoted f, X ↣ Y, using an arrow with a barbed tail. A function f that is not injective is sometimes called many-to-one, however, the injective terminology is also sometimes used to mean single-valued, i. e. each argument is mapped to at most one value. A monomorphism is a generalization of a function in category theory. Let f be a function whose domain is a set X, the function f is said to be injective provided that for all a and b in X, whenever f = f, then a = b, that is, f = f implies a = b. Equivalently, if a ≠ b, then f ≠ f, in particular the identity function X → X is always injective. If the domain X = ∅ or X has only one element, the function f, R → R defined by f = 2x +1 is injective. The function g, R → R defined by g = x2 is not injective, however, if g is redefined so that its domain is the non-negative real numbers [0, +∞), then g is injective. The exponential function exp, R → R defined by exp = ex is injective, the natural logarithm function ln, → R defined by x ↦ ln x is injective. The function g, R → R defined by g = xn − x is not injective, since, for example, g = g =0. More generally, when X and Y are both the real line R, then a function f, R → R is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the line test. Functions with left inverses are always injections and that is, given f, X → Y, if there is a function g, Y → X such that, for every x ∈ X g = x then f is injective. In this case, g is called a retraction of f, conversely, f is called a section of g. Conversely, every injection f with non-empty domain has an inverse g. Note that g may not be an inverse of f because the composition in the other order, f o g. In other words, a function that can be undone or reversed, injections are reversible but not always invertible
Injective function
–
An injective non-surjective function (injection not a
bijection)
23.
Factorization system
–
In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory, a factorization system for a category C consists of two classes of morphisms E and M of C such that, E and M both contain all isomorphisms of C and are closed under composition. Every morphism f of C can be factored as f = m ∘ e for some morphisms e ∈ E and m ∈ M. Two morphisms e and m are said to be orthogonal, denoted e ↓ m and this notion can be extended to define the orthogonals of sets of morphisms by H ↑ = and H ↓ =. Since in a factorization system E ∩ M contains all the isomorphisms, proof, In the previous diagram, take m, = i d, e ′, = i d and m ′, = m. Suppose e and m are two morphisms in a category C, then e has the left lifting property with respect to m when for every pair of morphisms u and v such that ve=mu there is a morphism w such that the following diagram commutes. The difference with orthogonality is that w is not necessarily unique, the class M is exactly the class of morphisms having the right lifting property wrt the morphisms of E. Every morphism f of C can be factored as f = m ∘ e for some morphisms e ∈ E and m ∈ M. Peter Freyd, journal of Pure and Applied Algebra. Notes, Factorization Systems, Emily Riehl,2008
Factorization system
–
This article includes a
list of references, related reading or
external links, but its sources remain unclear because it lacks
inline citations. Please
improve this article by introducing more precise citations. (August 2015)
24.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory
Group theory
25.
Prime factorization
–
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
Prime factorization
–
This image demonstrates the prime decomposition of 864. A shorthand way of writing the resulting prime factors is 2 5 × 3 3
26.
Algorithm
–
In mathematics and computer science, an algorithm is a self-contained sequence of actions to be performed. Algorithms can perform calculation, data processing and automated reasoning tasks, an algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391, English adopted the French term, but it wasnt until the late 19th century that algorithm took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu and it begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as, Algorism is the art by which at present we use those Indian figures, the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals. An informal definition could be a set of rules that precisely defines a sequence of operations, which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually, but humans can do something equally useful, in the case of certain enumerably infinite sets, They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. An enumerably infinite set is one whose elements can be put into one-to-one correspondence with the integers, the concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a set of axioms. In logic, the time that an algorithm requires to complete cannot be measured, from such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete and abstract usage of the term. Algorithms are essential to the way computers process data, thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Although this may seem extreme, the arguments, in its favor are hard to refute. Gurevich. Turings informal argument in favor of his thesis justifies a stronger thesis, according to Savage, an algorithm is a computational process defined by a Turing machine. Typically, when an algorithm is associated with processing information, data can be read from a source, written to an output device. Stored data are regarded as part of the state of the entity performing the algorithm. In practice, the state is stored in one or more data structures, for some such computational process, the algorithm must be rigorously defined, specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be dealt with, case-by-case
Algorithm
–
Alan Turing's statue at
Bletchley Park.
Algorithm
–
Logical NAND algorithm implemented electronically in
7400 chip
27.
Complex conjugate
–
In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign. For example, the conjugate of 3 + 4i is 3 − 4i. In polar form, the conjugate of ρ e i ϕ is ρ e − i ϕ and this can be shown using Eulers formula. Complex conjugates are important for finding roots of polynomials, according to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients, so is its conjugate. The complex conjugate of a number z is written as z ¯ or z ∗. The first notation avoids confusion with the notation for the transpose of a matrix. The second is preferred in physics, where dagger is used for the conjugate transpose, If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the conjugate of a previous known number is abbreviated as c. c. A significant property of the conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero. The conjugate of the conjugate of a number z is z. The ultimate relation is the method of choice to compute the inverse of a number if it is given in rectangular coordinates. Exp = exp ¯ log = log ¯ if z is non-zero If p is a polynomial with real coefficients, thus, non-real roots of real polynomials occur in complex conjugate pairs. In general, if ϕ is a function whose restriction to the real numbers is real-valued. The map σ = z ¯ from C to C is a homeomorphism and antilinear, even though it appears to be a well-behaved function, it is not holomorphic, it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension C / R and this Galois group has only two elements, σ and the identity on C. Thus the only two field automorphisms of C that leave the real numbers fixed are the identity map and complex conjugation. Similarly, for a fixed complex unit u = exp, the equation z − z 0 z ¯ − z 0 ¯ = u determines the line through z 0 in the direction of u
Complex conjugate
–
Geometric representation of z and its conjugate z̅ in the complex plane. The complex conjugate is found by
reflecting z across the real axis.
28.
Radical expression
–
A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using numbers, as in fourth root, twentieth root. For example,2 is a root of 4, since 22 =4. −2 is also a root of 4, since 2 =4. A real number or complex number has n roots of degree n. While the roots of 0 are not distinct, the n nth roots of any real or complex number are all distinct. If n is odd and x is real, one nth root is real and has the sign as x. Finally, if x is not real, then none of its nth roots is real. Roots are usually using the radical symbol or radix or √, with x or √ x denoting the square root, x 3 denoting the cube root, x 4 denoting the fourth root. In the expression x n, n is called the index, is the sign or radix. For example, −8 has three roots, −2,1 + i √3 and 1 − i √3. Out of these,1 + i √3 has the least argument,4 has two square roots,2 and −2, having arguments 0 and π respectively. So 2 is considered the root on account of having the lesser argument. An unresolved root, especially one using the symbol, is often referred to as a surd or a radical. Nth roots can also be defined for complex numbers, and the roots of 1 play an important role in higher mathematics. The origin of the root symbol √ is largely speculative, some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī, legend has it that it was taken from the Arabic letter ج, which is the first letter in the Arabic word جذر. However, many scholars, including Leonhard Euler, believe it originates from the letter r, the symbol was first seen in print without the vinculum in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician
Radical expression
–
Roots of integer numbers from 0 to 10. Line labels = x. x -axis = n. y -axis = n th root of x.
29.
Thomas Harriot
–
Thomas Harriot — or spelled Harriott, Hariot, or Heriot — was an English astronomer, mathematician, ethnographer, and translator. He is sometimes credited with the introduction of the potato to the British Isles, Harriot was the first person to make a drawing of the Moon through a telescope, on 26 July 1609, over four months before Galileo. After graduating from St Mary Hall, Oxford, Harriot travelled to the Americas, accompanying the 1585 expedition to Roanoke island funded by Sir Walter Raleigh and led by Sir Ralph Lane. Harriot was a member of the venture, having translated and learned the Carolina Algonquian language from two Native Americans, Wanchese and Manteo. On his return to England he worked for the 9th Earl of Northumberland, at the Earls house, he became a prolific mathematician and astronomer to whom the theory of refraction is attributed. Born in 1560 in Oxford, England, Thomas Harriot attended St Mary Hall and his name appears in the halls registry dating from 1577. Prior to his expedition with Raleigh, Harriot wrote a treatise on navigation, in addition, he made efforts to communicate with Manteo and Wanchese, two Native Americans who had been brought to England. Harriot devised an alphabet to transcribe their Carolina Algonquian language. Harriot and Manteo spent many days in one company, Harriot interrogated Manteo closely about life in the New World. In addition, he recorded the sense of awe with which the Native Americans viewed European technology, Many things they sawe with us. as mathematical instruments, as the only Englishman who had learned Algonkin prior to the voyage, Harriot was vital to the success of the expedition. His account of the voyage, named A Briefe and True Report of the New Found Land of Virginia, was published in 1588. The True Report contains an account of the Native American population encountered by the expedition, it proved very influential upon later English explorers. He wrote, Whereby it may be hoped, if means of government be used, that they may in short time be brought to civility. At the same time, his views of Native Americans industry and capacity to learn were later largely ignored in favour of the parts of the True Report about extractable minerals and resources. As a scientific adviser during the voyage, Harriot was asked by Raleigh to find the most efficient way to stack cannonballs on the deck of the ship. His ensuing theory about the close-packing of spheres shows a resemblance to atomism and modern atomic theory. His correspondence about optics with Johannes Kepler, in which he described some of his ideas, Harriott was employed for many years by Henry Percy, 9th Earl of Northumberland, with whom he resided at Syon House, which was run by Henry Percys cousin Thomas Percy. Harriot himself was interrogated and briefly imprisoned but was soon released, Walter Warner, Robert Hues, William Lower, and other scientists were present around the Earl of Northumberlands mansion as they worked for him and assisted in the teaching of the familys children
Thomas Harriot
–
Portrait often claimed to be Thomas Harriot (1602), which hangs in
Oriel College, Oxford.
Thomas Harriot
–
Watercolor by
John White of
Roanoke Indians
Thomas Harriot
–
Lord Egremont unveils a Plaque commemorating Thomas Harriot at Syon House, West London (July 2009)
Thomas Harriot
–
The Thomas Harriot Plaque in the grounds of Syon House (W. London).
30.
Robert Recorde
–
Robert Recorde was a Welsh physician and mathematician. He invented the sign and also introduced the pre-existing plus sign to English speakers in 1557. A member of a family of Tenby, Wales, born in 1512, Recorde entered the University of Oxford about 1525. Having adopted medicine as a profession, he went to the University of Cambridge to take the degree of M. D. in 1545 and he afterwards returned to Oxford, where he publicly taught mathematics, as he had done prior to going to Cambridge. It appears that he went to London, and acted as physician to King Edward VI and to Queen Mary. He was also controller of the Royal Mint and served as Comptroller of Mines and Monies in Ireland, after being sued for defamation by a political enemy, he was arrested for debt and died in the Kings Bench Prison, Southwark, by the middle of June 1558. The Pathway to Knowledge, containing the First Principles of Geometry, a book explaining Ptolemaic astronomy while mentioning the Copernican heliocentric model in passing. The Whetstone of Witte, whiche is the seconde parte of Arithmeteke, containing the extraction of rootes, the practise, with the rule of equation. This was the book in which the sign was introduced. With the publication of this book Recorde is credited with introducing algebra into England, a medical work, The Urinal of Physick, frequently reprinted. Sherburne states that Recorde also published Cosmographiae isagoge, and that he wrote books entitled De Arte faciendi Horologium, recordes chief contributions to the progress of algebra were in the way of systematising its notation. This article incorporates text from a now in the public domain, Chisholm, Hugh, ed. Recorde. The World of Mathematics Vol.1 Commentary on Robert Recorde Jourdain, the Nature of Mathematics Roberts, Gareth, and Fenny Smith, eds. Robert Recorde, The Life and Times of a Tudor Mathematician 232 pages Williams, Jack, Robert Recorde, Tudor Polymath, Expositor, the Mathematical Gazette Vol.60 No.411 Mar 1976 p 59-61 Roberts, Gordon, Robert Recorde, Tudor Scholar and Mathematician
Robert Recorde
–
Robert Recorde (1512–1558)
Robert Recorde
–
Recorde's introduction of "="
31.
FOIL method
–
In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials—hence the method may be referred to as the FOIL method. The order of the four terms in the sum is not important, the FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra, the term appears in William Betzs 1929 text, Algebra for Today, where he states. First terms, outer terms, inner terms, last terms, many students and educators in the United States now use the word FOIL as a verb meaning to expand the product of two binomials. This neologism has not gained acceptance in the mathematical community. The method is most commonly used to multiply linear binomials, in the second step, the distributive law is used to simplify each of the two terms. Note that this involves a total of three applications of the distributive property. In contrast to the FOIL method, the method using distributive can be applied easily to products with terms such as trinomials. The FOIL rule converts a product of two binomials into a sum of four monomials, the reverse process is called factoring or factorization. In particular, if the proof above is read in reverse it illustrates the technique called factoring by grouping, a visual memory tool can replace the FOIL mnemonic for a pair of polynomials with any number of terms. Make a table with the terms of the first polynomial on the left edge, the table equivalent to the FOIL rule looks like this. To multiply, the table would be as follows, thus = a w + a x + a y + a z + b w + b x + b y + b z + c w + c x + c y + c z. The FOIL rule cannot be applied to expanding products with more than two multiplicands, or multiplicands with more than two summands. However, applying the law and recursive foiling allows one to expand such products
FOIL method
–
A visual representation of the FOIL rule. Each colored line represents two terms that must be multiplied.
32.
Relatively prime
–
In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + by =1, the integer b has a multiplicative inverse modulo a, there exists an integer y such that by ≡1. In other words, b is a unit in the ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true
Relatively prime
–
Figure 1. The numbers 4 and 9 are coprime. Therefore, the diagonal of a 4 x 9 lattice does not intersect any other
lattice points
33.
Difference of two squares
–
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
Difference of two squares
–
Contents
34.
Algebraic number
–
An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients. All integers and rational numbers are algebraic, as are all roots of integers, the same is not true for all real and complex numbers because they also include transcendental numbers such as π and e. Almost all real and complex numbers are transcendental, the rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x = a/b is the root of bx − a. The quadratic surds are algebraic numbers, if the quadratic polynomial is monic then the roots are quadratic integers. The constructible numbers are numbers that can be constructed from a given unit length using straightedge. These include all quadratic surds, all numbers, and all numbers that can be formed from these using the basic arithmetic operations. Any expression formed from algebraic numbers using any combination of the arithmetic operations. Polynomial roots that cannot be expressed in terms of the arithmetic operations. This happens with many, but not all, polynomials of degree 5 or higher, gaussian integers, those complex numbers a + bi where both a and b are integers are also quadratic integers. Trigonometric functions of rational multiples of π, that is, the trigonometric numbers, for example, each of cos π/7, cos 3π/7, cos 5π/7 satisfies 8x3 − 4x2 − 4x +1 =0. This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers. Likewise, tan 3π/16, tan 7π/16, tan 11π/16, tan 15π/16 all satisfy the irreducible polynomial x4 − 4x3 − 6x2 + 4x +1, and so are conjugate algebraic integers. Some irrational numbers are algebraic and some are not, The numbers √2 and 3√3/2 are algebraic since they are roots of polynomials x2 −2 and 8x3 −3, the golden ratio φ is algebraic since it is a root of the polynomial x2 − x −1. The numbers π and e are not algebraic numbers, hence they are transcendental, the set of algebraic numbers is countable. Hence, the set of numbers has Lebesgue measure zero. Given an algebraic number, there is a monic polynomial of least degree that has the number as a root. This polynomial is called its minimal polynomial, if its minimal polynomial has degree n, then the algebraic number is said to be of degree n. An algebraic number of degree 1 is a rational number, a real algebraic number of degree 2 is a quadratic irrational
Algebraic number
–
Algebraic numbers on the
complex plane colored by degree (red=1, green=2, blue=3, yellow=4)
35.
Cosine
–
In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
Cosine
–
Trigonometric functions in the complex plane
Cosine
–
Trigonometry
Cosine
Cosine
36.
Binomial theorem
–
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. For example,4 = x 4 +4 x 3 y +6 x 2 y 2 +4 x y 3 + y 4, the coefficient a in the term of a xb yc is known as the binomial coefficient or. These coefficients for varying n and b can be arranged to form Pascals triangle and these numbers also arise in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set. Special cases of the theorem were known from ancient times. Greek mathematician Euclid mentioned the case of the binomial theorem for exponent 2. There is evidence that the theorem for cubes was known by the 6th century in India. Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement, were of interest to the ancient Hindus. The earliest known reference to this problem is the Chandaḥśāstra by the Hindu lyricist Pingala. The commentator Halayudha from the 10th century A. D. explains this method using what is now known as Pascals triangle. By the 6th century A. D. the Hindu mathematicians probably knew how to express this as a quotient n. k. the binomial theorem as such can be found in the work of 11th-century Persian mathematician Al-Karaji, who described the triangular pattern of the binomial coefficients. He also provided a proof of both the binomial theorem and Pascals triangle, using a primitive form of mathematical induction. The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, the binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, in 1544, Michael Stifel introduced the term binomial coefficient and showed how to use them to express n in terms of n −1, via Pascals triangle. Blaise Pascal studied the eponymous triangle comprehensively in the treatise Traité du triangle arithmétique, however, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin. Isaac Newton is generally credited with the binomial theorem, valid for any rational exponent. This formula is also referred to as the formula or the binomial identity. Using summation notation, it can be written as n = ∑ k =0 n x n − k y k = ∑ k =0 n x k y n − k. A simple variant of the formula is obtained by substituting 1 for y
Binomial theorem
–
The
binomial coefficients appear as the entries of
Pascal's triangle where each entry is the sum of the two above it.
37.
Quadratic polynomial
–
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
Quadratic polynomial
–
A quadratic polynomial with two
real roots (crossings of the x axis) and hence no
complex roots. Some other quadratic polynomials have their
minimum above the x axis, in which case there are no real roots and two complex roots.
38.
Quadratic formula
–
In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the equation instead of using the quadratic formula, such as factoring, completing the square. Using the quadratic formula is often the most convenient way, the general quadratic equation is a x 2 + b x + c =0. Here x represents an unknown, while a, b, and c are constants with a not equal to 0, one can verify that the quadratic formula satisfies the quadratic equation, by inserting the former into the latter. With the above parameterization, the formula is, x = − b ± b 2 −4 a c 2 a. Each of the solutions given by the formula is called a root of the quadratic equation. Geometrically, these represent the x values at which any parabola, explicitly given as y = ax2 + bx + c. The quadratic formula can be derived with an application of technique of completing the square. For this reason, the derivation is sometimes left as an exercise for students, the explicit derivation is as follows. Divide the quadratic equation by a, which is allowed because a is non-zero, subtract c/a from both sides of the equation, yielding, x 2 + b a x = − c a. The quadratic equation is now in a form to which the method of completing the square can be applied, accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain,2 = b 2 −4 a c 4 a 2. The square has thus been completed, taking the square root of both sides yields the following equation, x + b 2 a = ± b 2 −4 a c 2 a. Isolating x gives the formula, x = − b ± b 2 −4 a c 2 a. The plus-minus symbol ± indicates that both x = − b + b 2 −4 a c 2 a and x = − b − b 2 −4 a c 2 a are solutions of the quadratic equation. There are many alternatives of this derivation with minor differences, mostly concerning the manipulation of a and these result in slightly different forms for the solution, but are otherwise equivalent. A lesser known quadratic formula, as used in Mullers method, without going into parabolas as geometrical objects on a cone, a parabola is any curve described by a second-degree polynomial, i. e. The first and foremost geometrical application of the formula is that it will define the points along the x-axis where the parabola will cross it. If this distance term were to decrease to zero, the axis of symmetry would be the x value of the zero, algebraically, this means that √b2 − 4ac =0, or simply b2 − 4ac =0, for its term to be reduced to zero
Quadratic formula
–
Euclid in
Raphael 's
School of Athens
39.
Field (mathematics)
–
In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Associativity of addition and multiplication For all a, b, and c in F, the following hold, a + = + c. Commutativity of addition and multiplication For all a and b in F, the following hold, a + b = b + a. Existence of additive and multiplicative identity elements There exists an element of F, called the identity element and denoted by 0, such that for all a in F. Likewise, there is an element, called the identity element and denoted by 1, such that for all a in F. To exclude the trivial ring, the identity and the multiplicative identity are required to be distinct. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element −a in F, similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 =1. In other words, subtraction and division operations exist, distributivity of multiplication over addition For all a, b and c in F, the following equality holds, a · = +. A simple example of a field is the field of numbers, consisting of numbers which can be written as fractions a/b, where a and b are integers. The additive inverse of such a fraction is simply −a/b, to see the latter, note that b a ⋅ a b = b a a b =1. In addition to number systems such as the rationals, there are other. The following example is a field consisting of four elements called O, I, A and B, the notation is chosen such that O plays the role of the additive identity element, and I is the multiplicative identity. One can check that all field axioms are satisfied, for example, A · = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity. This field is called a field with four elements
Field (mathematics)
–
Given 0, 1, r 1 and r 2, the construction yields r 1 · r 2
40.
Cubic function
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In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d, where a is nonzero. Setting f =0 produces an equation of the form. The solutions of this equation are called roots of the polynomial f, If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the equation can be found algebraically. The roots can also be found trigonometrically, alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newtons method. The coefficients do not need to be complex numbers, much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, the problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, in the 3rd century, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. In the 11th century, the Persian poet-mathematician, Omar Khayyám, in an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution, in the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of an equation, x3 + 12x = 6x2 +35. He used what would later be known as the Ruffini-Horner method to approximate the root of a cubic equation. He also developed the concepts of a function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the equation to find algebraic solutions to certain types of cubic equations. Leonardo de Pisa, also known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x =20
Cubic function
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Niccolò Fontana Tartaglia
Cubic function
41.
Quartic function
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Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square, having the form f = a x 4 + c x 2 + e. The derivative of a function is a cubic function. Since a quartic function is defined by a polynomial of even degree, If a is positive, then the function increases to positive infinity at both ends, and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum, in both cases it may or may not have another local maximum and another local minimum. The degree four is the highest degree such that every polynomial equation can be solved by radicals, the solution of the quartic was published together with that of the cubic by Ferraris mentor Gerolamo Cardano in the book Ars Magna. Depman claimed that even earlier, in 1486, Spanish mathematician Valmes was burned at the stake for claiming to have solved the quartic equation, inquisitor General Tomás de Torquemada allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding. However Beckmann, who popularized this story of Depman in the West, said that it was unreliable, beckmanns version of this story has been widely copied in several books and internet sites, usually without his reservations and sometimes with fanciful embellishments. Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed. The notes left by Évariste Galois prior to dying in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus and it follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics. Here are example of geometric problems whose solution amounts of solving a quartic equation. In computer-aided manufacturing, the torus is a shape that is associated with the endmill cutter. In optics, Alhazens problem is Given a light source and a spherical mirror and this leads to a quartic equation. Finding the distance of closest approach of two ellipses involves solving a quartic equation, the eigenvalues of a 4×4 matrix are the roots of a quartic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a linear difference equation or differential equation is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending, intersections between spheres, cylinders, or other quadrics can be found using quartic equations. Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line, one of those regions is disjointed into sub-regions of equal area. The possible cases for the nature of the roots are as follows, If ∆ >0 then either the equations four roots are all real or none is
Quartic function
42.
Algebraic formula
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In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations. For example,3 x 2 −2 x y + c is an algebraic expression, since taking the square root is the same as raising to the power 12,1 − x 21 + x 2 is also an algebraic expression. By contrast, transcendental numbers like π and e are not algebraic, a rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations. In other words, an expression is an expression which may be constructed from the variables. Thus,3 x 2 −2 x y + c y 3 −1 is a rational expression, a rational equation is an equation in which two rational fractions of the form P Q are set equal to each other. These expressions obey the rules as fractions. The equations can be solved by cross-multiplying, division by zero is undefined, so that a solution causing formal division by zero is rejected. Such a solution of an equation is called an algebraic solution, but the Abel-Ruffini theorem states that algebraic solutions do not exist for all such equations if n ≥5. By convention, letters at the beginning of the alphabet are used to represent constants. They are usually written in italics, by convention, terms with the highest power, are written on the left, for example, x 2 is written to the left of x. When a coefficient is one, it is usually omitted, likewise when the exponent is one, and, when the exponent is zero, the result is always 1. The table below summarizes how algebraic expressions compare with other types of mathematical expressions by the type of elements they may contain. A rational algebraic expression is an expression that can be written as a quotient of polynomials. An irrational algebraic expression is one that is not rational, such as √x +4, Algebraic equation Algebraic function Analytical expression Arithmetic expression Closed-form expression Expression Polynomial Term James, Robert Clarke, James, Glenn
Algebraic formula
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1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, - variables
43.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
Real number
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A symbol of the set of real numbers (ℝ)
44.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
Complex number
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A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an
Argand diagram, representing the
complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the
imaginary unit which satisfies i 2 = −1.
45.
Completing the square
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In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form a x 2 + b x + c to the form a 2 + constant. In this context, constant means not depending on x, the expression inside the parenthesis is of the form. Thus a x 2 + b x + c is converted to a 2 + k for some values of h and k, in mathematics, completing the square is often applied in any computation involving quadratic polynomials. Completing the square is used to derive the quadratic formula. There is a formula in elementary algebra for computing the square of a binomial,2 = x 2 +2 p x + p 2. For example,2 = x 2 +6 x +92 = x 2 −10 x +25, in any perfect square, the coefficient of x is twice the number p, and the constant term is equal to p2. Consider the following polynomial, x 2 +10 x +28. This quadratic is not a square, since 28 is not the square of 5,2 = x 2 +10 x +25. However, it is possible to write the original quadratic as the sum of this square and this is called completing the square. Given any monic quadratic x 2 + b x + c and this square differs from the original quadratic only in the value of the constant term. Therefore, we can write x 2 + b x + c =2 + k and this operation is known as completing the square. For example, x 2 +6 x +11 =2 +2 x 2 +14 x +30 =2 −19 x 2 −2 x +7 =2 +6. Given a quadratic polynomial of the form a x 2 + b x + c it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial. Example,3 x 2 +12 x +27 =3 =3 =32 +15 This allows us to any quadratic polynomial in the form a 2 + k. The result of completing the square may be written as a formula. For the general case, a x 2 + b x + c = a 2 + k, specifically, when a=1, x 2 + b x + c =2 + k, where h = − b 2 and k = c − b 24. In analytic geometry, the graph of any function is a parabola in the xy-plane. Given a quadratic polynomial of the form 2 + k or a 2 + k the numbers h and k may be interpreted as the Cartesian coordinates of the vertex of the parabola
Completing the square
46.
Partition (number theory)
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In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition, a summand in a partition is also called a part. The number of partitions of n is given by the function p. The notation λ ⊢ n means that λ is a partition of n, Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general. For example, the partition 2 +2 +1 might instead be written as the tuple or in the more compact form where the superscript indicates the number of repetitions of a term. There are two common methods to represent partitions, as Ferrers diagrams, named after Norman Macleod Ferrers. Both have several possible conventions, here, we use English notation, with diagrams aligned in the upper-left corner. The partition 6 +4 +3 +1 of the positive number 14 can be represented by the diagram, The 14 circles are lined up in 4 rows. The diagrams for the 5 partitions of the number 4 are listed below, rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. As a type of shape made by adjacent squares joined together, by convention p =1, p =0 for n negative. The first few values of the function are,1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,1575,1958,2436,3010,3718,4565,5604. As of June 2013, the largest known prime number that counts a number of partitions is p, the generating function for p is given by, ∑ n =0 ∞ p x n = ∏ k =1 ∞. Expanding each factor on the side as a geometric series. The xn term in this product counts the number of ways to write n = a1 + 2a2 + 3a3 +, where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the generating function. More generally, the function for the partitions of n into numbers from a set A can be found by taking only those terms in the product where k is an element of A. This result is due to Euler, the formulation of Eulers generating function is a special case of a q-Pochhammer symbol and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function
Partition (number theory)
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Young diagrams associated to the partitions of the positive integers 1 through 8. They are arranged so that images under the reflection about the main diagonal of the square are conjugate partitions.
47.
Prime factor
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In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of an integer is a list of the integers prime factors, together with their multiplicities. The fundamental theorem of arithmetic says that every integer has a single unique prime factorization. To shorten prime factorizations, factors are expressed in powers. For example,360 =2 ×2 ×2 ×3 ×3 ×5 =23 ×32 ×5, in which the factors 2,3 and 5 have multiplicities of 3,2 and 1, respectively. For a prime factor p of n, the multiplicity of p is the largest exponent a for which pa divides n exactly. For a positive n, the number of prime factors of n. Perfect square numbers can be recognized by the fact all of their prime factors have even multiplicities. For example, the number 144 has the prime factors 144 =2 ×2 ×2 ×2 ×3 ×3 =24 ×32. These can be rearranged to make the more visible,144 =2 ×2 ×2 ×2 ×3 ×3 = × =2 =2. Because every prime factor appears a number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on, positive integers with no prime factors in common are said to be coprime. Two integers a and b can also be defined as if their greatest common divisor gcd =1. Euclids algorithm can be used to determine whether two integers are coprime without knowing their prime factors, the runs in a time that is polynomial in the number of digits involved. The integer 1 is coprime to every integer, including itself. This is because it has no prime factors, it is the empty product and this implies that gcd =1 for any b ≥1. The function, ω, represents the number of prime factors of n, while the function, Ω. If n = ∏ i =1 ω p i α i, for example,24 =23 ×31, so ω =2 and Ω =3 +1 =4
Prime factor
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This image demonstrates how to find the prime factorization of 864. A shorthand way of writing the resulting prime factors is 2 5 × 3 3
48.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an
EAN-13 bar code
49.
William Burnside
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William Burnside was an English mathematician. He is known mostly as a researcher in the theory of finite groups. Burnside was born in London, and attended St. Johns and Pembroke Colleges at the University of Cambridge and he lectured at Cambridge for the following ten years, before being appointed professor of mathematics at the Royal Naval College in Greenwich. While this was a little outside the centres of British mathematical research, Burnside remained a very active researcher. Burnsides early research was in applied mathematics and this work was of sufficient distinction to merit his election as a fellow of the Royal Society in 1893, though it is little remembered today. Around the same time as his election his interests turned to the study of finite groups and this was not a widely studied subject in Britain in the late 19th century, and it took some years for his research in this area to gain widespread recognition. One of Burnsides best known contributions to theory is his paqb theorem. In 1897 Burnsides classic work Theory of Groups of Finite Order was published, the second edition was for many decades the standard work in the field. A major difference between the editions was the inclusion of character theory in the second and he received an honorary doctorate from the University of Dublin in June 1901. In addition to his work, Burnside was a noted rower. While he was a lecturer at Cambridge, he coached the rowing crew team. In fact, his obituary in The Times took more interest in his athletic career and he is buried at the West Wickham Parish Church in South London. William Burnside at the Mathematics Genealogy Project
William Burnside
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William Burnside
50.
Felix Klein
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His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day. Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents, his father, Kleins mother was Sophie Elise Klein. He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, at that time, Julius Plücker held Bonns chair of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plückers interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868, Plücker died in 1868, leaving his book on the foundations of line geometry incomplete. Klein was the person to complete the second part of Plückers Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch. Klein visited Clebsch the following year, along with visits to Berlin, in July 1870, at the outbreak of the Franco-Prussian War, he was in Paris and had to leave the country. For a short time, he served as an orderly in the Prussian army before being appointed lecturer at Göttingen in early 1871. Erlangen appointed Klein professor in 1872, when he was only 23, in this, he was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day. Klein did not build a school at Erlangen where there were few students, in 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. After five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig, there his colleagues included Walther von Dyck, Rohn, Eduard Study and Friedrich Engel. Kleins years at Leipzig,1880 to 1886, fundamentally changed his life, in 1882, his health collapsed, in 1883–1884, he was plagued by depression. Nonetheless his research continued, his work on hyperelliptic sigma functions dates from around this period. Klein accepted a chair at the University of Göttingen in 1886, from then until his 1913 retirement, he sought to re-establish Göttingen as the worlds leading mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own role as the leader of a school of geometry, at Göttingen, he taught a variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory. The research center Klein established at Göttingen served as a model for the best such centers throughout the world and he introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein hired David Hilbert away from Königsberg, this appointment proved fateful, under Kleins editorship, Mathematische Annalen became one of the very best mathematics journals in the world. Founded by Clebsch, only under Kleins management did it first rival then surpass Crelles Journal based out of the University of Berlin, Klein set up a small team of editors who met regularly, making democratic decisions. The journal specialized in analysis, algebraic geometry, and invariant theory
Felix Klein
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Felix Klein