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Difference Of Two Squares
In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity :a^2-b^2 = (a+b)(a-b) in elementary algebra. Proof The proof of the factorization identity is straightforward. Starting from the left-hand side, apply the distributive law to get :(a+b)(a-b) = a^2+ba-ab-b^2 By the commutative law, the middle two terms cancel: :ba - ab = 0 leaving :(a+b)(a-b) = a^2-b^2 The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables. The proof holds in any commutative ring. Conversely, if this identity holds in a ring ''R'' for all pairs of elements ''a'' and ''b'', then ''R'' is commutative. To see this, apply the distributive law to the right-hand side of the equation and get :a^2 + ba - ab - b^2. For this to be equal to a^2 - b^2, we must have :ba - ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Inner Product Space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the picture ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rhombus Understood Analytically
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from grc, ῥόμβος, rhombos, meaning something that spins, which derives from the verb , romanized: , meaning "to turn round and round." The word was used both by Eucli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Miller–Rabin Primality Test
The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen primality test. It is of historical significance in the search for a polynomial-time deterministic primality test. Its probabilistic variant remains widely used in practice, as one of the simplest and fastest tests known. Gary L. Miller discovered the test in 1976; Miller's version of the test is deterministic, but its correctness relies on the unproven extended Riemann hypothesis. Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm in 1980. Mathematical concepts Similarly to the Fermat and Solovay–Strassen tests, the Miller–Rabin primality test checks whether a specific property, which is known to hold for prime values, holds for the number under testing. Strong probable primes The property is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fermat Primality Test
The Fermat primality test is a probabilistic test to determine whether a number is a probable prime. Concept Fermat's little theorem states that if ''p'' is prime and ''a'' is not divisible by ''p'', then :a^ \equiv 1 \pmod. If one wants to test whether ''p'' is prime, then we can pick random integers ''a'' not divisible by ''p'' and see whether the equality holds. If the equality does not hold for a value of ''a'', then ''p'' is composite. This congruence is unlikely to hold for a random ''a'' if ''p'' is composite. Therefore, if the equality does hold for one or more values of ''a'', then we say that ''p'' is probably prime. However, note that the above congruence holds trivially for a \equiv 1 \pmod, because the congruence relation is compatible with exponentiation. It also holds trivially for a \equiv -1 \pmod if ''p'' is odd, for the same reason. That is why one usually chooses a random ''a'' in the interval 1 < a < p - 1. Any ''a'' such that : [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quadratic Sieve
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties. It was invented by Carl Pomerance in 1981 as an improvement to Richard Schroeppel, Schroeppel's linear sieve. Basic aim The algorithm attempts to set up a congruence of squares modular arithmetic, modulo ''n'' (the integer to be factorized), which often leads to a factorization of ''n''. The algorithm works in two phases: the ''data collection'' phase, where it collects information that may lead to a congruence of squares; and the ''data processing'' phase, where it puts all the data it has collected into a Matrix (mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fermat Factorization Method
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: :N = a^2 - b^2. That difference is algebraically factorable as (a+b)(a-b); if neither factor equals one, it is a proper factorization of ''N''. Each odd number has such a representation. Indeed, if N=cd is a factorization of ''N'', then :N = \left(\frac\right)^2 - \left(\frac\right)^2 Since ''N'' is odd, then ''c'' and ''d'' are also odd, so those halves are integers. (A multiple of four is also a difference of squares: let ''c'' and ''d'' be even.) In its simplest form, Fermat's method might be even slower than trial division (worst case). Nonetheless, the combination of trial division and Fermat's is more effective than either. Basic method One tries various values of ''a'', hoping that a^2-N = b^2, a square. FermatFactor(N): ''// N should be odd'' a ← b2 ← a*a - N repeat until b2 is a square: a ← a + 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Base (exponentiation)
In exponentiation, the base is the number b in an expression of the form bn. Related terms The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". For example, the fourth power of 10 is 10,000 because . The term ''power'' strictly refers to the entire expression, but is sometimes used to refer to the exponent. Radix is the traditional term for ''base'', but usually refers then to one of the common bases: decimal (10), binary (2), hexadecimal (16), or sexagesimal (60). When the concepts of variable and constant came to be distinguished, the process of exponentiation was seen to transcend the algebraic functions. In his 1748 ''Introductio in analysin infinitorum'', Leonhard Euler referred to "base a = 10" in an example. He referred to ''a'' as a "constant number" in an extensive consideration of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Square Number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, \sqrt = 3, so 9 is a squ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Square Root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'', which is denoted by \sqrt, where the symbol \sqrt is called the '' radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case 9. For nonnegative , the principal square root can also be written in exponent notation, as . Every positive number has two square roots: \sqrt, which is positive, and -\sqrt, which is negative. The two roots can be written more concisely using the ± sign as \plusmn\sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Nth Root
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'': :r^n = x, where ''n'' is a positive integer, sometimes called the ''degree'' of the root. 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 ordinal numbers, as in ''fourth root'', ''twentieth root'', etc. The computation of an th root is a root extraction. For example, 3 is a square root of 9, since 3 = 9, and −3 is also a square root of 9, since (−3) = 9. Any non-zero number considered as a complex number has different complex th roots, including the real ones (at most two). The th root of 0 is zero for all positive integers , since . In particular, if is even and is a positive real number, one of its th roots is real and positive, one is negative, and the others (when ) are non-real complex numbers; if is even and is a negative real numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |