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Nested Radical
In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include :\sqrt, which arises in discussing the regular pentagon, and more complicated ones such as :\sqrt Denesting Some nested radicals can be rewritten in a form that is not nested. For example, \sqrt = 1+\sqrt\,, \sqrt = \frac \,. Rewriting a nested radical in this way is called denesting. This is not always possible, and, even when possible, it is often difficult. Two nested square roots In the case of two nested square roots, the following theorem completely solves the problem of denesting. If and are rational numbers and is not the square of a rational number, there are two rational numbers and such that :\sqrt = \sqrt\pm\sqrt if and only if a^2-c~ is the square of a rational number . If the nested radical is real, and are the two numbers :\frac2~ and ~\frac2~,~ where ~d=\sqrt~ is a ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Solution
A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of th roots (square roots, cube roots, and other integer roots). A well-known example is the solution :x=\frac of the quadratic equation :ax^2 + bx + c =0. There exist more complicated algebraic solutions for cubic equations and quartic equations. The Abel–Ruffini theorem,Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, and, more generally Galois theory, state that some quintic equations, such as :x^5-x+1=0, do not have any algebraic solution. The same is true for every higher degree. However, for any degree there are some polynomial equations that have algebraic solutions; for example, the equation x^ = 2 can be solved as x=\pm\sqrt 0. The eight other solutions are nonreal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Radicals
In computational complexity theory, there is an open problem of whether some information about a sum of radicals may be computed in polynomial time depending on the input size, i.e., in the number of bits necessary to represent this sum. It is of importance for many problems in computational geometry, since the computation of the Euclidean distance between two points in the general case involves the computation of a square root, and therefore the perimeter of a polygon or the length of a polygonal chain takes the form of a sum of radicals. The sum of radicals is defined as a finite linear combination of radicals: :\sum_^n k_i\sqrt _i where n, r_i are natural numbers and k_i, x_i are real numbers. Most theoretical research in computational geometry of combinatorial character assumes the computational model of infinite precision real RAM, i.e., an abstract computer in which real numbers and operations on them are performed with infinite precision and the input size of a real numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Convergence Theorem
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum. Convergence of a monotone sequence of real numbers Lemma 1 If a sequence of real numbers is increasing and bounded above, then its supremum is the limit. Proof Let (a_n)_ be such a sequence, and let \ be the set of terms of (a_n)_ . By assumption, \ is non-empty and bounded above. By the least-upper-bound property of real numbers, c = \sup_n \ exists and is finite. Now, for every \varepsilon > 0, there exists N such that a_N > c - \varepsilon , since otherwise c - \varepsilon is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonnegative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (having both signs). Whenever not specifically mentioned, this article adheres to the first convention. In some contexts, it makes sense to consider a signed zero (such as floating-point representations of real numbers within computers). In mathematics and physics, the phrase "change of sign" is associated with the generation of the additive inverse (negation, or multiplication by −1) of any object that allows for this construction, and is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate other binary aspects of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plastic Number
In mathematics, the plastic number (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, ) is a mathematical constant which is the unique real solution of the cubic equation : x^3 = x + 1. It has the exact value : \rho = \sqrt \sqrt Its decimal expansion begins with . Properties Recurrences The powers of the plastic number satisfy the third-order linear recurrence relation for . Hence it is the limiting ratio of successive terms of any (non-zero) integer sequence satisfying this recurrence such as the Padovan sequence (also known as the Cordonnier numbers), the Perrin numbers and the Van der Laan numbers, and bears relationships to these sequences akin to the relationships of the golden ratio to the second-order Fibonacci and Lucas numbers, akin to the relationships between the silver ratio and the Pell numbers. The plastic number satisfies the nested radical recurrence : \rho = \sqrt Numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viète's Formula
In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant : \frac2\pi = \frac2 \cdot \frac2 \cdot \frac2 \cdots It can also be represented as: \frac2\pi = \prod_^ \cos \frac The formula is named after François Viète, who published it in 1593. As the first formula of European mathematics to represent an infinite process, it can be given a rigorous meaning as a limit expression, and marks the beginning of mathematical analysis. It has linear convergence, and can be used for calculations of , but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses, and as a motivating example for the concept of statistical independence. The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan's Lost Notebook
Ramanujan's lost notebook is the manuscript in which the Indian mathematician Srinivasa Ramanujan recorded the mathematical discoveries of the last year (1919–1920) of his life. Its whereabouts were unknown to all but a few mathematicians until it was rediscovered by George Andrews in 1976, in a box of effects of G. N. Watson stored at the Wren Library at Trinity College, Cambridge. The "notebook" is not a book, but consists of loose and unordered sheets of paper described as "more than one hundred pages written on 138 sides in Ramanujan's distinctive handwriting. The sheets contained over six hundred mathematical formulas listed consecutively without proofs." have published several books in which they give proofs for Ramanujan's formulas included in the notebook. Berndt says of the notebook's discovery: "The discovery of this 'Lost Notebook' caused roughly as much stir in the mathematical world as the discovery of Beethoven’s tenth symphony would cause in the musical world ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( or \phi) denotes the golden ratio. The constant \varphi satisfies the quadratic equation \varphi^2 = \varphi + 1 and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of \varphi—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casus Irreducibilis
In algebra, ''casus irreducibilis'' (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of ''casus irreducibilis'' is in the case of cubic polynomials that have three real roots, which was proven by Pierre Wantzel in 1843. One can see whether a given cubic polynomial is in so-called ''casus irreducibilis'' by looking at the discriminant, via Cardano's formula. The three cases of the discriminant Let : ax^3+bx^2+cx+d=0 be a cubic equation with a\ne0. Then the discriminant is given by : D := \bigl((x_1-x_2)(x_1-x_3)(x_2-x_3)\bigr)^2 = 18abcd - 4ac^3 - 27a^2d^2 + b^2c^2 -4b^3d~. It appears in the algebraic solution and is the square of the pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
