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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 pentagon, regular pentagon, and more complicated ones such as \sqrt[3]. Denesting Some nested radicals can be rewritten in a form that is not nested. For example, \sqrt = 1+\sqrt\,, \sqrt[3] = \frac \,. Another simple example, \sqrt[3] = \sqrt[6] 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Solution
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of th roots ( square roots, cube roots, etc.). A well-known example is the quadratic formula :x=\frac, which expresses the solutions of the quadratic equation :ax^2 + bx + c =0. There exist algebraic solutions for cubic equations and quartic equations, which are more complicated than the quadratic formula. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Radicals
In mathematics, a sum of radicals is defined as a finite linear combination of th roots: :\sum_^n k_i\sqrt _i where n, r_i are natural numbers and k_i, x_i are real numbers. A particular special case arising in computational complexity theory is the square-root sum problem, asking whether it is possible to determine the sign of a sum of square roots, with integer coefficients, in polynomial time. This 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. In 1991, Blömer proposed a polynomial time Monte Carlo algorithm for determining whether a sum of radicals is zero, or more generally whether it represents a rational number.. Blömer's result applies more generally than the square-root sum problem, to sums of radicals that are not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product (mathematics), product of multiplying bases: b^n = \underbrace_.In particular, b^1=b. The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ". The above definition of b^n immediately implies several properties, in particular the multiplication rule:There are three common notations for multiplication: x\times y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variable (mathematics), variables are used; x\cdot y is used for emphasizing that one ta ... [...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 good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers a_1 \le a_2 \le a_3 \le ...\le K converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded. For sums of non-negative increasing sequences 0 \le a_ \le a_ \le \cdots , it says that taking the sum and the supremum can be interchanged. In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non ... [...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 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish between a positive and a negative zero. In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which 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 binary aspects of mathematical or scientific objects, such as odd and even ( sign of a permutation), sense of orientation or rotation ( cw/ccw), one sided limits, and other concepts described in below. Sign of a number Numbers from various number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plastic Ratio
In mathematics, the plastic ratio is a geometrical aspect ratio, proportion, given by the unique real polynomial root, solution of the equation Its decimal expansion begins as . The adjective ''plastic'' does not refer to Plastic, the artificial material, but to the formative and sculptural qualities of this ratio, as in ''plastic arts''. Definition Three quantities are in the plastic ratio if \frac =\frac =\frac The ratio is commonly denoted Substituting b=\rho c \, and a=\rho b =\rho^2 c \, in the middle fraction, \rho =\frac. It follows that the plastic ratio is the unique real solution of the cubic equation \rho^3 -\rho -1 =0. Solving with Cubic equation#Cardano's formula, Cardano's formula, \begin w_ &=\frac12 \left( 1 \pm \frac13 \sqrt \right) \\ \rho &=\sqrt[3] +\sqrt[3] \end or, using the Cubic equation#Trigonometric and hyperbolic solutions, hyperbolic cosine, :\rho =\frac \cosh \left( \frac \operatorname \left( \frac \right) \right). is the superstabl ... [...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 Multiplicative inverse, reciprocal of the mathematical constant pi, : \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 (mathematics), 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. Alt ... [...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 worl ... [...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 functions. Their multiplicative inverse, reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding Inverse trigonometric functions, inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-an ... [...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 summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \frac = \frac = \varphi, where the Greek letter Phi (letter), phi ( or ) denotes the golden ratio. The constant satisfies the quadratic equation 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; it also goes by 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 Straightedge and compass construction, construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casus Irreducibilis
() is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be ''reduced'' to the computation of square and cube roots. Cardano's formula for solution in radicals of a cubic equation was discovered at this time. It applies in the ''casus irreducibilis'', but, in this case, requires the computation of the square root of a negative number, which involves knowledge of complex numbers, unknown at the time. The casus irreducibilis occurs when the three solutions are real and distinct, or, equivalently, when the discriminant is positive. It is only in 1843 that Pierre Wantzel proved that there cannot exist any solution in real radicals in the ''casus irreducibilis''. 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)^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
