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Biquadratic
In algebra, a quartic function is a function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A ''quartic equation'', or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form :ax^4+bx^3+cx^2+dx+e=0 , where . The derivative of a quartic function is a cubic function. Sometimes the term biquadratic is used instead of ''quartic'', but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form :f(x)=ax^4+cx^2+e. Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. 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 ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourth-degree Equation
In algebra, a quartic function is a function (mathematics), function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of Degree of a polynomial, degree four, called a quartic polynomial. A ''quartic equation'', or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form :ax^4+bx^3+cx^2+dx+e=0 , where . The derivative of a quartic function is a cubic function. Sometimes the term biquadratic is used instead of ''quartic'', but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form :f(x)=ax^4+cx^2+e. Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If ''a'' is positive, then the function increases to positive infinity at both ends; and thus the function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Right Triangle Theorem
Fermat's right triangle theorem is a non-existence mathematical proof, proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has many equivalent formulations, one of which was stated (but not proved) in 1225 by Fibonacci. In its geometry, geometric forms, it states: *A right triangle in the Euclidean plane for which all three side lengths are rational numbers cannot have an area that is the square (algebra), square of a rational number. The area of a rational-sided right triangle is called a congruent number, so no congruent number can be square. *A right triangle and a square with equal areas cannot have all sides Commensurability (mathematics), commensurate with each other. *There do not exist two Pythagorean triple, integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle. More abstractly, as a result about Diophantine equa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Sum Of Powers Conjecture
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers is itself a th power, then is greater than or equal to : a_1^k + a_2^k + \dots + a_n^k = b^k \implies n \ge k The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case : if a_1^k + a_2^k = b^k, then . Although the conjecture holds for the case (which follows from Fermat's Last Theorem for the third powers), it was disproved for and . It is unknown whether the conjecture fails or holds for any value . Background Euler was aware of the equality involving sums of four fourth powers; this, however, is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number or the taxicab number 1729. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers. Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers. Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Computer arithmetic deals with the specificities of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see Order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20 The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of ''Arithmetica''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, Wiles's proof of Fermat's Last Theorem, the first success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in ''n'' dimensions is equal to \sqrt. An ''n''-dimensional hypercube is more commonly referred to as an ''n''-cube or sometimes as an ''n''-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The hypercube is the special case of a hyperrectangle (also called an ''n-orthotope''). A ''unit hypercube'' is a hypercube whose side has length one unit. Often, the hypercube whose corners (or ''vertices'') are the 2''n'' points in R''n'' with each coordinate equal to 0 or 1 i ... [...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] |
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] |
Cube (algebra)
In arithmetic and algebra, the cube of a number is its third exponentiation, power, that is, the result of multiplying three instances of together. The cube of a number is denoted , using a superscript 3, for example . The cube Mathematical operation, operation can also be defined for any other expression (mathematics), mathematical expression, for example . The cube is also the number multiplied by its square (algebra), square: :. The ''cube function'' is the function (mathematics), function (often denoted ) that maps a number to its cube. It is an odd function, as :. The volume of a Cube (geometry), geometric cube is the cube of its side length, giving rise to the name. The Inverse function, inverse operation that consists of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. The graph of a function, graph of the cube function is known as the cubic para ... [...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] |
