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For example: "An even number is an integer which is divisible by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -8..." In logic, the extension of a predicate is the set of all things for which the predicate is true. [48]
Denotes square root and is read as the square root of. Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example, √2. √ (radical symbol) 1. Denotes square root and is read as the square root of. For example, +. 2.
The rule states that if the nonzero terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number.
A root is a simple root if = or a multiple root if . Simple roots are Lipschitz continuous with respect to coefficients but multiple roots are not. In other words, simple roots have bounded sensitivities but multiple roots are infinitely sensitive if the coefficients are perturbed arbitrarily.
In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers) variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).
The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number. In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length | | can be constructed with compass and straightedge in a finite number of steps.
In mathematics, a law is a formula that is always true within a given context. [1] Laws describe a relationship, between two or more expressions or terms (which may contain variables), usually using equality or inequality, [2] or between formulas themselves, for instance, in mathematical logic.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
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