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The theorem extends to the other trigonometric functions as well. [2] For rational values of θ, the only rational values of the sine or cosine are 0, ±1/2, and ±1; the only rational values of the secant or cosecant are ±1 and ±2; and the only rational values of the tangent or cotangent are 0 and ±1. [3]
The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial ( x – r ) can be factored out of the polynomial using polynomial long division , resulting in a polynomial of lower degree ...
In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational ...
Any rational variety is thus, by definition, stably rational. Examples constructed by Beauville et al. (1985) show, that the converse is false however. Schreieder (2019) showed that very general hypersurfaces V ⊂ P N + 1 {\displaystyle V\subset \mathbf {P} ^{N+1}} are not stably rational, provided that the degree of V is at least log 2 N ...
In mathematics, an irrationality measure of a real number is a measure of how "closely" it can be approximated by rationals. If a function f ( t , λ ) {\displaystyle f(t,\lambda )} , defined for t , λ > 0 {\displaystyle t,\lambda >0} , takes positive real values and is strictly decreasing in both variables, consider the following inequality :
In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a / b is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory.
If v is a valuation corresponding to an absolute value, then one frequently writes to mean that v is an infinite place and to mean that it is a finite place. Considering all the places of the field together produces the adele ring of the number field. The adele ring allows one to simultaneously track all the data available using absolute values.
Dedekind cuts can be generalized from the rational numbers to any totally ordered set by defining a Dedekind cut as a partition of a totally ordered set into two non-empty parts A and B, such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element.
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