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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 ...
The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number p / q is a "good" approximation of a real number α if the absolute value of the difference between p / q and α may not decrease if p / q is replaced by another rational number with a smaller denominator.
If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. [3] In other words, A contains every rational number less than the cut, and B contains every rational number greater than or equal to the cut. An ...
Values of trigonometric functions of rational multiples of π (except when undefined): for example, cos π / 7 , cos 3 π / 7 , and cos 5 π / 7 satisfy 8x 3 − 4x 2 − 4x + 1 = 0. This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers.
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x 1/n).
Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous. Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
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