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The set of elements of between the positive and negative rationals is non-open. This is because the set consists of all the infinitesimals, which is just the set {} when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. Observe that in both cases, the set of ...
A binary relation between monoids M and N is a rational relation if the graph of the relation, regarded as a subset of M×N is a rational set in the product monoid. A function from M to N is a rational function if the graph of the function is a rational set. [4]
Some programming languages provide a built-in (primitive) rational data type to represent rational numbers like 1/3 and −11/17 without rounding, and to do arithmetic on them. Examples are the ratio type of Common Lisp , and analogous types provided by most languages for algebraic computation , such as Mathematica and Maple .
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 ...
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If X is a curve of genus 1 with a k-rational point p 0, then X is called an elliptic curve over k. In this case, X has the structure of a commutative algebraic group (with p 0 as the zero element), and so the set X(k) of k-rational points is an abelian group.