Imaginary Point

Imaginary Point

In number theory, a rational point is a point in space each of whose coordinates are rational; that is, the coordinates of the point are elements of the field of rational numbers, as well as being elements of larger fields that contain the rational numbers, such as the real numbers and the complex numbers.

For example, (3, −67/4) is a rational point in 2 dimensional space, since 3 and −67/4 are rational numbers. A special case of a rational point is an integer point, that is, a point all of whose coordinates are integers. E.g., (1, −5, 0) is an integral point in 3-dimensional space. On the other hand, more generally, a K-rational point is a point in a space where each coordinate of the point belongs to the field K, as well as being elements of larger fields containing the field K. This is analogous to rational points, which, as stated above, are contained in fields larger than the rationals. A corresponding special case of K-rational points are those that belong to a ring of algebraic integers existing inside the field K.