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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 ...
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
A number field, also called an algebraic number field, is a finite-degree field extension of the field of rational numbers. non-abelian The non-abelian class field theory is an extension of the class field theory (which is about abelian extensions of number fields) to non-abelian extensions; or at least the idea of such a theory. The non ...
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
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."
Number theory began with the manipulation of numbers, that is, natural numbers (), and later expanded to integers and rational numbers (). Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations. [15]