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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 ...
In decimal numbers greater than 1 (such as 3.75), the fractional part of the number is expressed by the digits to the right of the separator (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375 / 100 , or as a mixed number, 3 + 75 / 100 .
The naive height or classical height of a vector of rational numbers is the maximum absolute value of the vector of coprime integers obtained by multiplying through by a lowest common denominator. This may be used to define height on a point in projective space over Q , or of a polynomial, regarded as a vector of coefficients, or of an ...
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
It is the ring of integers in the number field () of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, [] is a Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain. [4]
In a context where rational numbers are formally defined, it is entirely correct and appropriate to say that a rational number *is* an equivalence class of Z×Z etc. etc.. In an informal definition, too, it is quite correct and appropriate to say that a rat.num. *is* a number that can be expressed as a quotient a/b etc.; and since a /1 is ...
This category represents all rational numbers, that is, those real numbers which can be represented in the form: ...where and are integers and is ...
The general approach of Diophantine geometry is illustrated by Faltings's theorem (a conjecture of L. J. Mordell) stating that an algebraic curve C of genus g > 1 over the rational numbers has only finitely many rational points. The first result of this kind may have been the theorem of Hilbert and Hurwitz dealing with the case g = 0. The ...