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Diophantine approximation. In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria . 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 ...
The set of natural numbers is a subset of , which in turn is a subset of the set of all rational numbers, itself a subset of the real numbers. [ a ] Like the set of natural numbers, the set of integers Z {\displaystyle \mathbb {Z} } is countably infinite .
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator p and a non-zero denominator q. [ 1] For example, is a rational number, as is every integer (e.g., ). The set of all rational numbers, also referred to as " the rationals ", [ 2] the field of rationals[ 3 ...
A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc.
A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a / b or , where a and b are both integers. [ 9] As with other fractions, the denominator ( b) cannot be zero. Examples include 1 2 , − 8 5 , −8 5 , and 8 −5 .
Continued fraction. A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this ...
Greatest common divisor. In mathematics, the greatest common divisor ( GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted . For example, the GCD of 8 and 12 is 4, that is, gcd (8, 12) = 4. [ 1][ 2]
The rational numbers form an initial totally ordered set which is dense in the real numbers. Moreover, the reflexive reduction < is a dense order on the rational numbers. The real numbers form an initial unbounded totally ordered set that is connected in the order topology (defined below). Ordered fields are totally ordered by definition. They ...
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