In mathematics, a rational number is a number that can be expressed as the quotient or fraction ⁠ p q {\displaystyle {\tfrac {p}{q}}} ⁠ of two integers...

In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though...

mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of...

proof that π cannot be rational; Legendre (1794) completed the proof and showed that π is not the square root of a rational number. Liouville (1840) showed...

In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example...

positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations...

number that it represents. In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers, rational numbers...

Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers. The set of all Gaussian rationals forms the...

decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits....

primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for...

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...

rational number (for example 2 2 {\displaystyle \textstyle {\frac {\sqrt {2}}{2}}} ), and even do not represent any number (for example the rational fraction...

transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients...

finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating. It can be shown that a number is rational if...

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example...

inequality implies that Liouville numbers possess an excellent sequence of rational number approximations. In 1844, Joseph Liouville proved a bound showing that...

numbers: typically "0" (zero) and "1" (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral...

representation for a real number is finite if and only if it is a rational number. In contrast, the decimal representation of a rational number may be finite, for...

the radial coordinate r given the polar angle θ (though when k is a rational number, a rose curve can be expressed in Cartesian coordinates since those...

containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more...

which case it is not a normal number). For bases r and s with log r / log s rational (so that r = bm and s = bn) every number normal in base r is normal...

often, a computer will use a rational approximation to a real number. The most general data type for a rational number (a number that can be expressed as...

allowing modular division to be transformed into multiplication. Every rational number can be represented as a sum of distinct unit fractions; these representations...

Look up rational in Wiktionary, the free dictionary. Rational may refer to: Rational number, a number that can be expressed as a ratio of two integers...

of rational numbers Q {\displaystyle \mathbb {Q} } , i.e. 1 Q ( x ) = 1 {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=1} if x is a rational number and...

mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle...

{\displaystyle y} is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately...

numbers, the prime numbers and the polynomials whose coefficients are rational number have the same cardinality as the natural numbers. The set of the real...

real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system...

\mathbb {Q} } ): Real numbers that are not rational. Imaginary numbers: Numbers that equal the product of a real number and the imaginary unit i {\displaystyle...