Question

define a rational number and write the properties of a rational number

Answer 1

any fraction in the form p/q where p and q are integers and are q is not equal to 0

Answer 2

Hi,
Here is the answer to your query:-

In mathematics, a rational number is any number that can be expressed as the quotientor fraction p/q of two integers, a numerator pand a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold {\displaystyle \mathbb {Q} }, Unicode ℚ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal).

A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.

Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q)such that q ≠ 0, for the equivalence relationdefined by (p1, q1) ~ (p2, q2) if, and only ifp1q2 = p2q1. With this formal definition, the fraction p/q becomes the standard notation for the equivalence class of (p2, q2).

Rational numbers together with addition and multiplication form a field which contains the integers and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Qis the field of algebraic numbers.[3]

In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals.

Properties

A diagram illustrating the countability of the positive rationals

The set Q, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers Z.

The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of Q. The rational numbers are therefore the prime fieldfor characteristic zero.

The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the algebraic numbers.

The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.

The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that
a/b<c/d

(where { b,d} are positive), we have

a/b<ad+bc/2bd<c/d

Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.

Hope it helps you...
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