Rational Number

In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number.

The decimal expansion of a rational number always either terminates after finitely many 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 binary, hexadecimal, or any other integer base.

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

The rational numbers can be formally defined as the equivalence classes of the quotient set Z × N / ~, where the cartesian product Z × N is the set of all ordered pairs (m,n) where m is integer and n is natural number (n ≠ 0), and "~" is the equivalence relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0.

In abstract algebra, the rational numbers together with certain operations of addition and multiplication form a field. This is the archetypical field of characteristic zero, and is the field of fractions for the ring of integers. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.

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 either Cauchy sequences or Dedekind cuts.

Zero divided by any other integer equals zero, therefore zero is a rational number (although division by zero itself is undefined).

Two rational numbers a/b and c/d are equal if, and only if, ad = bc.

Two fractions are added as follows





The rule for multiplication is



Additive and multiplicative inverses exist in the rational numbers





It follows that the quotient of two fractions is given by

A rational number is a number that can be expressed as a fraction or ratio.

The numerator and the denominator of the fraction are both integers.

When the fraction is divided out, it becomes a terminating or repeating decimal.

(The repeating decimal portion may be one number or a billion numbers.)

Rational numbers can be ordered on a number line.

Examples of rational numbers are :

6 or 6/1 can also be written as 6.0

-2 or -2/1 can also be written as -2.0

1/2 can also be written as 0.5

-5/4 can also be written as -1.25

2/3 can also be written as 0.666666666...

21/55 can also be written as 0.38181818...

53/83 can also be written as 0.62855421687..


Reference:
http://en.wikipedia.org/wiki/Rational_number
http://regentsprep.org/Regents/math/ALGEBRA/AOP1/Lrat.htm