InstructorPriyanka Mishra
TypeOnline Course
DateSep 2, 2016
Buy NowBook Now

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and 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“, is usually denoted by a boldface Q  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.

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.

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

Embedding of integers

Any integer n can be expressed as the rational number n/1.


  {\frac {a}{b}}={\frac {c}{d}} if and only if a d = b c .  ad=bc.


Where both denominators are positive:

  {\frac {a}{b}}<{\frac {c}{d}} if and only if  ad<bc.

If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations:

  {\frac {-a}{-b}}={\frac {a}{b}}


  {\frac {a}{-b}}={\frac {-a}{b}}.


Two fractions are added as follows:

  {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.


  {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.


The rule for multiplication is:

  {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.


Where c ≠ 0:

  {\frac {a}{b}}\div {\frac {c}{d}}={\frac {ad}{bc}}.

Note that division is equivalent to multiplying by the reciprocal of the divisor fraction:

  {\frac {ad}{bc}}={\frac {a}{b}}\times {\frac {d}{c}}.


Additive and multiplicative inverses exist in the rational numbers:

  -\left({\frac {a}{b}}\right)={\frac {-a}{b}}={\frac {a}{-b}}\quad {\mbox{and}}\quad \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}{\mbox{ if }}a\neq 0.

Exponentiation to integer power

If n is a non-negative integer, then

  \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}

and (if a ≠ 0):

  \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.

Continued fraction representation

Main article: Continued fraction

A finite continued fraction is an expression such as

   a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},

where an are integers. Every rational number a/b has two closely related expressions as a finite continued fraction, whose coefficients an can be determined by applying the Euclidean algorithm to (a,b).