Rational Numbers ( rational )

Definition

An instance q of type rational is a rational number where the numerator and the denominator are both of type integer.

#include < LEDA/numbers/rational.h >

Creation

rational	q	creates an instance q of type rational.
rational	q(integer n)	creates an instance q of type rational and initializes it with the integer n.
rational	q(integer n, integer d)	creates an instance q of type rational and initializes it to the rational number n/d.
rational	q(double x)	creates an instance q of type rational and initializes it with the value of x.

Operations

The arithmetic operations +, -, *, /, + =, - =, * =, / =, -(unary), + +, - -, the comparison operations <, < =, >, > =, = =, ! = and the stream operations are all available.

void	q.negate()	negates q.
void	q.invert()	inverts q.
rational	q.inverse()	returns the inverse of q.
integer	q.numerator()	returns the numerator of q.
integer	q.denominator()	returns the denominator of q.
rational&	q.simplify(const integer& a)
		simplifies q by a. Precondition a divides the numerator and the denominator of q.
rational&	q.normalize()	normalizes q.
double	to_float()	returns a double floating point approximation of q. If the q is approximable by a normalized, finite floating point number, the error is 3ulps, i.e., three units in the last place.
string	q.to_string()	returns a string representation of q.

Non-member functions

int	sign(const rational& q)	returns the sign of q.
rational	abs(const rational& q)	returns the absolute value of q.
rational	sqr(const rational& q)	returns the square of q.
integer	trunc(const rational& q)	returns the integer with the next smaller absolute value.
rational	pow(const rational& q, int n)
		returns the n-th power of q.
rational	pow(const rational& q, integer a)
		returns the a-th power of q.
integer	floor(const rational& q)	returns the next smaller integer.
integer	ceil(const rational& q)	returns the next bigger integer.
integer	round(const rational& q)	rounds q to the nearest integer.
rational	small_rational_between(const rational& p, const rational& q)
		returns a rational number between p and q whose denominator is as small as possible.
rational	small_rational_near(const rational& p, rational eps)
		returns a rational number between p - eps and p + eps whose denominator is as small as possible.

Implementation

A rational is implemented by two integer numbers which represent the numerator and the denominator. The sign is represented by the sign of the numerator.