Rational numbers

The rational numbers are an ordered field $((QQ, +, *), <=)$ , where

$QQ = (ZZ xx ZZ^+)/~$
$(x, y) ~ (z, w) <=> x * w = z * y$ ( $~$ is an equivalence relation).
$[(x, y)] <= [(z, w)] <=> x * w <= z * y$ ( $[(x, y)]$ means the equivalence class of $(x, y)$ )
$forall x, z in ZZ, y, w in ZZ^+: [(x, y)] + [(z, w)] = [(x * w + y * z, y * w)]$
$forall x, z in ZZ, y, w in ZZ^+: [(x, y)] * [(z, w)] = [(x * z, y * w)]$

Theory

It follows that:

$[(1, 1)]$ is the multiplicative identity element.
$[(0, 1)]$ is the additive identity element.
The subset consisting of the elements $[(x, 1)]$ is isomorphic to the ordered ring of integers.
$forall x in ZZ \setminus {0}, y in ZZ^+: [(x, y)]^-1 = [("signum"(x) * y, "signum"(x) * x)]$

Most often, one works with the elements of $ZZ xx ZZ^+$ , rather than with the rational numbers themselves. It must then be remembered that the corresponding equivalence classes are implicitly meant instead.

Practice

The Rational class template can be used in conjuction with integer types to provide a rational number. Rationals have the potential to provide exact arithmetic under restricted situations.

template <typename Integer>
class Rational;

There are several values with a special meaning:

$0 / 0$ is called Not-A-Number (NaN).
$x / 0$ for any $x > 0$ is called Infinity(x).
$y / 0$ for any $y < 0$ is called MinusInfinity(y).

It holds that:

Infinity(x) = Infinity(y), for any $x, y != 0$ .
MinusInfinity(x) = -Infinity(y), for any $x, y != 0$ .

Thus we can simply speak of Infinity and -Infinity. The Rational class maintains two invariants:

Rational numbers

Theory

Practice

Learn more

Files

Print a rational number to a stream

Rational module

Rational number

Testing for Rational