Rational Number

A number that can be expressed as a Fraction $p/q$ where $p$ and $q$ are Integers and $q\not=0$, is called a rational number with Numerator $p$ and Denominator $q$. Numbers which are not rational are called Irrational Numbers. Any rational number is trivially also an Algebraic Number.

For $a$, $b$, and $c$ any different rational numbers, then

$${1\over(a-b)^2}+{1\over(b-c)^2}+{1\over(c-a)^2}$$

is the Square of a rational number (Honsberger 1991). The probability that a random rational number has an Even Denominator is 1/3 (Beeler et al. 1972, Item 54).

See also Algebraic Integer, Algebraic Number, Anomalous Cancellation, Denominator, Dirichlet Function, Fraction, Integer, Irrational Number, Numerator, Quotient, Transcendental Number

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Courant, R. and Robbins, H. ``The Rational Numbers.'' §2.1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 52-58, 1996.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 52-53, 1991.