# Can the right answer ever be irrational

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Rational Numbers: A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q not equal to 0. A rational number p/q is said to have numerator p and denominator q . Numbers that are not rational are called irrational numbers. The real line consists of the union of the rational and irrational numbers. The set of rational numbers is of measure zero on the real line, so it is "small" compared to the irrationals and the continuum. The set of all rational numbers is referred to as the "rationals," and forms a field that is denoted Q. Here, the symbol Q derives from the German word Quotient, which can be translated as "ratio," and first appeared in Bourbaki's Algèbre (reprinted as Bourbaki 1998, p. 671). Any rational number is trivially also an algebraic number. Example of rational numbers include -7, 0, 1, 1/2, 22/7, 12345/67, and so on. Farey sequences provide a way of systematically enumerating all rational numbers. The set of rational numbers is denoted Rationals in Mathematica, and a number can be tested to see if it is rational using the command Element[x, Rationals]. The elementary algebraic operations for combining rational numbers are exactly the same as for combining fractions. It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable. For a, b, and c any different rational numbers, then 1/(a-b)^2 + 1/(b-c)^2 + 1/(c-a)^2 is the square of the rational number. The probability that a random rational number has an even denominator is 1/3 (Salamin and Gosper 1972). It is conjectured that if there exists a real number x for which both 2^x and 3^x are integers, then is rational. This result would follow from the four exponentials conjecture (Finch 2003). Irrational Numbers: An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q. Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational. The most famous irrational number is root 2, sometimes called Pythagoras's constant. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of root 2 while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans . Other examples include root3, e,pi , etc. The Erdos-Borwein constant. Numbers of the form n^1/m are irrational unless n is the mth power of an integer. Numbers of the form logm to the base n , where log is the logarithm, are irrational if m and n are integers, one of which has a prime factor which the other lacks. e^r is irrational for rational r not equal to 0. Cos r is irrational for every rational number r not equal to 0 (Niven 1956, Stevens 1999), and cos thita (for measured in degrees) is irrational for every rational thita between 0 to 90 with the exception of thita=60 (Niven 1956).tan r is irrational for every rational r not equal to 0 (Stevens 1999). Quadratic surds are irrational numbers which have periodic continued fractions. Hurwitz's irrational number theorem gives bounds of the form mod alpha-p/q less than 1/lnq^2 for the best rational approximation possible for an arbitrary irrational number alpha , where the I to the base n are called Lagrange numbers and get steadily larger for each "bad" set of irrational numbers which is excluded.

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