Rational or Irrational?

No, I am not talking about people this time. It is numbers. They are strange.

For those of you who have only a rudimentary knowlwdge of numbers, this paragraph is for you. For the rest of you, skip to paragraph three. Numbers come in all kinds. There are the rational numbers, that is, numbers that can descirbed as p/q, where p and q are whole numbers (integers). Then there are the irrational numbers, those that cannot be expressed as such. Rationals have a decimal sequence which either ends or repeats in a cycle. Irrationals have non-repeating and non-ending decimal sequences. Both sets of numbers can be found as solutions to general algebraic equations. A third class of numbers, the "transcendentals", are irrational, but they are "above" the others. That is, transcendental numbers are those numbers that can never be found as solutions to algebraic equations.

One of the more common irrational numbers is the square root of 2, or approximately 1.4142... In fact,the square root of any odd number will almost certainly give an irrational number as the answer.

Perhaps the most well-known irrational number is pi, the constant number that is the result of dividing any circle's circumference by its diameter. Its value is approximately 3.1415926... This number is so famous that the computer programmers have calculated it up to 300 or 400 million decimal places, with a billion well in sight. Less well-known, but maybe more important in higher mathematics, is the number e. "e" stands simply for "exponential", and is approximately 2.718281828... Not surprisingly, both numbers have been proved, many years ago, to be transcendental as well.

Mathematicians have combined the two in various ways. (pi + e) is irrational. (pi - e) is irrational. (pi * e) is also irrational. All are easily proved.

Here's the rub:~ e raised to the power pi has been proved to be irrational, and is suspected to be transcendental, but not proven (yet).

Astonishing? Maybe.

Try this one then :~ pi raised to the power e. A number approximately equal to 8.74248131559. Nobody has any idea whether this is even rational or irrational.

I find this absolutely fascinating.