Mathematics

An irrational number is any real number that is not a rational number—that is, it is a number which cannot be expressed as a fraction $m/n$ where $m$ and $n$ are integers, with $n$ non-zero. Equivalently, irrational numbers are real numbers that cannot be represented as simple fractions. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational. Perhaps the best known irrational numbers are $\sqrt{2}$, $\pi$ and $e$ and $phi$.

When the ratio of lengths of two line segments is irrational, the line segments are also described as incommesurable, meaning that they share no measure in common. A measure of a line segment $I$ in this sense is a line segment $J$ that "measure" $I$ in the sense that some whole number of copies of $J$ laid end-to-end occupy the same length as $I$.

A transcendental number is a number that is not algebraic, that is, not a solution of a non-constant polynomial equation with rational coefficients, The most common examples of transcendental numbers are $\pi$, and $e$ and $\phi$. Less known are $e^\pi$ and 2^\sqrt{2}. Only a few classes of transcendental numbers are known, for it is extremely difficult to prove that a given number is transcendental.

Transcendental numbers can be viewed as those which cannot be precisely identified to the point where the next number in the sequence can be known from the previous numbers in the sequence.

The inexactitude of the transcendental numbers and other factors may be essential to the structure, evolution and maintainance of the universe.