Irrational number

An irrational number is a real number that cannot be expressed as the ratio of two integers. Equivalently, an irrational number, when expressed in decimal notation, never terminates nor repeats. Examples are $\pi, \sqrt{2}, e, \sqrt{32134},$ etc.

Because the rational numbers are countable while the reals are uncountable, one can say that the irrational numbers make up "almost all" of the real numbers.

There are two types of irrational numbers: algebraic and transcendental.

History

Arguably the first irrational to be discovered was $\sqrt{2}$ . The Pythagoreans- an ancient Greek philosophical university and religious brotherhood- stumbled upon $\sqrt{2}$ as the length of a diagonal of a square with side lengths 1 in the sixth century $B.C$ . The Pythagoreans lived by the doctrine that all is number, or that all things could be explained by relationships between numbers. And so when it was proven that this number did not fit the doctrine, the man who proved it was killed.

The Greeks then decided $\sqrt{2}$ was an anomaly of the square. This is ridiculous, and the Greeks soon discovered it was. $\sqrt{3}$ , for example.

In 1844 Joseph Liouville discovered the existence of transcendental numbers. Transcendental numbers are usually the most famous- $\pi, e,$ etc.

$\pi$ is the ratio of a circle's diameter to its radius. Since ancient times mathematicians have been obsessed with finding $\pi$ . Archimedes of Syracuse suggested a method of exhaustion for finding $\pi$ . Beginning with regular hexagons (whose $\pi$ is 3 exactly), Archimedes calculated the limits for $\pi$ , until he got to a pair of regular 96-gons. For the 96-gons, $\pi$ 's limits were:

$3\frac{10}{71}$ $to$ $3\frac{1}{7}$ .

In India, some interesting values of $\pi$ began to emerge. In 499, Aryabatha published $\pi = 3.1416...$ ; Born in 598, Brahmagupta published $\pi = \sqrt{10} = 3.1622...$ ; and Bhaskara, born 1114, said that $\pi = 3.14156...$