What Are Irrational Numbers? | References - Livescience.com

Reference Article: Facts about irrational numbers by Adam Mann, journalist specializing in astronomy and physics stories.

Irrational numbers are real numbers that, when expressed as a decimal, go on forever after the decimal and never repeat.
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Irrational numbers are numbers that cannot be expressed as the ratio of two whole numbers. This is opposed to rational numbers, like 2, 7, one-fifth and -13/9, which can be, and are, expressed as the ratio of two whole numbers. When expressed as a decimal, irrational numbers go on forever after the decimal point and never repeat.

Who figured out irrational numbers? 
The Greek mathematician Hippasus of Metapontum is credited with discovering irrational numbers in the 5th century B.C., according to an article from the University of Cambridge. While working on a separate problem, Hippasus is said to have stumbled on the fact that an isosceles right triangle whose two base sides are 1 unit in length will have a hypotenuse that is √2, which is an irrational number. (This can be shown using the famous Pythagorean theorem of a^2 + b^2 = c^2.)
As a reward for his great discovery, legend has it that Hippasus was thrown into the sea. This is because he was a member of the Pythagoreans, a quasi-religious order who believed that "All is number" and that the universe was made from whole numbers and their ratios. Disturbed by Hippasus' discovery, the group sentenced him to death by drowning...

The majority of real numbers are irrational. The German mathematician Georg Cantor proved this definitively in the 19th century, showing that the rational numbers are countable but the real numbers are uncountable. That means there are more reals than rationals, according to a website on history, math and other topics from educational cartoonist Charles Fisher Cooper.
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Source: Livescience.com