Short answers, pulled from the story.
Greek mathematicians led by Pythagoras discovered the existence of real numbers around 500 BC. This discovery occurred when they realized the diagonal of a square with side length one could not be expressed as a ratio of two integers.
René Descartes introduced the term real in the 17th century to distinguish these numbers from imaginary numbers. He used the adjective real to describe roots of a polynomial, creating a linguistic boundary between the tangible and the abstract.
The year 1872 marked a turning point when two independent definitions of real numbers were published by Richard Dedekind and Georg Cantor. Dedekind defined real numbers as cuts in the rational numbers, while Cantor defined them as equivalence classes of Cauchy sequences.
Georg Cantor showed that the set of all real numbers is uncountably infinite in 1874. He published his famous diagonal argument in 1891, which provided a more elegant demonstration of the same result.
Paul Cohen proved in 1963 that the continuum hypothesis is an axiom independent of the other axioms of set theory. This result implies that the real numbers are far more numerous than the integers.
The cardinality of the set of all real numbers is called the cardinality of the continuum. It equals the cardinality of the power set of the natural numbers.