I would like to know why the real numbers are called “the real numbers.” I would also like to know the meaning of “real” in the phrase “real number.” Further questions and clarifications: I’d like to know both why the phrase is used now and why it was initially introduced (should these two diffe...

I believe the distinction between real and imaginary number was introduced first by Descartes; e.g., Au reste, tant les vraies racines que les fausses ne sont pas toujours r´eelles, mais quelquefois seulement imaginaires (see here, page 47). The idea of arbitrary number (rational or otherwise) ...