We are all familiar with the real numbers yet certain great scientists such as Albert Einstein do argue that there is nothing real in mathematics.
Reality is only a name. Real numbers is the name that we assigned to certain mathematical structure. And a mathematical structure is a set which elements are subject to well defined axioms. And the axioms are conventions necessary to communicate between scientists. Therefore, mathematics is a universal language, and in this context, real numbers is only the name of this kind on number.
rationals are ratios of integers. then some others can be easily determined by rule and compass, and are not rational. these had a special status, their existence was provable, but for a very long time if was not well understood exactly what set was it.
put them all on a line, which you can, by rule and compass, and I guess you want to give the status of "reality" to the whole line.
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