What is the cardinality of the real numbers, if they include all possible number magnitudes?

$2^{\aleph_0}$. The notation literally means the set of all functions from the natural numbers to the 2-element set {0,1}. You can think of it as a binary sequence with a binary point in front, representing a real number between 0 and 1.

Now the funny thing is that if you ask, well what number is $2^{\aleph_0}$, mathematicians have no idea. It might be $\aleph_1$ or $\aleph_2$ or $\aleph_{47}$ or some transfinite cardinal much larger than that. Nobody knows. More accurately, the answer is independent of the standard axioms of set theory, and nobody's found a plausible, universally acceptable axiom that settles the matter one way or another.