To be a little bit more precise (sorry, this is one of my favorite topics). N (the natural numbers), Z (the integers), and Q (the rational numbers) are all the same size (countably infinite, with cardinality aleph0. R (the real numbers) cannot be put into one-to-one correspondence with any countably infinite set (Cantor originally used N): instead they have cardinality c,1 the name of which is derived from "continuum."2 Note that the set of reals between 0 and 1 -- that is, the open interval set (0,1) -- has the same cardinality as R, which is somewhat interesting. I leave finding a mapping from R to (0,1) to the reader.
1: The Continuum Hypothesis would state that c = aleph1 = 2aleph0. 2: This is how we got into the Continuum Hypothesis and transfinite set theory (which Hilbert called the "paradise of the infinite").
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1: The Continuum Hypothesis would state that c = aleph1 = 2aleph0.
2: This is how we got into the Continuum Hypothesis and transfinite set theory (which Hilbert called the "paradise of the infinite").