Constructing $\mathbb{R}$ from $\mathbb{Z}$? I have been told that the real number line $\mathbb{R}$ can be constructed from the cartesian product $\mathbb{Z} \times [0,1)$.
How exactly is that true? Surely, the cartesian product $\mathbb{Z} \times [0,1)$ would give a set of ordered pairs of numbers? How is this equivalent to $\mathbb{R}$?


I have been told that the real number line $\mathbb{R}$ can be constructed from the cartesian product $\mathbb{Z} \times [0,1)$.

"constructed from" is perhaps relatively vaguely defined, but I assume you mean "has the same cardinality as" or, equivalently, "a bijection exists to".
$$ f((n, r)) = n + r $$ is exactly such a bijection.

How exactly is that true? Surely, the cartesian product $\mathbb{Z} \times [0,1)$ would give a set of ordered pairs of numbers? How is this equivalent to $\mathbb{R}$?

"Equivalent" makes no sense in this case. One could define equivalence in many distinct ways. When you say "constructed from" I assume you are searching for a bijection.
