I was thinking about how many proofs involving complex numbers attempt to prove that a particular number is equal to its own complex conjugate, $c=\bar{c}$, in order to show that $c\in\mathbb{R}$.
I thought this was pretty clever and wondered if there were other ways of 'collapsing' number systems into their subsets. Here's a list I've got so far:
- $c=\bar{c}$ $(\mathbb{C}\rightarrow\mathbb{R})$
- $???$ $(\mathbb{R}\rightarrow\mathbb{Q})$
- $???$ $(\mathbb{Q}\rightarrow\mathbb{Z})$
- $n=|n|$ $(\mathbb{Z}\rightarrow\mathbb{N})$
- $n=-n$ $(\mathbb{C},\mathbb{R},\mathbb{Q},\mathbb{Z}\rightarrow\{0\})$
Now I know for $(\mathbb{Q}\rightarrow\mathbb{Z})$, for example, I could just say $p/q\ \wedge q=1$ and maybe somthing similar with $(\mathbb{R}\rightarrow\mathbb{Q})$ regarding limits of terms but my intention is to get some simple/common function that you can apply to a variable $x$ that would collapse the number system, rather than putting it into a form like $p/q$ first.
Does anyone else have any other ways of collapsing these or other number systems?
$n=n+\sin\pi n$ collapses the reals to the integers.