What identities collapse number systems?

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I was thinking about how many proofs involving complex numbers attempt to prove that a particular number is equal to its own complex conjugate, $c=barc$, in order to show that $cinmathbbR$.



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:



  • $c=barc$ $(mathbbCrightarrowmathbbR)$

  • $???$ $(mathbbRrightarrowmathbbQ)$

  • $???$ $(mathbbQrightarrowmathbbZ)$

  • $n=|n|$ $(mathbbZrightarrowmathbbN)$

  • $n=-n$ $(mathbbC,mathbbR,mathbbQ,mathbbZrightarrow0)$

Now I know for $(mathbbQrightarrowmathbbZ)$, for example, I could just say $p/q wedge q=1$ and maybe somthing similar with $(mathbbRrightarrowmathbbQ)$ 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?




complex-numbers real-numbers integers rational-numbers number-systems




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    Rings with involution and their subrings of fixed points.
    – Randall
    Jul 29 at 1:43














up vote
1
down vote

favorite
1












I was thinking about how many proofs involving complex numbers attempt to prove that a particular number is equal to its own complex conjugate, $c=barc$, in order to show that $cinmathbbR$.



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:



  • $c=barc$ $(mathbbCrightarrowmathbbR)$

  • $???$ $(mathbbRrightarrowmathbbQ)$

  • $???$ $(mathbbQrightarrowmathbbZ)$

  • $n=|n|$ $(mathbbZrightarrowmathbbN)$

  • $n=-n$ $(mathbbC,mathbbR,mathbbQ,mathbbZrightarrow0)$

Now I know for $(mathbbQrightarrowmathbbZ)$, for example, I could just say $p/q wedge q=1$ and maybe somthing similar with $(mathbbRrightarrowmathbbQ)$ 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?




complex-numbers real-numbers integers rational-numbers number-systems




share|cite|improve this question

















  • 1




    Rings with involution and their subrings of fixed points.
    – Randall
    Jul 29 at 1:43












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





I was thinking about how many proofs involving complex numbers attempt to prove that a particular number is equal to its own complex conjugate, $c=barc$, in order to show that $cinmathbbR$.



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:



  • $c=barc$ $(mathbbCrightarrowmathbbR)$

  • $???$ $(mathbbRrightarrowmathbbQ)$

  • $???$ $(mathbbQrightarrowmathbbZ)$

  • $n=|n|$ $(mathbbZrightarrowmathbbN)$

  • $n=-n$ $(mathbbC,mathbbR,mathbbQ,mathbbZrightarrow0)$

Now I know for $(mathbbQrightarrowmathbbZ)$, for example, I could just say $p/q wedge q=1$ and maybe somthing similar with $(mathbbRrightarrowmathbbQ)$ 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?




complex-numbers real-numbers integers rational-numbers number-systems




share|cite|improve this question













I was thinking about how many proofs involving complex numbers attempt to prove that a particular number is equal to its own complex conjugate, $c=barc$, in order to show that $cinmathbbR$.



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:



  • $c=barc$ $(mathbbCrightarrowmathbbR)$

  • $???$ $(mathbbRrightarrowmathbbQ)$

  • $???$ $(mathbbQrightarrowmathbbZ)$

  • $n=|n|$ $(mathbbZrightarrowmathbbN)$

  • $n=-n$ $(mathbbC,mathbbR,mathbbQ,mathbbZrightarrow0)$

Now I know for $(mathbbQrightarrowmathbbZ)$, for example, I could just say $p/q wedge q=1$ and maybe somthing similar with $(mathbbRrightarrowmathbbQ)$ 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?





complex-numbers real-numbers integers rational-numbers number-systems





share|cite|improve this question












share|cite|improve this question




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edited Jul 29 at 5:16
























asked Jul 29 at 1:38









Ozaner Hansha

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  • 1




    Rings with involution and their subrings of fixed points.
    – Randall
    Jul 29 at 1:43












  • 1




    Rings with involution and their subrings of fixed points.
    – Randall
    Jul 29 at 1:43







1




1




Rings with involution and their subrings of fixed points.
– Randall
Jul 29 at 1:43




Rings with involution and their subrings of fixed points.
– Randall
Jul 29 at 1:43










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$n=n+sinpi n$ collapses the reals to the integers.






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  • That's a good one!
    – Ozaner Hansha
    Jul 29 at 5:31










  • Or $n=lfloor nrfloor$.
    – Batominovski
    Jul 29 at 6:59










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up vote
1
down vote













$n=n+sinpi n$ collapses the reals to the integers.






share|cite|improve this answer





















  • That's a good one!
    – Ozaner Hansha
    Jul 29 at 5:31










  • Or $n=lfloor nrfloor$.
    – Batominovski
    Jul 29 at 6:59














up vote
1
down vote













$n=n+sinpi n$ collapses the reals to the integers.






share|cite|improve this answer





















  • That's a good one!
    – Ozaner Hansha
    Jul 29 at 5:31










  • Or $n=lfloor nrfloor$.
    – Batominovski
    Jul 29 at 6:59












up vote
1
down vote










up vote
1
down vote









$n=n+sinpi n$ collapses the reals to the integers.






share|cite|improve this answer













$n=n+sinpi n$ collapses the reals to the integers.







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answered Jul 29 at 2:57









Gerry Myerson

142k7143292




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  • That's a good one!
    – Ozaner Hansha
    Jul 29 at 5:31










  • Or $n=lfloor nrfloor$.
    – Batominovski
    Jul 29 at 6:59
















  • That's a good one!
    – Ozaner Hansha
    Jul 29 at 5:31










  • Or $n=lfloor nrfloor$.
    – Batominovski
    Jul 29 at 6:59















That's a good one!
– Ozaner Hansha
Jul 29 at 5:31




That's a good one!
– Ozaner Hansha
Jul 29 at 5:31












Or $n=lfloor nrfloor$.
– Batominovski
Jul 29 at 6:59




Or $n=lfloor nrfloor$.
– Batominovski
Jul 29 at 6:59












 

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