## hartnn 4 years ago inverse of mod function? like y= x mod p, 1=5 mod 2 how do i get 5 from {1,2}

1. anonymous

doesnot exist as inverse only possible for one-one function

2. anonymous

y=|x| ? inverse of this will not be a function

3. anonymous

inverse exist x=|y| but its not a function

4. ganeshie8

mod is not 1-1 function

5. anonymous

modulus is a many-one function hence it's inverse not possible

6. hartnn

c= a mod b = -> corresponds to

7. anonymous

NO PROBLEMAS SENIOUR @hartnn ! Let B be the set of ordered pairs of pairs-of-numbers, in other words pairs of points in the plane: $b \in B \,\,\iff b=(P_1, P_2)$ then the inverse function of $Abs^{-1}(y) = ((y,y),(y,-y))$ and a medal , naturally. forgive my humbleness...

8. hartnn

whats and how ? $$Abs^{-1}(y) = ((y,y),(y,-y))$$

9. hartnn

if i have y=x mod 7, how would i find its inverse ?

10. anonymous

The Abs is of course |x| not the algebraic coincidence of names.

11. hartnn

12. hartnn

like 5 mod 2 is 1

13. anonymous

if modulus is with only number then it is one-one and inverse will exist

14. anonymous

x = sgn(x) .|x|

15. anonymous

Anyway also for remainder possible to invert $mod:\,r_{Ideal} \rightarrow r \\ mod^{-1}: r \rightarrow r_{Ideal}$

16. anonymous

and a , the action that should not be asked but anyway received :)

17. hartnn

but i didn't understand....

18. anonymous

thx oh exalted one !

19. anonymous

Learn about Ideals even if you dont have them :) http://en.wikipedia.org/wiki/Ideal_(ring_theory)

20. anonymous

MEdals, Medals gentleman for the wondering scholar ....! Keep them falling !

21. hartnn

hmmm...didn't study ideals, i'll go through them, thanks

22. anonymous

Everybody that did - pls gratify

23. anonymous

i think the question is maybe something different, are you asking how do you solve $x\equiv y (mod n)$ for $$y$$ if you know $$x$$ ?

24. hartnn

yes, i wrote somewhere in between = -> correcponds to

25. anonymous

If that was the question I would not solve it - but it was not!

26. anonymous

so for example solve $$5\equiv x\text{ mod } 3$$

27. anonymous

No uniqueness - only an IDEAL OF ANSWERS

28. hartnn

there are many (or infinite ?) values of x in 5=x mod 3 ??

29. anonymous

Yes - they form an IDEAL

30. anonymous

ideals live in arbitrary rings, and are not necessary for understanding elementary number theory

31. anonymous

Anyway the set pf answers here IS an Ideal.

32. anonymous

By the way the only proper way to make equivalence necessary for "REMAINDER CALCULUS" is bu using ideals. Elementary - well yes.

33. hartnn

ok, thank you @Mikael i'll go through IDEALS and ask u if i have any doubts. thanks to @satellite73 also :)

34. anonymous

Firstly is the "cyclical group". It will give you the applications. Only then that. This takes you further

35. hartnn

any good reference for such things(other than wikipedia) where these things are explained in lucid manner ?

36. anonymous
37. anonymous

38. hartnn

because of those ... i cannot get the reference

39. hartnn

only last one i couldn't get

40. anonymous

Just ggle "elementary remainder group theory"