Results 1 to 7 of 7

Math Help - which numbers exist in modular arithmetic? negative exponents???

  1. #1
    Newbie
    Joined
    May 2011
    Posts
    13

    Question which numbers exist in modular arithmetic? negative exponents???

    I have a question about which numbers exist and which ones don't in modular arithmetic.

    The questions from the textbook are in red. The solutions from the manual are in blue (but got scanned as purple). My comments are green.

    Thanks for any help you can give!

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    In \mathbb{Z}_n, the inverse x^{-1} of x is an element of \mathbb{Z}_n such that if you multiply it by x (mod n), the answer is 1.

    So in \mathbb{Z}_5 the inverse of 2 is 2^{-1} = 3, because when you multiply 2 by 3 you get 6, which is equal to 1 (mod 5).

    But in \mathbb{Z}_4, 2^{-1} does not exist, because whatever you multiply 2 by you will always get an even number, so the answer can never be equal to 1 (mod 4).

    In general, an element x in \mathbb{Z}_n will have an inverse provided that the numbers x and n have no common factor.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2011
    Posts
    13
    Quote Originally Posted by Opalg View Post
    In \mathbb{Z}_n, the inverse x^{-1} of x is an element of \mathbb{Z}_n such that if you multiply it by x (mod n), the answer is 1.
    What is the logic and/or intuition behind this rule?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    Working in \mathbb{Z}_5, the notation 3^{-1} simply means the inverse of 3, which means the number s, such that 3s\equiv 1\pmod{5}. In this case, we have s=2, since 3\cdot 2 = 6, which is 1 mod 5. Short answer, 3^{-1}=2.

    Generally, when working mod n, the number x has an inverse, iff x and n are relatively prime.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by kablooey View Post
    Quote Originally Posted by Opalg View Post
    In \mathbb{Z}_n, the inverse x^{-1} of x is an element of \mathbb{Z}_n such that if you multiply it by x (mod n), the answer is 1.
    What is the logic and/or intuition behind this rule?
    It's the natural definition of an an inverse. The inverse of any number is what you multiply the number by in order to get 1. Like the inverse of 7 is \tfrac17, because 7*\tfrac17 = 1. The only difference from ordinary arithmetic is that in \mathbb{Z}_n, "multiplication" means "multiplication mod n".
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,450
    Thanks
    790
    it all has to do with the prime factorization of n.

    for example, in Z4, 2 is what is called a "zero divisor", because (2)(2) = 4 = 0 (mod 4).

    and, as you might expect, "dividing by a zero divisor" is like "dividing by 0", it just doesn't work.

    but 5 is prime, which means that every non-zero element of Z5 will have an inverse:

    (1)(1) = 1 (mod 5), so 1 is it's own inverse.
    (2)(3) = 6 = 1 (mod 5) so 2 and 3 are inverses.
    (4)(4) = 16 = 1 (mod 5), so 4 is it's own inverse.

    that is: "1/2" can be defined in Z5: it is just 3.

    but "1/2" cannot be defined in Z4:

    (1)(2) = 2 (mod 4), which is not 1.
    (2)(2) = 4 = 0 (mod 4), not 1 either.
    (3)(2) = 6 = 2 (mod 4), again, not 1.

    no matter what we multply 2 by in Z4, we never get 1. multiplication by 2 in Z4 is a "one-way street", we can't go backwards and "undo it" .

    suppose we know that 2x = 2 (mod 4). well, we can't say for sure if x = 1, or if x = 3. this happens precisely because 2 and 4 share a common factor.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    Quote Originally Posted by kablooey View Post
    What is the logic and/or intuition behind this rule?
    I'm not sure exactly what you're asking. If you're asking what the use is of this notion of inverse, then the original post is a fine example: You can use inverses to solve modular equations such as 3x = 4 (mod 5).

    You know that 2*3 = 1 (mod 5). Hence you can multiply both sides of the congruence 3x = 4 (mod 5) by 2 to obtain

    2*3x = 8 (mod 5),

    so since 2*3 = 1 (mod 5), and since 8 = 3 (mod 5), this reduces to

    1*x = 3 (mod 5), or

    x = 3 (mod 5).

    Thus knowing the inverse of 3 helped solve the problem.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. proof that square root of a negative does not exist.
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: September 7th 2011, 06:03 PM
  2. Modular arithmetic
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: May 3rd 2011, 01:37 PM
  3. Modular arithmetic
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: March 21st 2011, 03:40 AM
  4. Replies: 1
    Last Post: October 2nd 2007, 07:15 PM
  5. negative fractions to negative exponents
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 24th 2006, 06:02 PM

Search Tags


/mathhelpforum @mathhelpforum