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Muddled Equations

  1. Oct 16, 2006 #1
    This one is experimental. I'm trying to get at a notation that allows me to investigate equations where there are multiple modular operations form different bases.

    Define

    A natural variable x or a can take on any natural value and zero if the context allows.

    And then

    1) [itex]\underline{x}[/itex] deal with x as a constant

    2) [itex]x_{[\underline{a}]}=x mod \underline{a}[/itex] equals the set of residues modulo a

    3) [itex]x_{[a]}=x mod a[/itex] equals the set of natural numbers since a is assumed to be a variable in this notation

    4) [itex]\underline x_{[\underline{a}]}=\underline{x} mod \underline{a}[/itex] is a generic way of writing one specific residue of a, the following are equivalent 3 mod 4, 7 mod 4, etc.

    5) [tex]\underline x_{[a]}=\underline{x} mod a[/itex] is equivalent to the set [itex]\{0,1,...,\underline{x}\}[/itex].


    operations and multiple base maps.

    Consider [itex]x_{[\underline{a},\underline{b}]} =_{[\underline{c}]} y_{[\underline{d}]}[/itex]

    It says first evaluate x mod a then mod b, and evaluate y mod d, then solve the resulting equation mod c.

    Multiplication now requires a symbol, juxtiposition no longer works. [itex]x \times_{[\underline{a}]} y = \underline{c}[/itex] means x and y are natural variables so find natural numbers that can be multiplied mod a to yield c. Since the base quantifier is missing on the equals sign there may not be a solution. Whereas [itex]x \times_{[\underline{a}]} y =_{[\underline{a}]} \underline{c}[/itex] means solkve the equation mod a as well.

    The same works for addition [itex]x +_{[\underline{a}]} y = \underline{c}[/itex]

    The obvious question is why make such a notation? Are there any problems where solving muddled equations is necessary? Does the notation help facilitate either the understanding of the problems or their solution?
     
    Last edited: Oct 16, 2006
  2. jcsd
  3. Oct 16, 2006 #2
    Implications of multiple base operations.

    1) [itex]x_{[\underline{a},\underline{b}]}=x[/itex] if x<a and x<b

    2) [itex]x_{[\underline{a},\underline{b}]}=x mod \underline{a}[/itex] is x moda < b

    and of course it is always true that it equals x mod a mod b, but the operations may not actually alter x. In fact it must be true that if a and b are realtively prime then there exist x such that x mod a mod b is equal to every residue of b. But if a and b have common divisors then classes of residues in mod a are mapped to specific classes mod b. for instance consider a=6 and b=3, then

    N -> 1 2 3 4 5 6 7 8 9 10 11 12

    mod 6 1 2 3 4 5 0 1 2 3 4 5 0

    mod 3 1 2 0 1 2 0 1 2 0 1 2 0

    {0,3} mod 6 goes to {0} mod 3
    {1,4} mod 6 goes to {1} mod 3
    {2,5} mod 6 goes to {2} mod 3


    Order of operations matters since taking mod 3 first gives

    N -> 1 2 3 4 5 6 7 8 9 10 11 12

    mod 3 1 2 0 1 2 0 1 2 0 1 2 0

    mod 6 1 2 0 1 2 0 1 2 0 1 2 0

    {0} mod 3 goes to {0} mod 6
    {1} mod 3 goes to {1} mod 6
    {2} mod 3 goes to {2} mod 6

    The notion reminds me of permutations as taught in a first course in Abstract Algebra.
     
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