Can Muddled Equations and Modular Operations Enhance Problem Solving?

[Playdo]

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) \underline{x} deal with x as a constant

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

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

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

5) \underline x_{[a]}=\underline{x} mod a[/itex] is equivalent to the set \{0,1,...,\underline{x}\}.<br /> <br /> <br /> operations and multiple base maps.<br /> <br /> Consider x_{[\underline{a},\underline{b}]} =_{[\underline{c}]} y_{[\underline{d}]}<br /> <br /> It says first evaluate x mod a then mod b, and evaluate y mod d, then solve the resulting equation mod c.<br /> <br /> Multiplication now requires a symbol, juxtiposition no longer works. x \times_{[\underline{a}]} y = \underline{c} 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 x \times_{[\underline{a}]} y =_{[\underline{a}]} \underline{c} means solkve the equation mod a as well.<br /> <br /> The same works for addition x +_{[\underline{a}]} y = \underline{c}<br /> <br /> 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?

 

[Playdo]

Implications of multiple base operations.

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

2) x_{[\underline{a},\underline{b}]}=x mod \underline{a} 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.