Exploring Muddled Equations & Modular Operations

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


operations and multiple base maps.

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

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. $x \times_{[\underline{a}]} y = \underline{c}$ 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.

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

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.

 

[tacman]

This notation does seem to have potential for exploring equations with multiple modular operations from different bases. It allows for a clearer representation of the operations being performed and can potentially make it easier to understand the problem at hand. Additionally, it may also make it easier to find solutions to these types of equations, as it provides a clear structure for evaluating each operation.

As for the use of this notation in practical applications, it may be useful in certain mathematical problems or in computer science and coding, where modular operations are commonly used. It could also have applications in cryptography, where solving equations with modular operations is crucial.

However, it is important to note that this notation is still experimental and may not be widely used or accepted yet. It may also have limitations in certain situations, and further research and testing may be necessary to fully understand its potential and usefulness. But overall, it seems like a promising tool for exploring and solving equations with multiple modular operations.