MHB Conservation of Symbols Law of Algebra

[Ackbach]

So, I'm trying to think of some way to formulate a "Conservation of Symbols" Law of Mathematics. Something like this:

By "symbol" I mean any atomic variable, constant, digit, operator, bracket, etc., that is present in a syntactically correct expression, equation, or inequality. So the beautiful $e^{i\pi}+1=0$ has precisely 7 symbols: $e, i, \pi, +, 1, =,$ and $0$. The equation $1.25+x=4.67$ has exactly 11 symbols in it: $1, ., 2, 5, +, x, =, 4, ., 6,$ and $7$.

It is unlawful to omit or introduce any symbol or combination of symbols from one line of a derivation to another, unless it is specifically allowed by a valid and relevant mathematical property. That is, symbols are conserved in mathematical derivations.

I'm posting in the algebra forum, because it seems to me that algebra is by far and away the area of mathematics most prone to violations of this rule.

So, my question is this: how could this law be sharpened? Also, how could it be made useful to students? My goal is to help students understand the importance of attention to detail in algebraic manipulations.

Thank you!

 

[I like Serena]

It seems to me it's not so much symbols that are conserved, but "units".
Algebraic rules allow merging and conversion of symbols, but their "unit" remains, just like in physics.

For instance $e+2e = 3e$ shows how the symbols are merged, but the $e$ doesn't go away.
It's only through $\pi\ln (3e) = (\ln 3 + 1)\pi$ that $e$ can disappear, but only because the final unit is $\pi$ and does not include $e$.