MHB Conservation of Symbols Law of Algebra

AI Thread Summary
The discussion proposes a "Conservation of Symbols" Law in mathematics, emphasizing that symbols in equations and expressions should not be omitted or altered without valid mathematical justification. It highlights the importance of maintaining the integrity of symbols during algebraic manipulations to prevent errors. The conversation suggests that while symbols can be merged or converted, their underlying "units" must remain consistent, similar to principles in physics. The goal is to enhance students' understanding of detail-oriented approaches in algebra. Ultimately, the focus is on reinforcing the significance of symbol conservation in mathematical derivations.
Ackbach
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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!
 
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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 $e$.
 
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