Why is an equation that originally had a division through 0, still not defined in that point - after it was removed

Question

Relearning the basics. When I have the equation: $$2x^2+2x+2=2x^2+2x+2$$ $$0=0.$$ I can say that x is any number.

But when i simplify this equation from the following one: $$2x(x+1+ \frac{1}{x})=2(x^2+x)+2$$ $$\iff 2x^2+2x+2=2x^2+2x+2.$$ $$\iff0=0.$$ I suddenly can't say that x can equal 0, as it isn't defined in my first equation. Why does it matter what my original equation was? Am i destroying information by changing the terms?

Answer 1

It is actually $$ 2x\left( x + 1 + \frac{1}{x} \right) = 2 (x^2+x)+2 \wedge x \ne 0 \iff \\ 2x^2+2x+2=2x^2+2x+2 \wedge x \ne 0 \iff \\ 0=0 \wedge x \ne 0 \iff \\ x \ne 0 $$

Answer 2

You're not destroying information, instead you're just leaving out an assumption you've made: The first equivalence only holds if x does not equal 0 - otherwise the first expression is undefined.

So rather than destroying information, you're assuming that x does not equal zero and then conclude from that, through arithmetic equivalences, that x can take on any value - except, of course, 0.

mvw gave a nice rigorous mathematical notation of what I described.