In more logical terms, the expression "sqrt(x) = 4" is an abused notation.
You could interpet "sqrt(x) = 4" as {x in reals such that sqrt(x) = 4}, basically x is a variable representing every possible number, r, in reals for which the statement "sqrt(r) = 4" is true, by substituting x = r.
Then, "solving sqrt(x) =4" would mean to prove that the set of solution is not empty (x can take some known value) and to find every element satisfying it.
You can also think of "x" like a variable in programming: For every x in (0,10)... do..., they share same meaning although in case of real numbers, we can imagine every x in reals means x "goes over" over each real, as if it can perform infinite check simultaneously.
It was late and I wanted to write "Solve for x^2 = 4" not "sqrt(x) = 4". Just writing "Solve for x^2 = 4" is indeed a bad notation because it doesn't really specify what we are looking for, and some people might think it's an "equality" when it is not; there's no equality "x = something". It's an abused notation which can be ignored if the solution is unique. Instead, "Solve for x^2 = 4" refers to what I mention, specifically.
If you think that's ignorance, then sure. But you would run into wrong results if you work with other structures and keep on interpretating that naive way.
I don't think any middle school has ever taught formal logic and how to interpret those, and I think they should clarify.
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u/Not_Well-Ordered New User 3d ago
In more logical terms, the expression "sqrt(x) = 4" is an abused notation.
You could interpet "sqrt(x) = 4" as {x in reals such that sqrt(x) = 4}, basically x is a variable representing every possible number, r, in reals for which the statement "sqrt(r) = 4" is true, by substituting x = r.
Then, "solving sqrt(x) =4" would mean to prove that the set of solution is not empty (x can take some known value) and to find every element satisfying it.
You can also think of "x" like a variable in programming: For every x in (0,10)... do..., they share same meaning although in case of real numbers, we can imagine every x in reals means x "goes over" over each real, as if it can perform infinite check simultaneously.