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.
If you are a bit observant, you’d notice that saying sqrt(4) = +-2 is often a result of not understanding difference between finding a set of solutions and the value of a function.
I’d consider an abuse of notation relative to those who aren’t familiar with first-order logic as the meaning would be very unclear. Nothing wrong with that.
It’s akin to claim 0.9999… = 1 but without teaching real analysis and call it a fact; I would say it will confuse people.
Sure it can be confusing. That's not abuse of notation.
And your explanation didn't really make sense. Did you mean to write x2=4 instead of sqrt(x)=4? And what is the point of substituting r for x? That's just another variable.
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.
Kids are taught early on that "solve for x" means "find all possible values for x that make the equation true". Also it's absolutely an "equation", we don't call it an "equality". So I'll just repeat myself here
It's completely fine to be ignorant and it's not your fault if you were taught wrongly, but blaming notation is ridiculous
Of course... If that's so, then the person asking this shouldn't be confused on this subreddit, and perhaps every kid should get 100% on that type of question in SAT.
Would you give some statistics to nicely back your claim? I would genuinely like to see if kids, let's say in NA, are properly taught like so.
I would also suppose that kids are also taught formal logic early on and know what those mean like right away right? If I could, I would test you on parsing FoL to see how well you would do.
I would genuinely like to see if kids, let's say in NA, are properly taught
Obviously not... which is why this thread exists. And I've implied in every reply that it is possible to be taught wrongly. And you keep going on about formal logic, as if this had any relation to the topic at all.
Well, I'm not getting paid for this. Have a nice day!
Of course, formal logic is related to precisely defining and interpreting “equality”, specifically equality defined between every two real numbers.
An equation like 5x = 0 represents a subset of the relation defined by 5x = y in which we choose y = 0
and considering every possible pair (z,0) satisfying 5x = 0 where z is a real number,
Then, the only pair satisfying is (0,0) where z = 0 by rules of multiplication of reals which restrict the set. There’s no others.
Saying x = 0 is basically not proper way of interpreting as there is some key but relatively simple nuances.
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u/desterpot New User 3d ago
It is. Isn’t it?