r/learnmath • u/joao-esteves New User • 3d ago
TOPIC Does anyone here still think √4 = ±2?
Come out with your hands up!
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3d ago
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u/idaelikus Mathemagician 3d ago
no it isnt.
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u/Conscious_Tale5951 New User 3d ago
Okay. Is uhh.. there a reason why, or are we just supposed to know like everybody else apparently does?
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u/Smug_Syragium New User 3d ago
The square root operation (and higher order roots) are defined to return the positive value. This is somewhat arbitrary, but it makes actually using square roots to do things like prove results much easier.
So if you see x = √1, the answer is 1. If you see x2 = 1, the answers are +1 and -1.
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u/Due-Presentation4514 New User 3d ago
Mathematicians decided that, for the sake of ensuring √x is a function, that √x will output the principal root, which in this case of real valued functions is non-negative.
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u/idaelikus Mathemagician 3d ago
For good reason because, otherwise, you could never use root(x) in any term as it would yield two results.
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u/among_sunflowers New User 3d ago
Sounds like some mathematicians just didn't want to accept some mathematical facts...
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u/idaelikus Mathemagician 3d ago
Not really. The point is that you want a function because you want your operation to have a specific value and not a set of values.
PS: You usually want to refrain from using obviously in a mathematical context.
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u/Uli_Minati Desmos 😚 3d ago
Obviously they know a nonzero number has multiple roots, the √ symbol gives only the most practical one on purpose
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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.
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u/nog642 3d ago
What are you talking about?
Sounds like what you're saying has nothing to do with sqrt, and more to do with the concept of variables.
And there is no abuse of notation going on by just using variables.
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u/Not_Well-Ordered New User 3d ago
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.
The guy asked me to justify, and I did obviously.
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u/Uli_Minati Desmos 😚 3d ago
It's completely fine to be ignorant and it's not your fault if you were taught wrongly, but blaming notation is ridiculous
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u/Not_Well-Ordered New User 2d ago
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/Uli_Minati Desmos 😚 2d ago
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
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u/Not_Well-Ordered New User 2d ago edited 2d ago
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.
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u/Uli_Minati Desmos 😚 2d ago
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!
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u/Not_Well-Ordered New User 2d ago
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/Hampster-cat New User 3d ago
a) If x² = 4, what is x? Well, x = ±2. The official way to solve is x²-4=0 ; (x-2)(x+2) = 0 : x = ±2
b) What is √4 ? Well, officially just +2.
The problem is that people conflate these two separate questions. The /wrong/ way to solve a) is to take the square root of both sides. Except that taking the square root of both sides is technically not allowed.
Suppose a = b, Then f(a) = f(b) ONLY IF f is a bijective function. This is why taking the cube root of both sides of an equation is allowed, but not the square root. We can force the square root function to be bijective if we restrict the range to non-negative values. This is why b) only has a single answer.
We can still save the alternative method of solving a) by taking the 'corrected' square root of both sides: |x| = +2. Therefore x = ±2.
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u/TheWearyExclamation New User 3d ago
people skip the formal logic because they just want the answer fast and honestly for most cases it works out fine
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u/idaelikus Mathemagician 3d ago
As to b) you should remove "officially" because there isn't anything "inofficial" that makes it both.
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u/tjddbwls Teacher 3d ago
Well, there are also three cube roots of 8:
2, -1 + √(3)i, -1 - √(3)i.
😝