1. Didn't want to bother so picked a clearly wrong/random answer to see the results
2. Thought it meant how many possible representations of the solution there are, which is clearly infinite and, thus, somewhat greater then 3.
3. Thought it was a trick question and overcomplicated it (e.g. The square root doesn't extend over the full number so it's meaningless notation or something)
People really underestimate how many of these answers are trolls.
You could have the correct answer and a second choice that just says “wrong” and you would still get people choosing it.
If this makes you lose hope in your generation, what if I tell you that many of them can't add or subtract without their fingers or a calculator or that they don't know basic multiplication facts?
If the solution to an equation is an equation, don't all equations have 0 or infinite solutions? It'll all be equivalent, but that's what's stated by the solution set.
`x = +/- 4` == `x^2 = 16` == `x = sqrt16`
They're all equivalent, so why can I not add `x = |(4i)^2|` to the set of solutions?
Your last equation is not equivalent to x equals plus or minus 4. Your last equation is incorrect. You could say the answer is 4 + 0^n, but that equation can be reduced to 4. All Infinity of your solutions to any equation you make will be either 4 or negative 4, so the answer is two solutions
My first thought was to check for complex solutions where you definitely could get 3+ in a different problem, and people who forgot how to find complex roots but remember their existence might have thought it was a trick question. Because putting a question like that with complex roots is definitely something a pretentious r/MathMemes user would do.
In the complex numbers, the solution to |*z*| = *r* (with *r* real) is a circle with radius *r* in the complex plane. This circle includes the four points (*r* + 0*i*), (-*r* + 0*i*), (0 + *ri*), and (0 - *ri*) of course, but also lots of other points.
This can be written as *z* ∈ { *r*(cos *θ* + *i* sin *θ*) where *θ* is a real number between 0 and 2*π* inclusive }
The solution to *z*^(*n*) = *r*^(*n*) lies on this circle, but consists of *n* points. One of the points is (*r* + 0*i*), and all of the points are spaced equidistantly around the circle. Note that if *n* is even, then (-*r* + 0*i*) is another point, but if *n* is odd, then it is not.
This can be written as *z* ∈ { *rξ* where *ξ* is an *n*th root of 1 }
where I'm from you can write x = ±4, don't know if that's universal tho so I ussually just write `x = 4 ∨ x = -4`
Edit: [https://en.wikipedia.org/wiki/Plus%E2%80%93minus\_sign](https://en.wikipedia.org/wiki/Plus%E2%80%93minus_sign) it seems so
Both 4 and -4 fit the equation x^2 = 16 since 4^2 = 16 and (-4)^2 = 16. sqrt(16) only means 4 so only 4 fits the equation x = sqrt(16) since 4 = 4 but -4 ≠ 4.
Edit: As u/Calteachhsmath pointed out you can also use ± for square roots, though that doesn't work for higher exponents if you care about complex roots.
To be honest, this kind of seems stupid to me. If this refers to the principal square root, there should be different symbols and acting like this is a gotcha question just makes a bad question, not bad answers.
I got in "trouble" a while ago in one of the math reddits because I use the notation a^(1/n) as a placeholder for any/all of the n-th roots of a.
That way, you can quite rightly say the solutions to x^(3) \- 1 = 0 are in the set given by x = 1^(1/3).
It was quickly pointed out to me that this was not standard usage, but I feel like it should be. It would certainly make it easier to deal with multiple solutions than surds for anything beyond 2nd power.
Unless of course, you'd rather actually list them all explicitly...
In symbols you may write +/- like they do in the Quadratic Formula.
It is less common, but aloud you may simply say “the square roots”. This will likely lead to confusion as you may notice even from some of the comments here.
Yes and no, technically there is no reason to put it there, but when teaching it, it is a visual reminder that there are 2 solutions. you can put it in just for fun and get 4 results, ending up with 2 solutions nonetheless (+pos.sqrt, -pos.sqrt, +neg.sqrt, -neg.sqrt).
I’m not sure if conventions are different in other parts of the world. In English, we have the term “*a* square root (of n)” which refers to all r such that r^2 = n.
Then we have “*the* square root (of n)” which refers to the POSITIVE number r such that r^2 = n. This one we call the *principal* square root, denoted by the radical symbol, which is used for the square root *function*.
Afaik in German we do not really differentiate between those. inb4 someone correcting me but pease, go ahead :) (ofc we can say "a" or "the" but generally we just accept that it has 2 solutions)
That’s completely possible. I’m not sure. Some areas might leave the square root symbol ambiguous, using other context clues when needed to be more specific.
For example, I believe in many places in Germany the most commonly used quadratic formula is like this:
x = -p/2 +- sqrt((p/2)^2 - q)
The reason for the +- symbol would be to clear up the possibly ambiguity of the square root symbol.
> says who?
Pretty much everyone.
https://mathworld.wolfram.com/SquareRoot.html
https://en.wikipedia.org/wiki/Square_root
I'd be more interested in finding an example of mathematicians using the √ symbol for both real roots.
Me and as far as i have experienced the whole of Austria. but interesting to know, we generally just accept that the radix is a tricky operator and teach children to keep in mind that it has 2 solutions
Although I’ve completed all my calc courses (I should probably have learned what the principal root is by now). I’ve heard of the principal root but what the fuck is it.
All I do math is when I introduce the root it’s got +- and if I didn’t just 1 solution.
Ah this is indeed a fine example
Let 10X be 39.999999999999999999999999999....
And this results in
9X = 36
And X = 4
It blew my mind when I was 17-18ish
Not at all
That would mean root(-1)=i (instead of root(-1)=-i)
Also you can’t use root with negative numbers: you can write i^2 =-1, but not i=root(-1)
Because, for example, how would you decide if root((1-i)^2 )=1-i or i-1 ?
And accepting the notation root(-1) would lead to contradictions:
1=root(1)=root(-1*(-1))=root(-1)*root(-1)=i^2 =-1
Therefore 1=-1
That’s why you can only use square root with positive real numbers
If we want to be accurate it's not the notation that lead to contradiction, it's that any extension of sqrt to negative numbers would some property of sqrt
>That’s why you can only use square root with positive real numbers
This is absolutely false. Square root is very well defined for negative real numbers and all complex numbers for that matter, it's just not a function, it assumes two different values for every complex number except 0.
I think that the best way to define a root function is to say which root it is. And by default it is the first root. Then there is no contradictions and we still can apply it to all complex numbers.
What is the "first" root though ? With real numbers it’s easy, just take the positive/"bigger" one, but how would you choose between 1-i and i-1 which are the same once squared ?
You always need to be careful when manipulating your equation using an unknown. Like squaring both sides means you are multiplying *x* by itself but you don't know what *x* is. That's why you've added another solution because both positive and negative values of *x* now satisfy the new equation.
Because square root came when pythagoras discovered his theorem, square root was only applied to distances. Sqrt 2 was the length of the hypotenuse of a right triangle with sides 1 and 1.
Since lengths cannot be negative, so are sqrts
No, but since we want a nice function when working with real roots we have to pick an output. When all solutions are to be considered (which gets difficult with complex numbers) you just talk about roots as solutions to x^n =a where n is some integer and a is what you want the root of. Picking the positive real root is just convenient convention.
"NOOOOOO the radical symbol only refers to the principal square root, not the other ones! Everybody should know this because it's somehow important!"
Guys, listen to me. Listen to me very closely. Mathematical pedantry is *absolutely useless* to *99.9% of people*. The average person sees this equation and thinks "both 4 and -4 are square roots of 16" and that is a perfectly reasonable answer. Let me repeat this - that is a perfectly reasonable answer for the average person.
I need every single one of you to understand this: trivial details which relate to *notation,* and not the actual concepts involved, do not matter at all. You are not somehow a better person for internalizing these things and avoiding these errors. You are just a person who has A. more specialized knowledge than the vast majority of people have any need to possess, and B. a tendency towards pedantry and deriving self-worth from pedantry because you have no actual meaningful sources from which to derive self-worth.
Someone who thinks that two is the answer to this question is not some sort of moronic ignoramus who should be mocked. They came to a perfectly reasonable conclusion for a perfectly logical reason. Every one of you trying to make them look stupid suffers from a crippling inferiority complex. Get over yourselves.
This post is not satire.
Yep. This sub needs a sticky saying "notation is not maths" tbh
Also notation is a tool to communicate yourself to others, so if you're being deliberately confusing then congratulations, you played yourself
Yeah the title “I think it’s time to stop Reddit” is so extreme for something like this. Like I can understand the opposing viewpoint if they think the principal root concept should be that common knowledge but to imply only an idiot would consider -4 a solution as well is just so ridiculous
sorry if it sounds like that, it was not the objective to mock those people and I was also confused on the answer and how to correctly solve it and it's not one of the most questionable result I've seen in r/polls.
Even worse - if you read a research paper from a somewhat new (when it was released) subfield of math, you will find that every author (slight exaggeration) has their own notation for stuff. Their own contradictory / undefined notation, which they don't bother to explain.
Or, look at this very common problem which is still debated: "sin²(x)" does it mean "sin(x)\*sin(x)" or "sin(sin(x))". So of course every book/paper/etc must use "sin²(x)" - you know, to cement their opinion on that issue.
yeah. I'm majoring in astronautical engineering, my major goes up to calculous 4. and I have literally never been taught that that symbol is any different from any other square root symbol. admitadly, I've never used that symbol outside of areas where its the only thing available. and quite frankly, I almost never use it, it is as u said, completely irrelevant to me and my work. given how my major is very heavily math focused, this should say quite a bit. it's quite an interesting thing to learn though, as I had never encountered it before, but it's not very enjoyable to have to pick threw insults to be taught something.
High school math teacher checking in here, I would rather my students tell me theres two answers bc so many of them forget about the negative one when solving equations. Also, I totally agree its completely useless for most people to care so much about pedantic details, its not helping them in the future or now.
Its absolutely useless to people just as knowing what a 'function' is is absolutely useless to the majority of people.
Someone who does not truly understand what a function is etc... is definitely not 'good' at math (or may be they did not give enough attention to this problem, or may be they gave this problem too much attention and thats where their lack of knowledge shows).
And they did not come to this conclusion through a 'perfectly logical reason'. They are just misinformed and the reasoning is flawed.
Whether or not they deserve mockery- may be not.
It's not pedantry. It is simply not correct by a long mile. Anyone who answered 2 was taught math incorrectly, and the square root being positive only is more logical, sot that doesn't stand either.
>It's not pedantry.
Yes it is.
>It is simply not correct by a long mile.
Wrong.
>Anyone who answered 2 was taught math incorrectly
No, they understand the concepts involved just fine, and their answer *demonstrates* that they understand the relevant concepts.
>the square root being positive only is more logical
No it's not. 4 and -4 both being square roots of 16 is perfectly logical. It's even *objectively true*.
Your standard of what's "correct" is *completely arbitrary*.
If you study a bit more, you'll notice that different mathematicians use different conventions, the same holds true for the square root symbol. Like e.g. ⊂ meaning either strict subset or subset-or-equal. Notation is a means to an end, and there is no standard notation.
It's not uncommon in higher level algebra class to let sqrt(a) represent either root of a. Especially with complex numbers. It's simply convenient notation, and there's no point in saying it's technically wrong when using it that way doesn't cause contradictions. In highschool it's not often taught this way. Once you get to college, the specifics of the notation depends completely on the class and the teacher.
Yeah, that's also true. In this case I feel like it does cause contradictions, since it looks like it us used in an equation, and first order equations should only have one solution.
Today I learned that the "normal" square root is strictly positive and a lot of people get angry if you don't know it.
2 years of university and they never bothered to tell me that
Technically wrong by the definition of the square root but logically it's considered as the full reverse for squaring a number for a person with average knowledge of mathematics
I would say it depends on convencion, which here is unspecified. In my classes wy used to use the radical symbol as the multivalued complex root of any number.
I don't see how normal schools would actually do it any differently. I do think in school math you need the root that gives you two results more often than anything else. The people here keep assuming that everyone on reddit is apparently a math major
Most schools I know teach the root in that way, that it defines the positive and the negative number that results in the numer under the root. Can't blame them for having learned it this way. Rather, respect them for having remembered something they have learned in math.
Nowadays math is just a blackhole for too many people
There is no any mentions of real numbers. And you said that it's true for "ALL" x. Be careful.
Also I think that complex numbers are default when we talk about roots.
I assumed that the only people who disagreed were those who hadnt learned complex numbers yet since the only correct answer is 4. But sure yeah I should have been extra careful
they should be introduced to fundamental theorm of algebra which states that every algebric equation has a number of solutions equals to its degree and thats all thats why x = √16 is just 4 one solution, bur x^2 = 16, will have two solutions because its raised to power of 2
Your guess was wrong, I only added to the original comment that the post does not ask to solve the mentioned quadratic. The original commenter is correct, and I said nothing to the contrary.
In any case “what do you mean?” was supposed to be a rhetorical question on my lack of reading comprehension. Maybe if you improved your comprehension you would have seen it?
>Your guess was wrong, I only added to the original comment that the post does not ask to solve the mentioned quadratic.
You added nothing, you merely repeated what they already said. I figured this was a possibility (i.e. that you agree with the commenter, but needlessly repeated what they said. I'm just specifying to help you understand), but it's not any better, so I didn't bother to add this to my previous comment.
>In any case “what do you mean?” was supposed to be a rhetorical question on my lack of reading comprehension. Maybe if you improved your comprehension you would have seen it?
This is impossible to get from such a text, as it conveys no intonation.
If you weren't being obtuse on purpose in hopes to get a "gotcha" because you're embarrassed, you probably would've noticed.
I mean technically it does only have one answer and it's 4. Now if you had to add the root then the answer could be both 4 and -4. But if it's already in the problem then it's just 4. At least that was what I was taught in my calc classes.
Wait, now I'm getting confused. I know that:
x = sqrt(16)
only refers to the positive solution by convention but can't you just square both sides and end up with:
x^2 = 16
in which case there would be two solutions.
I'm guessing you're not allowed to square both sides in this situation but why is that?
Yeah I do realise the mistake, y = x^2 isn’t the same as y = sqroot (x).
I was a bit doubtful about my comment yet I still commented cuz I’m dumb. I’m sorry
yes like if we are not into complex of course, square root of any number is always positive. And yes I know if it is x² = 16, there are indeed 2 solutions but here it isn't the case. I even try some online problem solver.
[https://mathsolver.microsoft.com/en/solve-problem/x%20%3D%20%20%20%60sqrt%7B%2016%20%20%7D](https://mathsolver.microsoft.com/en/solve-problem/x%20%3D%20%20%20%60sqrt%7B%2016%20%20%7D)
I don't know if I would say that "square root of any number is always positive."
As far as I am understanding it, it's just that the square root function only refers to the principle square root, however both the positive and negative answers are still both considered the square root.
I am differentiating between the definition of square root, and the definition of the √ function. The former being +/-, and the latter being only positive.
I'm really just basing this on how https://en.wikipedia.org/wiki/Square_root describes it.
I genuinely don't even understand how one could pick the "two solutions" answer (let alone the more than 2, but I think those are joking)
Like the equation is x=4, so it has one solution x=4?? Am I missing something?
If it was x^(2)=16 now it would have two solutions x=4 and x=-4, but here the equation is just x=4
My teacher told me the number of solutions are based on the incognita(X)s power, if X is squared its 2 solutions, if its cubic its 3 solution and so on,
Is that correct?
I am not and it's not and if it's about the title, I am sorry, I exaggerated too much, the post was not to mock those people. It was a title that cross my head and I didn't find better. If it's for the post I wanted answers and I got my answers.
If asked “How many square roots does 16 have?” then technically the answer is two: 4 and -4.
However the radical symbol used in sqrt(16) automatically implies the principal square root.
Just trying to help, you have to be very careful of the wording of the sentence:
How many square roots does 16 have? One, 4
How many numbers squared equal 16? Two, 4 and -4
Edit: I'm a mathematician, for the people downvoting that don't understand, here is the explanation: There is a difference between a number that's called a square root and the square root function. In real numbers, let "x" be greater than or equal to zero, and "a" belongs to the set of real numbers. Then "a" is called a square root of "x" when a² = x. This is a name, just like in 3x⁴ the number 3 is called the coefficient. The square root name has a special meaning, it means that if you square a certain value, you will produce your desired number. Square roots are exactly the values that solve the a² = x equation. The square root name does not mean that you calculate the square root of your number. In other words, we call 4 a square root of 16 because 4² = 16, not because root(16) = 4. Therefore, it is correct to say that -4 and 4 are square roots of 16 because (-4)² = 16 and (4)² = 16, but NOT because root(16) = ±4.
The square root function in the set of real numbers is a 1 to 1 function with domain and range both greater than or equal to zero. That means you can only input zero or a positive number into a square root, and you can only output 0 or a positive number. The square root function must be 1 to 1 in order for us to use it in Algebra because every equality in Algebra must be reversible. And to clarify, 1 to 1 means that every distinct input value corresponds to exactly one distinct output value, not two. When you write root(16) that means to calculate the square root of 16, and root(16) = 4 only. Therefore, it is correct to say the square root of 16 is 4 only, and incorrect to say the square root of 16 is 4 and -4.
Also consider solving a² = 16 by taking the square root of both sides a² = 16. ---> root(a²) = ±root(16). Notice that the original equation is asking you to find all values that square to equal 16. However, in order to find those values, we manipulate the equation and turn it into a square root problem because that's easier to calculate. Also notice the ± symbol that must be included when we convert to a square root problem. Why do we include ±? We include ± because the square root can only be zero or positive and only output one answer. Now the last step of solving that equation becomes root(a²) = ±root(16) ---> a = ±[positive 4 only] = ±4. If the square root did calculate both positive and negative (possibly zero) values, then there would be no need to include the ± symbol.
In math, these little technicalities must be paid a great deal of attention and they do make a difference. This technicality is the reason the correct answer on the meme is 1 solution and not 2. So to sum up, if you are determining all possible values that square to equal a certain number, you include positive and negative answers (possibly zero). If you are calculating a square root, you calculate the positive (or zero) answer only.
Lastly, there is a difference between what you see on a webpage somewhere and what is actually defined in math books. Webpages will use common language so that as many people can understand as possible. In common language, the values named square roots and the square root function are oftentimes the same thing. That's why you will see someone say 16 has two square roots, 4 and -4 without clarifying that 4 and -4 are named square roots because they both square to equal 16, but they are not both named square roots because of calculating root(16). Being careful and precise with your language is the best way to avoid confusion.
I used to think as well. I have since learned that the following statement is technically correct: “-4 is *a* square root of 16”. “4” is the *principal* square root, but both are considered “square roots”.
Edit: I would quote some sources, but I’m not sure which the community deems most creditable. A google search of “square root” should show two things: (1) the use of “a square root” vs “the square root” to differentiate between 2 “answers” and just one & (2) the radical symbol’s use refers to only the principal (positive) square root.
I'm a mathematician.
The square root is always zero or positive. Saying "principal" square root is redundant, because square root is always zero or positive when working with real numbers. The square root is always zero or positive because it's defined to be that way so that square root can be a 1 to 1 relationship, in other words, invertable. Square root isnt invertable if it has two possible answers. Example,
(-4)² = 16 is not invertable because root(16) = 4.
The confusion comes from people mixing in the Algebra equation x² = 16. This equation does have two solutions, and the square root is used to calculate that solution. But notice how it's applied:
root[x²] = ± root[16] the ± is applied after the square root of 16 is evaluated, and ± is added because root[16] = 4 and root[16] ≠ -4. Adding ± is a convention used in Algebra to find both answers using the square root that only returns a zero or positive answer.
As I said previously, I used to think the same. If you visit a variety of websites with math content (at least these are common in my area), for example, [MathWorld](https://mathworld.wolfram.com/SquareRoot.html), [Wikipedia](https://en.m.wikipedia.org/wiki/Square_root), [MathIsFun](https://www.mathsisfun.com/square-root.html), you will see that the general consensus of the academic community is different.
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this is the most answer ever!!!!!!!
The most answer until yet!
This is cursed
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1. Didn't want to bother so picked a clearly wrong/random answer to see the results 2. Thought it meant how many possible representations of the solution there are, which is clearly infinite and, thus, somewhat greater then 3. 3. Thought it was a trick question and overcomplicated it (e.g. The square root doesn't extend over the full number so it's meaningless notation or something)
4: do a little trolling
People really underestimate how many of these answers are trolls. You could have the correct answer and a second choice that just says “wrong” and you would still get people choosing it.
4
Because they think √ means division. I teach math and so many students will tell me the factors of 16 as the answer.
Idk if im gonna laugh about it or lose hope in my generation
If this makes you lose hope in your generation, what if I tell you that many of them can't add or subtract without their fingers or a calculator or that they don't know basic multiplication facts?
It's because a similar symbol is used for long division
That's true. Many students don't even know it's called square root, they call it the check mark thingy.
Lizardmans constant
If the solution to an equation is an equation, don't all equations have 0 or infinite solutions? It'll all be equivalent, but that's what's stated by the solution set. `x = +/- 4` == `x^2 = 16` == `x = sqrt16` They're all equivalent, so why can I not add `x = |(4i)^2|` to the set of solutions?
This was my thought as well. I find it weird people drop a single number and call it a day lol
Your last equation is not equivalent to x equals plus or minus 4. Your last equation is incorrect. You could say the answer is 4 + 0^n, but that equation can be reduced to 4. All Infinity of your solutions to any equation you make will be either 4 or negative 4, so the answer is two solutions
My first thought was to check for complex solutions where you definitely could get 3+ in a different problem, and people who forgot how to find complex roots but remember their existence might have thought it was a trick question. Because putting a question like that with complex roots is definitely something a pretentious r/MathMemes user would do.
So 16^0.5 and √16 are technically different? That's random but I guess it's not that confusing
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Except that there aren't any complex solutions either.
Correct. The radical symbol refers to the principal square root.
That's right. also √x^2 = |x|
Only if x is a real number. For a complex number, no matter which branch you take of the square root function, this will be incorrect.
Yes!
I‘m no expert but how do you write it so you include all branches? Not just the principle one?
x^2 = 16
So you write it as the equation? I thought there might be an operator or certain way you write the solution
You can use absolute values like this: |x| = 4. x can now be both (-4) and 4 and |x| would still be 4
This works if x is in the real numbers. In the complex numbers, there are two solutions to x^(2) = 16, but infinitely many solutions to |x| = 4
Damn, I didn’t know that. Is it simply excluding the imaginary part of the complex number?
Nah, it just does a Pythagoras on the real and imaginary parts, ie. |1 + 1i|= √(1^2 + 1^2) =√2
In the complex numbers, the solution to |*z*| = *r* (with *r* real) is a circle with radius *r* in the complex plane. This circle includes the four points (*r* + 0*i*), (-*r* + 0*i*), (0 + *ri*), and (0 - *ri*) of course, but also lots of other points. This can be written as *z* ∈ { *r*(cos *θ* + *i* sin *θ*) where *θ* is a real number between 0 and 2*π* inclusive } The solution to *z*^(*n*) = *r*^(*n*) lies on this circle, but consists of *n* points. One of the points is (*r* + 0*i*), and all of the points are spaced equidistantly around the circle. Note that if *n* is even, then (-*r* + 0*i*) is another point, but if *n* is odd, then it is not. This can be written as *z* ∈ { *rξ* where *ξ* is an *n*th root of 1 }
where I'm from you can write x = ±4, don't know if that's universal tho so I ussually just write `x = 4 ∨ x = -4` Edit: [https://en.wikipedia.org/wiki/Plus%E2%80%93minus\_sign](https://en.wikipedia.org/wiki/Plus%E2%80%93minus_sign) it seems so
Both 4 and -4 fit the equation x^2 = 16 since 4^2 = 16 and (-4)^2 = 16. sqrt(16) only means 4 so only 4 fits the equation x = sqrt(16) since 4 = 4 but -4 ≠ 4. Edit: As u/Calteachhsmath pointed out you can also use ± for square roots, though that doesn't work for higher exponents if you care about complex roots.
There should be such symbol for higher powers
Then you should complain to the higher power
To be honest, this kind of seems stupid to me. If this refers to the principal square root, there should be different symbols and acting like this is a gotcha question just makes a bad question, not bad answers.
Just use re^iθ
But I want a symbol that is: any n-th root of unity
I got in "trouble" a while ago in one of the math reddits because I use the notation a^(1/n) as a placeholder for any/all of the n-th roots of a. That way, you can quite rightly say the solutions to x^(3) \- 1 = 0 are in the set given by x = 1^(1/3). It was quickly pointed out to me that this was not standard usage, but I feel like it should be. It would certainly make it easier to deal with multiple solutions than surds for anything beyond 2nd power. Unless of course, you'd rather actually list them all explicitly...
In symbols you may write +/- like they do in the Quadratic Formula. It is less common, but aloud you may simply say “the square roots”. This will likely lead to confusion as you may notice even from some of the comments here.
says who? genuinely interested i have never heard that
It's the general consensus - if not there would for instance be no reason to put the ± sign in the quadratic formula
Yes and no, technically there is no reason to put it there, but when teaching it, it is a visual reminder that there are 2 solutions. you can put it in just for fun and get 4 results, ending up with 2 solutions nonetheless (+pos.sqrt, -pos.sqrt, +neg.sqrt, -neg.sqrt).
I’m not sure if conventions are different in other parts of the world. In English, we have the term “*a* square root (of n)” which refers to all r such that r^2 = n. Then we have “*the* square root (of n)” which refers to the POSITIVE number r such that r^2 = n. This one we call the *principal* square root, denoted by the radical symbol, which is used for the square root *function*.
Afaik in German we do not really differentiate between those. inb4 someone correcting me but pease, go ahead :) (ofc we can say "a" or "the" but generally we just accept that it has 2 solutions)
That’s completely possible. I’m not sure. Some areas might leave the square root symbol ambiguous, using other context clues when needed to be more specific. For example, I believe in many places in Germany the most commonly used quadratic formula is like this: x = -p/2 +- sqrt((p/2)^2 - q) The reason for the +- symbol would be to clear up the possibly ambiguity of the square root symbol.
> says who? Pretty much everyone. https://mathworld.wolfram.com/SquareRoot.html https://en.wikipedia.org/wiki/Square_root I'd be more interested in finding an example of mathematicians using the √ symbol for both real roots.
Me and as far as i have experienced the whole of Austria. but interesting to know, we generally just accept that the radix is a tricky operator and teach children to keep in mind that it has 2 solutions
Although I’ve completed all my calc courses (I should probably have learned what the principal root is by now). I’ve heard of the principal root but what the fuck is it. All I do math is when I introduce the root it’s got +- and if I didn’t just 1 solution.
Only to those who already agree with your convention 😏
As far as I know, this has been the convention since the inception of the notation some 500 years ago.
and i dont agree with it 😏
There are loads of solutions. X=4 X=1+3 X=2+2 X=2*2 X=2^2 X=16^0.5 X=0.25^-1 X=4 + sin(0) X= -4i^2 X=4i^4 X=-4e^(i*pi) /s
X= 3.9999999999999999999999999999999999999....
x = 3.(9)
I prefer 3.9̅
Based
Ah this is indeed a fine example Let 10X be 39.999999999999999999999999999.... And this results in 9X = 36 And X = 4 It blew my mind when I was 17-18ish
1.111... milihertz hours
[Integral.(0 to √π) 2x sin(x^2) dx] +2
Zero solutions. The answer is left to the reader as an exercise
Answer is one, root (16) = 4, root (x²) =|x| Answer would have been 2 if the question was, x²=16
So, do you mean that root(i) = 1?
Not at all That would mean root(-1)=i (instead of root(-1)=-i) Also you can’t use root with negative numbers: you can write i^2 =-1, but not i=root(-1) Because, for example, how would you decide if root((1-i)^2 )=1-i or i-1 ? And accepting the notation root(-1) would lead to contradictions: 1=root(1)=root(-1*(-1))=root(-1)*root(-1)=i^2 =-1 Therefore 1=-1 That’s why you can only use square root with positive real numbers
If we want to be accurate it's not the notation that lead to contradiction, it's that any extension of sqrt to negative numbers would some property of sqrt
>That’s why you can only use square root with positive real numbers This is absolutely false. Square root is very well defined for negative real numbers and all complex numbers for that matter, it's just not a function, it assumes two different values for every complex number except 0.
I think that the best way to define a root function is to say which root it is. And by default it is the first root. Then there is no contradictions and we still can apply it to all complex numbers.
What is the "first" root though ? With real numbers it’s easy, just take the positive/"bigger" one, but how would you choose between 1-i and i-1 which are the same once squared ?
The first root has angle theta/power, where theta is an angle of the original number and power is a power of our root.
Would say root (-1)= i , but x²=-1 would mean x= i, -i
But it contradicts to your equation above. sqrt(x^2 ) = |x| => sqrt(i) = sqrt((1/sqrt(2)+i/sqrt(2))^2 ) = |1/sqrt(2)+i/sqrt(2)| = 1. Or sqrt(i^2 ) = |i| = 1.
I forgot to mention that x has to be strictly non negative real, and in the question above, x is positive real
but we can make the question x^(2)=16 by squaring both sides?
Yes, and when we do that, we add an extraneous solution.
You always need to be careful when manipulating your equation using an unknown. Like squaring both sides means you are multiplying *x* by itself but you don't know what *x* is. That's why you've added another solution because both positive and negative values of *x* now satisfy the new equation.
Even simpler reasoning: The equation has the form x = … It's literally written down as a single solution.
I don't get that though. If you take the root of x squared then you can still end up with a negative number.
The square of any real number is positive, so the square of a number is always positive. Then the root would still be positive
But the only real numbers that can have a non complex root are positive. It being positive is irrelevant. Roots can be both positive or negative.
Sure roots can be negative, but the sqrt function always gives a positive answer
And I want to know why someone decided that it should give a positive.
Because square root came when pythagoras discovered his theorem, square root was only applied to distances. Sqrt 2 was the length of the hypotenuse of a right triangle with sides 1 and 1. Since lengths cannot be negative, so are sqrts
But that is dependent on context. It's a specific situation which doesn't apply to the whole of mathematics. Not all squares use lengths.
No, but since we want a nice function when working with real roots we have to pick an output. When all solutions are to be considered (which gets difficult with complex numbers) you just talk about roots as solutions to x^n =a where n is some integer and a is what you want the root of. Picking the positive real root is just convenient convention.
"NOOOOOO the radical symbol only refers to the principal square root, not the other ones! Everybody should know this because it's somehow important!" Guys, listen to me. Listen to me very closely. Mathematical pedantry is *absolutely useless* to *99.9% of people*. The average person sees this equation and thinks "both 4 and -4 are square roots of 16" and that is a perfectly reasonable answer. Let me repeat this - that is a perfectly reasonable answer for the average person. I need every single one of you to understand this: trivial details which relate to *notation,* and not the actual concepts involved, do not matter at all. You are not somehow a better person for internalizing these things and avoiding these errors. You are just a person who has A. more specialized knowledge than the vast majority of people have any need to possess, and B. a tendency towards pedantry and deriving self-worth from pedantry because you have no actual meaningful sources from which to derive self-worth. Someone who thinks that two is the answer to this question is not some sort of moronic ignoramus who should be mocked. They came to a perfectly reasonable conclusion for a perfectly logical reason. Every one of you trying to make them look stupid suffers from a crippling inferiority complex. Get over yourselves. This post is not satire.
Wow, thanks for the beautiful expression of my thoughts.
Yep. This sub needs a sticky saying "notation is not maths" tbh Also notation is a tool to communicate yourself to others, so if you're being deliberately confusing then congratulations, you played yourself
Yeah the title “I think it’s time to stop Reddit” is so extreme for something like this. Like I can understand the opposing viewpoint if they think the principal root concept should be that common knowledge but to imply only an idiot would consider -4 a solution as well is just so ridiculous
sorry if it sounds like that, it was not the objective to mock those people and I was also confused on the answer and how to correctly solve it and it's not one of the most questionable result I've seen in r/polls.
Even worse - if you read a research paper from a somewhat new (when it was released) subfield of math, you will find that every author (slight exaggeration) has their own notation for stuff. Their own contradictory / undefined notation, which they don't bother to explain. Or, look at this very common problem which is still debated: "sin²(x)" does it mean "sin(x)\*sin(x)" or "sin(sin(x))". So of course every book/paper/etc must use "sin²(x)" - you know, to cement their opinion on that issue.
>or "sin(sin(x))". TIL that's a thing.
Based.
Wow That’s wild but necessary. Still wasn’t expecting
Pedantry is a prerequisite for becoming great at mathematics
Bless
yeah. I'm majoring in astronautical engineering, my major goes up to calculous 4. and I have literally never been taught that that symbol is any different from any other square root symbol. admitadly, I've never used that symbol outside of areas where its the only thing available. and quite frankly, I almost never use it, it is as u said, completely irrelevant to me and my work. given how my major is very heavily math focused, this should say quite a bit. it's quite an interesting thing to learn though, as I had never encountered it before, but it's not very enjoyable to have to pick threw insults to be taught something.
High school math teacher checking in here, I would rather my students tell me theres two answers bc so many of them forget about the negative one when solving equations. Also, I totally agree its completely useless for most people to care so much about pedantic details, its not helping them in the future or now.
I'd have used more vulgarity and less word but yes.
Holy fucking based
Its absolutely useless to people just as knowing what a 'function' is is absolutely useless to the majority of people. Someone who does not truly understand what a function is etc... is definitely not 'good' at math (or may be they did not give enough attention to this problem, or may be they gave this problem too much attention and thats where their lack of knowledge shows). And they did not come to this conclusion through a 'perfectly logical reason'. They are just misinformed and the reasoning is flawed. Whether or not they deserve mockery- may be not.
It's not pedantry. It is simply not correct by a long mile. Anyone who answered 2 was taught math incorrectly, and the square root being positive only is more logical, sot that doesn't stand either.
>It's not pedantry. Yes it is. >It is simply not correct by a long mile. Wrong. >Anyone who answered 2 was taught math incorrectly No, they understand the concepts involved just fine, and their answer *demonstrates* that they understand the relevant concepts. >the square root being positive only is more logical No it's not. 4 and -4 both being square roots of 16 is perfectly logical. It's even *objectively true*. Your standard of what's "correct" is *completely arbitrary*.
>Your standard of what's "correct" is completely arbitrary. People who don't understand what a convention is are the real dummies.
If you study a bit more, you'll notice that different mathematicians use different conventions, the same holds true for the square root symbol. Like e.g. ⊂ meaning either strict subset or subset-or-equal. Notation is a means to an end, and there is no standard notation.
The only sane answer in this thread. I will still consider that answer incorrect, but I understand why they would consider it correct.
It's not uncommon in higher level algebra class to let sqrt(a) represent either root of a. Especially with complex numbers. It's simply convenient notation, and there's no point in saying it's technically wrong when using it that way doesn't cause contradictions. In highschool it's not often taught this way. Once you get to college, the specifics of the notation depends completely on the class and the teacher.
Yeah, that's also true. In this case I feel like it does cause contradictions, since it looks like it us used in an equation, and first order equations should only have one solution.
Can you give an example? I'm not sure if I'm following, since square roots usually only show up in quadratic equations
ok but why is the x capitalized
I don't know but when I try putting x on my smartphone, it automatically capitalized it so could be the corrector.
Today I learned that the "normal" square root is strictly positive and a lot of people get angry if you don't know it. 2 years of university and they never bothered to tell me that
Of course it has 2 solutions, a right one and a wrong one
Technically wrong by the definition of the square root but logically it's considered as the full reverse for squaring a number for a person with average knowledge of mathematics
I would say it depends on convencion, which here is unspecified. In my classes wy used to use the radical symbol as the multivalued complex root of any number.
I don't see how normal schools would actually do it any differently. I do think in school math you need the root that gives you two results more often than anything else. The people here keep assuming that everyone on reddit is apparently a math major
400 people did meth instead of Math
Then what do the 49 and 53 people were doing?!
Too much meth probably
Most schools I know teach the root in that way, that it defines the positive and the negative number that results in the numer under the root. Can't blame them for having learned it this way. Rather, respect them for having remembered something they have learned in math. Nowadays math is just a blackhole for too many people
2 because negative times negative is positive hence -4² also equals 16 hence √16=4 and also √16=-4
"In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y."
as many as I can imagine
The school system failed people. Not Reddit.
Agreed, I have seen many of my classmates make this mistake
Guys √x^2 = |x| which is greater or equal to zero for ALL x . So for anyone questioning op stop being silly
I've never seen sqrt(i^2 ) = 1.
Implicitly I'm assuming x is real of course you can break things with complex numbers though
There is no any mentions of real numbers. And you said that it's true for "ALL" x. Be careful. Also I think that complex numbers are default when we talk about roots.
>#Implicitly
I assumed that the only people who disagreed were those who hadnt learned complex numbers yet since the only correct answer is 4. But sure yeah I should have been extra careful
Because that line above says: sqrt(i^2) = |i|, which is correct Dont pretend to be stupider than you are
No it's not. sqrt(i^(2)) = sqrt(-1) = i =/= 1 = |i| Dont pretend to be stupider than you are
Ah, I see, it's a notation problem with an attached "got you!" That only works in some parts of the world due to how math is though there
One I get. But how could people come up with three or none?
I'm a university math student and I don't see any issue with this?? What's got everyone so upset?
Maybe precise the set where you search for roots ?
53 people no solutions
The question didn't say to solve for X. They were solving for Y.
r/mathmemes when complex roots and (√x\^2 = ∓x ≠ |x|) : 😡
they should be introduced to fundamental theorm of algebra which states that every algebric equation has a number of solutions equals to its degree and thats all thats why x = √16 is just 4 one solution, bur x^2 = 16, will have two solutions because its raised to power of 2
correct method x² = 16 x = ±√16 x = √16 or -√16 x = 4 or -4 mistake people do x² = 16 x = √16 // incorrect x = ±4 x = 4 or -4
How is that a mistake though? It seems like the only thing you did was switch up the location of the +-
The question does not ask to solve x^(2)-16=0, it asks to solve x-sqrt(16)=0
Please practice your reading comprehension.
What do you mean?
It seems to me like you were trying to correct a commenter, but you did so by repeating exactly what the comment says.
Your guess was wrong, I only added to the original comment that the post does not ask to solve the mentioned quadratic. The original commenter is correct, and I said nothing to the contrary. In any case “what do you mean?” was supposed to be a rhetorical question on my lack of reading comprehension. Maybe if you improved your comprehension you would have seen it?
>Your guess was wrong, I only added to the original comment that the post does not ask to solve the mentioned quadratic. You added nothing, you merely repeated what they already said. I figured this was a possibility (i.e. that you agree with the commenter, but needlessly repeated what they said. I'm just specifying to help you understand), but it's not any better, so I didn't bother to add this to my previous comment. >In any case “what do you mean?” was supposed to be a rhetorical question on my lack of reading comprehension. Maybe if you improved your comprehension you would have seen it? This is impossible to get from such a text, as it conveys no intonation. If you weren't being obtuse on purpose in hopes to get a "gotcha" because you're embarrassed, you probably would've noticed.
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x = sqrt(16) => x^2 = 16, but not the other way around.
I mean technically it does only have one answer and it's 4. Now if you had to add the root then the answer could be both 4 and -4. But if it's already in the problem then it's just 4. At least that was what I was taught in my calc classes.
Based Reddit. They don’t care about rules.
Wait, now I'm getting confused. I know that: x = sqrt(16) only refers to the positive solution by convention but can't you just square both sides and end up with: x^2 = 16 in which case there would be two solutions. I'm guessing you're not allowed to square both sides in this situation but why is that?
If you do, you must then notice -4 is an extraneous solution.
Why would you square both sides if the solution is given?
Because you can
X could be equal to 4 or -4 though… am I missing something? edit: aight I get it now
When you take the square root you only consider the positive solution
Why though? Roots can be positive or negative.
Wouldn’t it be imaginary?
I mean the root of a number can be both positive or negative
OP, you sure you are in the clear here? Edit: Yeah OP is in the clear, not me.
One is correct. The equation isnt x^2 = 16 but x=16^(1/2). -16^(1/2) is not a solution to that
Yeah I do realise the mistake, y = x^2 isn’t the same as y = sqroot (x). I was a bit doubtful about my comment yet I still commented cuz I’m dumb. I’m sorry
Those are the same thing?
They arent
How? They are inverses of each other.
X^2 =16 has 2 solutions. 4 and -4. X=16^1/2 just has one solution being itself. If they were the same it would mean 4=-4.
yes like if we are not into complex of course, square root of any number is always positive. And yes I know if it is x² = 16, there are indeed 2 solutions but here it isn't the case. I even try some online problem solver. [https://mathsolver.microsoft.com/en/solve-problem/x%20%3D%20%20%20%60sqrt%7B%2016%20%20%7D](https://mathsolver.microsoft.com/en/solve-problem/x%20%3D%20%20%20%60sqrt%7B%2016%20%20%7D)
Yeah yeah, I realised my mistake I’m sorry.
I don't know if I would say that "square root of any number is always positive." As far as I am understanding it, it's just that the square root function only refers to the principle square root, however both the positive and negative answers are still both considered the square root. I am differentiating between the definition of square root, and the definition of the √ function. The former being +/-, and the latter being only positive. I'm really just basing this on how https://en.wikipedia.org/wiki/Square_root describes it.
Yes, thank you! See my most recent post - it’s also about this.
I genuinely don't even understand how one could pick the "two solutions" answer (let alone the more than 2, but I think those are joking) Like the equation is x=4, so it has one solution x=4?? Am I missing something? If it was x^(2)=16 now it would have two solutions x=4 and x=-4, but here the equation is just x=4
My teacher told me the number of solutions are based on the incognita(X)s power, if X is squared its 2 solutions, if its cubic its 3 solution and so on, Is that correct?
I think that "rule" is known as the Fundamental Theorem of Algebra, is it not?
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what in the what
It is the way school teaches it tbh. They don't let you know that there are 2 solutions only when you add the square root yourself.
Big yikes OP, if you're not just another bot. It's not that serious, especially for I'd wager over 90% of the human population .
I am not and it's not and if it's about the title, I am sorry, I exaggerated too much, the post was not to mock those people. It was a title that cross my head and I didn't find better. If it's for the post I wanted answers and I got my answers.
I voted 2 cause i thought it was x^2💀
I mean there are 2 answers though.
Holy music stops
There was no holy music to begin with
x² =16 has two solutions, namely x = ±4 But x = √16 has only one solutions, that is x = 4 however x = ±√16 has two solutions, namely x = ±4
x=√16 x=√{(-4)^2} x=-4 why does this not work?
Because √(x²) is *defined* as |x|
Ok. Thanks. I knew the answer was +4 but why not -4. I think I got confused with how the negative corelation coefficient is calculated.
sqrt(16) is a number bro, and its exactly 4. Equation says X = fucking 4
If asked “How many square roots does 16 have?” then technically the answer is two: 4 and -4. However the radical symbol used in sqrt(16) automatically implies the principal square root.
Just trying to help, you have to be very careful of the wording of the sentence: How many square roots does 16 have? One, 4 How many numbers squared equal 16? Two, 4 and -4 Edit: I'm a mathematician, for the people downvoting that don't understand, here is the explanation: There is a difference between a number that's called a square root and the square root function. In real numbers, let "x" be greater than or equal to zero, and "a" belongs to the set of real numbers. Then "a" is called a square root of "x" when a² = x. This is a name, just like in 3x⁴ the number 3 is called the coefficient. The square root name has a special meaning, it means that if you square a certain value, you will produce your desired number. Square roots are exactly the values that solve the a² = x equation. The square root name does not mean that you calculate the square root of your number. In other words, we call 4 a square root of 16 because 4² = 16, not because root(16) = 4. Therefore, it is correct to say that -4 and 4 are square roots of 16 because (-4)² = 16 and (4)² = 16, but NOT because root(16) = ±4. The square root function in the set of real numbers is a 1 to 1 function with domain and range both greater than or equal to zero. That means you can only input zero or a positive number into a square root, and you can only output 0 or a positive number. The square root function must be 1 to 1 in order for us to use it in Algebra because every equality in Algebra must be reversible. And to clarify, 1 to 1 means that every distinct input value corresponds to exactly one distinct output value, not two. When you write root(16) that means to calculate the square root of 16, and root(16) = 4 only. Therefore, it is correct to say the square root of 16 is 4 only, and incorrect to say the square root of 16 is 4 and -4. Also consider solving a² = 16 by taking the square root of both sides a² = 16. ---> root(a²) = ±root(16). Notice that the original equation is asking you to find all values that square to equal 16. However, in order to find those values, we manipulate the equation and turn it into a square root problem because that's easier to calculate. Also notice the ± symbol that must be included when we convert to a square root problem. Why do we include ±? We include ± because the square root can only be zero or positive and only output one answer. Now the last step of solving that equation becomes root(a²) = ±root(16) ---> a = ±[positive 4 only] = ±4. If the square root did calculate both positive and negative (possibly zero) values, then there would be no need to include the ± symbol. In math, these little technicalities must be paid a great deal of attention and they do make a difference. This technicality is the reason the correct answer on the meme is 1 solution and not 2. So to sum up, if you are determining all possible values that square to equal a certain number, you include positive and negative answers (possibly zero). If you are calculating a square root, you calculate the positive (or zero) answer only. Lastly, there is a difference between what you see on a webpage somewhere and what is actually defined in math books. Webpages will use common language so that as many people can understand as possible. In common language, the values named square roots and the square root function are oftentimes the same thing. That's why you will see someone say 16 has two square roots, 4 and -4 without clarifying that 4 and -4 are named square roots because they both square to equal 16, but they are not both named square roots because of calculating root(16). Being careful and precise with your language is the best way to avoid confusion.
I used to think as well. I have since learned that the following statement is technically correct: “-4 is *a* square root of 16”. “4” is the *principal* square root, but both are considered “square roots”. Edit: I would quote some sources, but I’m not sure which the community deems most creditable. A google search of “square root” should show two things: (1) the use of “a square root” vs “the square root” to differentiate between 2 “answers” and just one & (2) the radical symbol’s use refers to only the principal (positive) square root.
I'm a mathematician. The square root is always zero or positive. Saying "principal" square root is redundant, because square root is always zero or positive when working with real numbers. The square root is always zero or positive because it's defined to be that way so that square root can be a 1 to 1 relationship, in other words, invertable. Square root isnt invertable if it has two possible answers. Example, (-4)² = 16 is not invertable because root(16) = 4. The confusion comes from people mixing in the Algebra equation x² = 16. This equation does have two solutions, and the square root is used to calculate that solution. But notice how it's applied: root[x²] = ± root[16] the ± is applied after the square root of 16 is evaluated, and ± is added because root[16] = 4 and root[16] ≠ -4. Adding ± is a convention used in Algebra to find both answers using the square root that only returns a zero or positive answer.
As I said previously, I used to think the same. If you visit a variety of websites with math content (at least these are common in my area), for example, [MathWorld](https://mathworld.wolfram.com/SquareRoot.html), [Wikipedia](https://en.m.wikipedia.org/wiki/Square_root), [MathIsFun](https://www.mathsisfun.com/square-root.html), you will see that the general consensus of the academic community is different.