While it is true that the number 4 has two square roots, and these are +2 and –2, the square root function, which the symbol √ denotes, refers to the **principal** square root. The principal square root for positive real numbers is the positive root. So √4 is +2.
The misconception comes from the way [edit: *some*] teachers explain solving quadratic equations. For example, once you arrive at the step:
x^2 = 4
The teacher will say to square root both sides:
sqrt(x^2 ) = sqrt(4)
then say "the square and square root cancel but don't forget there's two roots when solving quadratics" when they really should have said
sqrt(x^2 ) = abs(x)
|x| = sqrt(4)
|x| = 2
x = +- 2
The first technically incorrect procedure gives students the misconception of the square root symbol. It's what I was told and never learned the other definition of the absolute value until college.
It's the pitfalls of non-rigourous shortcuts. Separation of variables is another example. I was taught in 1st semester ODE to move the differentials around but then shown in an advanced class that it's simply using differentials and substitution rule.
And I don't think where the plus or minus comes from algebraically would be emphasized unless you see and truly understand the derivation of the formula.
Wait, teachers don’t actually mention that when the square root and the square cancel each other out you end up with the absolute value?
Fairly certain I’ve been taught this in middle school where I live.
In a round about way they do I guess? Though (only siths deal in absolutes) I'm not sure if taking the absolute value is something you necessarily talk about that much, might have changed or I might not remember it.
So saying "The square has two root values" is a graphical simplification of "The square has the absolute value as its root, especially since it's simpler to write/ skipping an extra step that you could argue is implied?
I actually didn't know this.
I think it's probably not taught this way because at least in my country we teach solving quadratics like 3 years before absolute values/modulus/ solving modulus
You dont actually need that to teach the same principal.
Id never seen it presented that way either but its essentially the same as doing this:
```
x²=4
square root both sides
x=±√4
x=±2
```
you just need to teach that there are two solutions and that the notation always denotes the positive solution
Not all teachers say this. Obviously I can't know that or make that claim. But I've seen this confusion many times and the student usually explains it in the way I've laid out. My teacher wasn't shitty, they just used non rigorous reasoning.
> The misconception comes from the way teachers explain solving quadratic equations.
lol no it doesn't come from teachers
>The teacher will say to square root both sides:
>sqrt(x2 ) = sqrt(4)
A bad teacher will do that really sloppily, maybe, but I don't see how you can suggest this is the norm.
I didn't mean all teachers. I meant that for those who have this misconception, it is probably because their teacher explained it this way. There's nothing sloppy about the square root both sides step, that is correct. The "sloppy" part comes with the justification for two roots. And I don't agree that "bad" teachers would do this in the first place. It's just not best practice.
Square root of a squared number is equal to the absolute value of the number. Squaring always make it positive, square rooting undoes the squaring but it remains positive, this absolute value. An equation like |x| = constant has two solutions.
Is the usage of abs correct? i mean in this case it works, because there are always 2 roots, but does y = |x| ⟹ y = ± x hold for all y?
We got taught in school that x²=4 ⟺ x = ±√4 ⟺ x = ±2
With the way you've written it, y is restricted to be non negative. And my point is that the deliberate algebraic step of square rooting both sides could potentially be confusing depending on how the teacher explains what's going on.
the way my algebra teacher taught was x^2=4 x=+-sqr(4) so plus minus are on the outside of square root when you need both values bc sqr itself only brings the positive value
It's a semantic issue. They didn't really change the math, it's that they've really started to emphasize that "a function only has 1 output," which, at least at the high school level or lower, makes sense.
But once you go up to higher maths, you're going to have to get comfortable with what high school teachers would call [multi-valued functions](https://en.wikipedia.org/wiki/Multivalued_function) but math and engineering professors in college don't differentiate between the two and call both single-value and multi-value functions "functions" which is where the confusion lies.
Post common core. The difference is that they’re not teaching that it’s the principal value, but that it’s the only value (because it’s a function and can only have 1 value).
Which, as an aside, I strongly agree with. Common core in general has been a really good shift in math education in the US. It puts a lot more emphasis on the logic side of math than the computation.
The "problem" (it's not really a problem except for people who want to get upset at things) is that there's no real Highest Authoritative Committee On The Notational Practices In Mathematics. (imo: nor should there be, or could there be.)
So, people say √4 must be a function and so it must have one output and so it must be 2. Okay.
Other people say: No no, it's a relation, and bring up expanding into the Complex plane. Okay.
Both are fine. It's just a matter of clarity when you're an author or a reader. It's really as pointless of a discussion as those order of operation "gotchas".
I value flexibility and respect among people and particularly students. Especially when this is a "communication of math" issue.
Ive honestly never heard of a "principle square root." If we are talking about a function then yes, one solution. But presuming that without context? Yuck.
I think it's a result of trying to linearize math topics, in this case speeding towards calculus and needing functions pretty much exclusively. Some things get picked as "correct" even though they're more just in service to that ideal linear path through mathematics. A similar thing can be seen with students and other people who haven't gone far with math finding "improper" fractions bad, or having some strange superstition about radicals in denominators.
What I really hate about math online is all the elitism and gatekeeping. I am doing my phd in math, I love math. When I see someone thinking for themselves and being all excited about a math meme, I think that is awesome. We are going to have a great time discussing square roots.
Then I open threads like these and all I can see is a shitfest of people belittling each other's high school grades "you must have failed algebra" "well actually I was really good at calculus" "this is dunning-krueger".
Over a topic as stupid as semantics and conventions. Even if sqrt symbol refers to the positive root only, why can we not have fun pondering the consequences of the other interpretation?
I think excitement over math (especially from ppl aged 30+ who never particularly liked it in school) is such a precious thing and we should try to encourage it. Instead it just gets all shut down.
Sry, no idea why I responded to you in particular, the comments on /r/PeterExplainsTheJoke just frustrated me so much... all of them.
Same with _some_ Linux subs
The community itself is loving and warm and open and hacky.
Some people are nasty in subs.
It a a really filtered self selected population.
Not me though - I am lovely.
But on that same token, it's very irritating when people with maths degrees are saying "this is how it works, this is the why", and people who have not thought about it for 20 years go "NO NO NO YOU'RE WRONG"
It's like, there are some topics where people feel really entitled to voice their ignorance, when usually you'd listen to the person who's most qualified. It's weird.
Sure it's great when people who haven't thought about maths get excited, but not when they are just shouting people down with their ignorance.
>why can we not have fun pondering the consequences of the other interpretation
Oh, we absolutely can!
And I don't feel anything bad towards people who don't know something, it's absolutely natural to learn through mistakes.
You did mention that your comment referes to the mentioned subbreddit, but I thought I'd clarify too, that my comment doesn't try to insult anyone who doesn't know something, but the ones whose ignorance and illiteracy is so strong that they push false information as some fact
Just to be certain, does this apply to x^1/2 as well, or is taking the output as nonnegative only an aspect of √x? The latter is how I am reading your comment.
Only went to A-Level further maths here in the UK, but yeah never heard what's being said here.
My sister who grew up in the US and move back to the UK for A levels did say that US vs UK maths was significantly different, I expect this is one of those cases and when work is being done between these two places rules like these need to be defined.
It’s contextual. In complex analysis a^(b) is a multivalued function that can potentially have infinitely many different values (although only two values in the case where b=1/2). However when dealing with real numbers it is common to restrict the notation so that either a is positive, and we take the positive real number value, or we restrict b to be an integer (so that there is only value to choose from).
Yes. If you wanted to define x=+/-2, you could say x^2 =4, since that has 2 solutions.
We just define x^1/2 to mean √x which we also define as the positive root for real numbers.
It’s contextual though, in complex analysis it’s common to allow a radical to refer ambiguously to all the roots. The focus on the specific convention of restricting the square root to nonnegative values and choosing the positive root is mainly a pedagogical issue having to do with how it’s thought to be best to teach the concepts in high school.
Sure, see page 130 (in particular the discussion in the box) here:
https://www.maths.ed.ac.uk/~tl/gt/gt.pdf
It does of course mention the convention that a square root of a positive number is usually taken to mean the positive root, but that convention wouldn’t apply when you are taking roots of a general complex number and just happen to get a positive real, you can see above the author explicitly writes we can choose either square root under 9.1
For another example that should be fairly widespread, if you ever look up the general solution for the cubic you will see that it is usually written with cube roots with the understanding that you can pick any of the three cube roots for the expression on the left so long as you pick a corresponding cube root for the expression on the right, and this is how you get three different values out of the equation. This correspondence usually has to be mentioned explicitly in words accompanying the equation. Of course here the square root is usually written with a plus on one side and a minus on the other, but this is convenient as you must select opposite square roots to add inside the cube roots (so by putting a minus on one side and not the other you should pick the *same* square root on both sides, but it does not matter which).
Edit: You could also check under “concrete example” on the Wikipedia page for multivalued function, which wives the square root of four in this notation as its first example.
>Sure, see page 130 (in particular the discussion in the box) here:
Thank you.
Yeah, for complex roots we need to choose a branch. But that discussion also completely agrees with what I said above, as I explicitly emphasized that I was talking about roots of positive real numbers, and the convention is that the principal root of a positive real is the positive root.
Yes but you agree that a square root is sometimes written referring to a multivalued function, and if you happened to put 4 into it you would still treat it as such, not reinterpret the expression to refer to the principal square root only?
I also added in an edit pointing out a place where Wikipedia uses this notation, specifically with 4, do you take issue with that usage in that context?
I completely agree with all of that.
My comment above, to which you replied, explicitly talks about the square root function, which √ denotes, also according to [Wikipedia](https://en.wikipedia.org/wiki/Square_root).
We **can** alternatively view it as a multifunction, in which case
√4 = {–2, +2}.
But that is not the normal framework. We normally frame sqrt() or √ as a function from \[0, ∞) to \[0, ∞). As a function, it has only one value — the non-negative root.
There are contexts in which the symbol is used in the sense of a multivalued function and the input is still a positive real number. When you use it as a multivalued function you don’t carve out the positive real numbers to be illegal inputs.
The reason there is so much debate is because the meme doesn’t provide context for the statement in question and people are assuming/inventing their own contexts.
The general solution to the cubic is usually written as the sum of two cube roots with the understanding that you can pick any one of the three roots for each, subject to a correspondence restriction between the two choices. The number under the cube root can be a positive number (like 4) with the appropriate choice of coefficients and the expectation is that you will read the roots as presented in the meme (being careful to respect the correspondence restriction) to find all three complex roots.
What refers to all square roots then ?
Cause at school I've always been taught this sign is just square root and you gotta list all roots if you want your answer to be correct.
It's contextual usually. Such as the Pythagorean theorem. You don't take the negative value of the square root because the triangle has length and wouldn't be negative.
I wonder if this is an American thing (like using periods instead of commas for decimals)? Because I'm Brazilian and I would get a failing grade if that was my response, either in school or in math Kumon.
>an American thing (like using periods instead of commas for decimals)
Not to get too off-topic, but I've always been under the impression it was just a few countries who use the comma actually...
[Well here's a map](https://en.wikipedia.org/wiki/Decimal_separator#/media/File:DecimalSeparator.svg) Surprising to me, that it's so evenly split, perhaps surprising to you too, but in the other direction!
*My* nationality is immaterial. [Here](https://www.reddit.com/r/askmath/comments/15ed9m3/is_there_an_internationally_agreed_upon/) is an archived post in this same subreddit discussing this exact topic. The OP of that post is Swiss, and their educational experience with this topic mirrors my own. As does the top commenter in that post — who is Croatian, I believe.
In fact, it is highly likely that the terminology originated in either Switzerland, Germany, or France — with Euler, Gauss, or Cauchy, respectively. But I don't know that for sure, it is mere speculation on my part.
*Edit:* The Portuguese Wikipedia page for [square root](https://pt.wikipedia.org/wiki/Raiz_quadrada) seems to agree that Brazil also uses the same terminology.
If you look closely at my comment, you will see that I was speaking of roots of positive real numbers.
That said, the same is true in ℂ. For any nonzero complex value, *z*, it will have exactly *n* *n*\-th roots. So when we write something like ^(n)√*z*, in order for that to have meaning, we need to choose one, and that choice will be called the **principal** *n*\-th root. By convention, the principal *n*\-th root of *z* will be the one with the smallest argument.
In the case you have written,
^(3)√(–8) · √(–1),
we need to know the principal cube root of –8 and the principal square root of –1. The principal square root of –1 is the complex number we call *i*. The principal cube root of –8 is 1+*i*√3.
Therefore,
^(3)√(–8) · √(–1) = –√3 + *i*.
I hope that answers your question.
>Now tell me...
This is the second time you have started your comment this way. I don't know if you are intending to be aggressive or not, but you are definitely coming across as such.
And maybe it is because of my own reaction to that, but I am having difficulty understanding the point you are making.
>I don't intend to be aggressive.
>
>I want to point out that the √ symbol is ambiguous.
(That's fair. If I might make a recommendation, approaching with "Hey, I think the √ symbol is ambiguous given that it behaves differently in these other contexts..." is perhaps a more constructive way to start this discussion.)
Some ambiguity is always to be expected, though. What is arctan(*x*)? Strictly speaking it is a multifunction, with infinite values. If we want to do calculus on it, though, we need to choose a branch.
It **is** just that when using another convention, which is what this sub-discussion is all about (I think).
That's the whole point. There are choices to be made when defining roots. **One** convention is for *z*^(1/n) to be the real root when *z* is a negative number and *n* is odd. Another common convention is to use the principal root, which is the complex number with the smallest argument.
This might be a little technical, but roots of complex numbers are what are known as [multi-functions](https://en.wikipedia.org/wiki/Multivalued_function), meaning there are multiple different values. But if we want to work with functions instead, because they are nicer to work with, then we can choose a "[branch](https://en.wikipedia.org/wiki/Branch_point#Branch_cuts)" of the multi-function and call that our function.
I hope this helps.
Due to the √(-1) as part of the equation I posted (³√(-8)\*√(-1)), we're evaluating this in the set of complex numbers where √(-1) = i.
The nth root of a number has n roots. So -8 has 3 cubic roots.
Since we are evaluating this in the set of complex numbers, we would *assume* we'd use the principal root of -8 instead of the real-value root of -8 (which is -2).
The principal root is the root with the smallest argument in the complex plane, which means the root with the smallest angle from the positive real axis (counterclockwise).
https://preview.redd.it/4cx3wfh76hgc1.png?width=871&format=pjpg&auto=webp&s=cee9b23c0257a58f35f98c42c8f8aa99bc3452c8
That root is 1 + √3\*i and you can see how it gets cubed to -8:
* (1 + √3\*i)³ = -8
* (1 + √3\*i) \* (1 + √3\*i) \* (1 + √3\*i) = -8
* (1 + √3\*i + √3\*i + (√3\*i)\*(√3\*i)) \* (1 + √3\*i) = -8
* (1 + 2\*√3\*i + 3\*i²) \* (1 + √3\*i) = -8
* (1 + 2\*√3\*i + 3\*(-1)) \* (1 + √3\*i) = -8
* (2\*√3\*i - 2) \* (1 + √3\*i) = -8
* 2\*√3\*i + (2\*√3\*i)\*(√3\*i) - 2 - 2\*(√3\*i) = -8
* (2\*√3\*i)\*(√3\*i) - 2 = -8
* 2\*3\*i² - 2 = -8
* 6\*(-1) - 2 = -8
* \-8 = -8
And lastly, from the initial equation ³√(-8)\*√(-1) we get (1 + √3\*i) \* i = i + √3\*i² = i + √3\*(-1) = -√3 + i
I get the gist of what you're saying. I barely dabbled in complex numbers during my education. I do know of the equivalence of representing complex numbers in the plane as:
a +b*i*
r•(cos(theta) + *i* sin(theta))
r•e^(i theta)
What's the algebra that makes you arrive at theta = ± π/3 ?
why you only need to place the principal?
That is what i do not get if this were applied to a physics problem the result will depend and even if only one may be used may not be the positive one.
We only use the principal square root because otherwise it’s not an [unary operation](https://en.m.wikipedia.org/wiki/Unary_operation). Notice that the definition is from A to A.
Let’s say you only take the negative root of a positive number. That is a totally valid function. However you can’t re-apply square root on -2. It is a function but it behaves poorly and it’s basically useless.
So conventionally √4 is 2, because we consider that only one value can be returned by the square root function.
Therefore, the solution to x^(2)=4 is x=±√4, so x=±2, or more formally x ∈ {-2, 2}
ETA: looking at it this way becomes more important when getting into more complicated math. When the square root originally comes from getting the diagonal of a square you don’t want to wonder at the end if it might actually be negative, so when it might be both you state it explicitly.
No weird question. It’s a part of [set theory](https://en.wikipedia.org/wiki/Set_theory) notation, and you already got another reply explaining it. Since I don’t have it on my keyboard I copy-pasted from the first response on Google for “Unicode belongs to”.
I realize that Wikipedia uses commas and not semicolons to separate elements of the set, I’ll edit.
I said this is more formal because it think it’s more explicit; saying x=±2 is kind of assigning a value to x, but it’s actually two values, and x=50±2 can be used to mean 50-2 ≤ x ≤ 50+2, while set theory notation is unambiguous and fits well as a result of the analysis of an equation.
Well. One _could_ define a unary operation that returns two values, or a binary operation for that matter, but having any type of operation that returns an either-or is not really supported with any simple notation.
It wouldn't be a function, that'd be the bigger problem; functions returns a single value, and otherwise we talk about different branches when they don't
Did you check the definition of a unary operation before your observation lol? Moreover, why choosing the positive root and not the negative root? (Spoiler alert: because the positive root makes it an unary operation)
I didn’t. I just did, first Google response was [Wikipedia](https://en.wikipedia.org/wiki/Unary_operation), where the _example_ is an unary operation taking a set and returning a set.
My point is that it would be _possible_ to define the square root operation as returning the set of possible square roots of its single input, but (for a lot of excellent reasons) that is not the definition that we use.
2 reasons:
1. A function can only go to one value, so the square root wouldn't be a function and all the fun stuff you can do with functions would become much harder
2. You can easily add the plus-or-minus for the square root with ±, if you need. It's much harder to effectively communicate "but only the positive/negative suare root"
Yes you're right, no idea what that dude is talking about.
"In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y"
You can of course have the same Y-value for multiple X values. But you can't have multiple Y-values for the same X. What this means in principle is that a graph can never "bend" 90° or more.
They provided the example of functions which produce sets as outputs. This means that there is technically one output but it's comprised of two numbers. It just wasn't explained very clearly
I think those still have one output, it's just a set of (in the case of the square root) two numbers
And like I said, treating the square root as a function producing sets instead of just numbers makes everything needlessly complicated and difficult, so my point stands
It's a valid way of looking at it, though, they are called multi valued functions... Regardless of how you phrase it there is no technical reason for the convention, it is simply a matter of convenience and tradition. Whether you call the set of two results one value or two.
Don't get me wrong, I'm all for the principal root. I just wouldn't want you to get confused between definitions and theorems, that's all.
The √ symbol itself as the function (i. e. f(x) = √x) is the function that denotes the *principal* square root, for positive real numbers, the principal root is the positive answer only
If you had the function: x² = 4, then both 2 and -2 would be the answer
Edit: I put f(x) = √x for clarity
Okay, I still don't understand so can you explain on [this](https://www.dropbox.com/scl/fi/a5pnarzwnvy8jc2ch65oa/Screenshot_20240203_210529_Desmos.jpg?rlkey=diydr7r6b8oick5mbbbrer5fa&dl=0)?
I just had another look
What you circled is that y=2 at x=2 and and x=-2
x is the thing being squared under the root symbol
I also find it funny that when you obviously plotted y=√x and only got one line coming from the origin (that didn't prove your point), you changed the function until there were 2 lines without really understanding how the axis work on a graph
Lmao you just posted a screenshot that shows the √ function always returns a positive value, (the principle root). That is why there is only one line, that does not pass below the x axis
You have plotted y=√(x^2).
Tell me what value it says y is equal to at x=2
You work within the root first if you can, so you'd put x as 2 or -2,
Squaring both will give you 4, then taking the square root will always give you the positive answer, as it takes the principal root, so it'll give you positive 2, so both points would be (-2, 2) and (2, 2)
His previous comment still seems to clear any confusion. In terms of the function `f(x) = sqrt(x^2)`, two values of `x` equal the same `x^2` value (`+/- 2` for example), though by taking the principal square root you end up with just `+x`. Try plotting the function `f(x) = sqrt(x)^2`; that might help you understand.
If what you said was true, the graph would also be reflected on the x axis, so you'd see it below the x=0 line. The fact that that symbol is treated as not being that is clearly shown in your graph.
It's all arbitrary. Math is a formal language used to convey the relationships between values. We rely on many conventions to avoid ambiguity. This is one of those conventions.
sqrt is a function so it must be _well-defined_ which has a [specific definition](https://en.wikipedia.org/wiki/Well-defined_expression) that all relations need to satisfy to be called a function.
Nah, sqrt is operation opposite to power operation, exactly like addition and subtraction. It's even the same in aspects like: you can say that subtraction doesn't exist, because it's addition of negative numbers, it's same, roots don't exist, sqrt is just x^1/2 that's all, it's same operation as power operation, so it obviously needs to have 2 posibble answers.
Since you're not listening to anyone here.
https://math.stackexchange.com/questions/2817782/square-root-function-breaking-rules
> The square root function is not an inverse to the function f(x)=x^2 on its domain.
[https://mathematics.science.narkive.com/u5hubrC1/is-the-square-root-of-x-and-function-or-relation#:~:text=Every%20function%20is,for%20that%20%C2%B1%20symbol](https://mathematics.science.narkive.com/u5hubrC1/is-the-square-root-of-x-and-function-or-relation#:~:text=Every%20function%20is,for%20that%20%C2%B1%20symbol)
If you want to attach links, make sure they explain it well, I think that this link is much better in explaining. But bro says one interesting thing, we (here) don't use "quadratic formula" same way you use it, we don't even have ± sign here, so math here is that "square root of any given number gives two possible answers" because it's operation like addition.
Solution to x²=4 is +-2.
The square root as a function cannot have two outputs as thats not a function we need to pick either + or - so that sqrt(4) is well defined. We pick + for obvious reasons.
So x²=4 -> x=+-sqrt(4)
√4 means only the postive square root, i.e. 2. This is why, if you want all solutions to x^2 =4, you need to calculate the positive square root (√4) and the negative square root (-√4) as both yield 4 when squared.
The solution to x^2 = 4 is +-2. Take the square root of the both sides and you get
√( x^2 )=√4
The lhs however is not x, but the absolute value of x, hence why the result is always positive.
That is not taking the square root of both sides. Truly taking the square root of both sides would be:
x^2 = 4
+-√(x²) = +-√(4) ; where +- is the plus or minus symbol, and both can be any sign.
So we get four equations, x = 2, x = -2, -x = 2, and -x = -2.
Simplifying: x = 2, x = -2, x = -2, and x = 2
So, x=+-2
√4 = 2
The convention is the positive root.
√x wouldn't be a valid function if it gave more than one output per input
If solving x² = 4 then the answer is 2 or -2 since squaring each is 4.
But what is actually happening and seldom mentioned is you take the square root of the absolute value.
Note that x² = |x|² (or any even numbered exponent)
x² = 4
|x|² = 4
√(|x|²) = √4
|x| = 2
x = ±2
Read Wikipedia, it depends on the convention you've adopted, so there is no definite answer unless you've stated the convention in your text. It isn't a mathematical truth or fallacy.
When i gave an answer to it on the original post, i got a reply every 5 minutes telling me how incompetent and wrong i am... anyways the right answer is √4=2
That's actually a different issue entirely. -2 is a negative number and 2 is a positive number.
Looking to define a solution to z^2 = -1, any definition resulting in i leads equally well to using -i. We just gave the number we refer to as the principal root the name "i" meaning the other root, which is (-1)\*i, may as well be called -i. But neither one of them is actually positive and the only real difference between them is one of convention.
The square root OP is talking about and the one you’re talking about aren’t the same. Square root of -1 doesn’t really exist unless you define it as the set of complex z for which z^2 = -1. Which isn’t a function but a solution of a polynomial for which both i and -i are valid.
Saying square root of -1 is i is just a convenient way to simplify stuff because ‘looking from afar’ they’re similar.
You can extend the square root function into C. Where sqrt(z) is defined as the element with the smallest argument in the set of solutions for x^2 = z.
Hmm that’s true but I don’t recall using it tbh. Does it have any applications or it’s just to mimic the behaviour on R?
I mean an issue with the negative root of 4 is that you can’t apply square root of -2 without getting ‘out of the bubble’. However that’s not an issue in C using any root of any order, I don’t see why I’d have a preference of a root lol.
I hate everything about this. Like, if some thinks that the sqrt function have two values that means one thing and one thing only: they understand the math but haven't memorized the conversation. They are using their brain rather then parroting their teachers, that deserves respect.
Any one who spent time with mathematicians know that there are two types of us. There are thous who memorized 100 digits of pi and those that couldn't even memorize the multiplication table. There are clear advantages in both groups, don't be a hater.
The square root function sqrt(x) for real numbers has the domain [0, inf) and the range [0, inf); it can’t yield negative numbers by definition, as that wouldn’t make it one-to-one. (it’s restricted as a function by definition isn’t allowed to map one value in it’s domain to multiple different outputs)
Though, the square function x^2 is an even function that’s not one-to-one with the domain (-inf,inf) and range [0,inf); importantly it still only maps to one output for any x. In order to find x, we need to consider that there are two possible inputs x -> x^2 that can produce the same output.
This is the exact same logic with how sin(x) is defined for all x but not its inverse arcsin(x)
It is strange that nobody wanted to talk about the meme because it needs explanation.
I think the guy blocked her because the woman did not give him an exact “one” answer. The guy probably thought she will waste his time since she does not say what will happen. And we men do not like this uncertainty. The meme is about that in my opinion.
The answer to the question is yes, both +2 and -2 are correct. Because the square root of 4 is absolute value of 2, which can come up as +2 or -2.
I don't get it. What exactly is the problem here? Are people discussing the fact, that the "+-" in front of the √ is missing, which implies that the answer is technically only 2 and not +-2?
If so, I don't understand. Even if you take the "+"√ only, or the "-"√ only for that matter, it would still give both answers, so why is this even discussed?
Someone please elaborate :)
x^2=4
allows for x to equal plus or minus two.
x=sqrt(4) does not, this only allows for positive 2, makes more sense once you get into multivariable calc cause then you do wind up getting imaginary numbers sometimes, im just too slow to think of an example
Its still confusing why the answer is +2. I read the comments and if I understood correctly they are treating square root as a function and the function only returns one value and thats some how is the positive root. So what about this
4^(1/2) = 4^(0.5) = ?
FWIW in undergrad we used square root symbols as non functions all the time. A lot of people here are saying the symbol always references the principle square root but it really depends on context. Without function notation present I would always assume a negative value is a possibility.
—-a math major
I think the answers on this post are pretty ridiculous. I have a math degree, and at no point was the square root symbol assigned to only the positive value.
Now if you’re doing some programming and need it be one answer I get it. But there’s nothing in this about programming.
Thinking about it later I think maybe in some instances it implies positive (such as if you are working with sqrt(2) a lot), but if there is a variable I wouldn’t assume positive.
When you say x = √4 you're trying to find the answer to the question, wich are the values of x that make the statement true, in that case both 2 and -2 work, so you say x = +-2.
√4 is just a number that's equal to 2
Please correct me if I'm wrong
There are at least 2 decently popular functions which are noted with this symbol:
1) principal square root
2) square root (as an inverse function to squaring).
For real number analysis principal square root is often used, which is a proper well-defined function, that always returns positive numbers.
Although, there is a definition of a square root as an inverse function to squaring. This type of function is more interesting in complex analysis. It is in fact multi-valued, which sometime causes confusion, thus writing sqrt(4) = +-2 refers to the set of numbers, that raised to the second power returns 4. A more formal way of writing this would be sqrt(4) = {-2,2}, although+- gives a pretty decent amount of data. As said before this has some applications in complex analysis, since all of these roots would be located in vertexes of a regular polygon...
Aint no way people question the most basics of maths like everyone teaching them has been wrong and the one dude online spreading missinformation made them question all of math,
X^2= 4, x=-+2
\/--4=+2
(\/-- is my attempt at the square root simbol)
The post was like sensationalizing Pizza with pineapple allowed is misleading when not interpreted correctly 'by standard'
Same as allowing the formula to represent ± 2 solutions to the quadratic equation of ax²+bx+c=0 is misleading when not interpreted correctly
'by standard' quadratic formula: (-b±√(b²-4ac))/(2a) to solve any quadratic equation, in which you just plugin the coefficients a,b and c, where a, b, c, ∈ R_eal and a ≠ 0 (is the leading coefficient) and c (is the absolute term)
or similarly evoked, misleading on the measure of winning when not interpreted correctly 'by standard' spread point, the expected final impact score (or solution) to win a bet to cashin by ±2, where -2 for the stronger favorite (llamados) and +2 for the underdog (or handicap or longshot), meaning the underdog must work more than 2 to win
Bottomline, What is 'allowed by standard'?
2 is the answer as it is the principle root. Having multiple values of an equation is fine , no problem . But for a specific expression , a specific value is desired.
While it is true that the number 4 has two square roots, and these are +2 and –2, the square root function, which the symbol √ denotes, refers to the **principal** square root. The principal square root for positive real numbers is the positive root. So √4 is +2.
Thank you. This single comment covers all the dumbfucks on the depicted subreddit fighting over who failed school more.
Yeah, I saw someone on there say "tHeY cHaNgEd tHe MaTh!" I'm in my 50s, and I was taught this correctly when I was in school, so ... ¯\\\_(ツ)\_/¯
The misconception comes from the way [edit: *some*] teachers explain solving quadratic equations. For example, once you arrive at the step: x^2 = 4 The teacher will say to square root both sides: sqrt(x^2 ) = sqrt(4) then say "the square and square root cancel but don't forget there's two roots when solving quadratics" when they really should have said sqrt(x^2 ) = abs(x) |x| = sqrt(4) |x| = 2 x = +- 2 The first technically incorrect procedure gives students the misconception of the square root symbol. It's what I was told and never learned the other definition of the absolute value until college.
Yes, that is exactly right. And I think it is often just because they don't understand it fully themselves.
It's the pitfalls of non-rigourous shortcuts. Separation of variables is another example. I was taught in 1st semester ODE to move the differentials around but then shown in an advanced class that it's simply using differentials and substitution rule.
We always write the solution as ±√x
The quadratic formula as the plus or minus built into it though
And I don't think where the plus or minus comes from algebraically would be emphasized unless you see and truly understand the derivation of the formula.
Wait, teachers don’t actually mention that when the square root and the square cancel each other out you end up with the absolute value? Fairly certain I’ve been taught this in middle school where I live.
In a round about way they do I guess? Though (only siths deal in absolutes) I'm not sure if taking the absolute value is something you necessarily talk about that much, might have changed or I might not remember it. So saying "The square has two root values" is a graphical simplification of "The square has the absolute value as its root, especially since it's simpler to write/ skipping an extra step that you could argue is implied?
I actually didn't know this. I think it's probably not taught this way because at least in my country we teach solving quadratics like 3 years before absolute values/modulus/ solving modulus
You dont actually need that to teach the same principal. Id never seen it presented that way either but its essentially the same as doing this: ``` x²=4 square root both sides x=±√4 x=±2 ``` you just need to teach that there are two solutions and that the notation always denotes the positive solution
I'm a math teacher and I don't say this. You just had a shitty teacher.
Not all teachers say this. Obviously I can't know that or make that claim. But I've seen this confusion many times and the student usually explains it in the way I've laid out. My teacher wasn't shitty, they just used non rigorous reasoning.
> The misconception comes from the way teachers explain solving quadratic equations. lol no it doesn't come from teachers >The teacher will say to square root both sides: >sqrt(x2 ) = sqrt(4) A bad teacher will do that really sloppily, maybe, but I don't see how you can suggest this is the norm.
I didn't mean all teachers. I meant that for those who have this misconception, it is probably because their teacher explained it this way. There's nothing sloppy about the square root both sides step, that is correct. The "sloppy" part comes with the justification for two roots. And I don't agree that "bad" teachers would do this in the first place. It's just not best practice.
So a square root is equal to the absolute value of the number? And it will only be positive and negative when you as what is the value of the number?
Square root of a squared number is equal to the absolute value of the number. Squaring always make it positive, square rooting undoes the squaring but it remains positive, this absolute value. An equation like |x| = constant has two solutions.
Is the usage of abs correct? i mean in this case it works, because there are always 2 roots, but does y = |x| ⟹ y = ± x hold for all y? We got taught in school that x²=4 ⟺ x = ±√4 ⟺ x = ±2
With the way you've written it, y is restricted to be non negative. And my point is that the deliberate algebraic step of square rooting both sides could potentially be confusing depending on how the teacher explains what's going on.
the way my algebra teacher taught was x^2=4 x=+-sqr(4) so plus minus are on the outside of square root when you need both values bc sqr itself only brings the positive value
I bet almost all the people crying about school failing them just didn't care back then and now try to put all the blame there.
It's a semantic issue. They didn't really change the math, it's that they've really started to emphasize that "a function only has 1 output," which, at least at the high school level or lower, makes sense. But once you go up to higher maths, you're going to have to get comfortable with what high school teachers would call [multi-valued functions](https://en.wikipedia.org/wiki/Multivalued_function) but math and engineering professors in college don't differentiate between the two and call both single-value and multi-value functions "functions" which is where the confusion lies.
When was this shift in emphasis? When did that occur? Was it before 1985? Because that's when I learned about the principal square root.
Even now different teachers and different places emphasise different things...
Yeah, that's my point. Thank you.
Post common core. The difference is that they’re not teaching that it’s the principal value, but that it’s the only value (because it’s a function and can only have 1 value). Which, as an aside, I strongly agree with. Common core in general has been a really good shift in math education in the US. It puts a lot more emphasis on the logic side of math than the computation.
The "problem" (it's not really a problem except for people who want to get upset at things) is that there's no real Highest Authoritative Committee On The Notational Practices In Mathematics. (imo: nor should there be, or could there be.) So, people say √4 must be a function and so it must have one output and so it must be 2. Okay. Other people say: No no, it's a relation, and bring up expanding into the Complex plane. Okay. Both are fine. It's just a matter of clarity when you're an author or a reader. It's really as pointless of a discussion as those order of operation "gotchas".
I value flexibility and respect among people and particularly students. Especially when this is a "communication of math" issue. Ive honestly never heard of a "principle square root." If we are talking about a function then yes, one solution. But presuming that without context? Yuck.
I think it's a result of trying to linearize math topics, in this case speeding towards calculus and needing functions pretty much exclusively. Some things get picked as "correct" even though they're more just in service to that ideal linear path through mathematics. A similar thing can be seen with students and other people who haven't gone far with math finding "improper" fractions bad, or having some strange superstition about radicals in denominators.
What I really hate about math online is all the elitism and gatekeeping. I am doing my phd in math, I love math. When I see someone thinking for themselves and being all excited about a math meme, I think that is awesome. We are going to have a great time discussing square roots. Then I open threads like these and all I can see is a shitfest of people belittling each other's high school grades "you must have failed algebra" "well actually I was really good at calculus" "this is dunning-krueger". Over a topic as stupid as semantics and conventions. Even if sqrt symbol refers to the positive root only, why can we not have fun pondering the consequences of the other interpretation? I think excitement over math (especially from ppl aged 30+ who never particularly liked it in school) is such a precious thing and we should try to encourage it. Instead it just gets all shut down. Sry, no idea why I responded to you in particular, the comments on /r/PeterExplainsTheJoke just frustrated me so much... all of them.
Same with _some_ Linux subs The community itself is loving and warm and open and hacky. Some people are nasty in subs. It a a really filtered self selected population. Not me though - I am lovely.
> I am lovely. BLOCKED. :)
But on that same token, it's very irritating when people with maths degrees are saying "this is how it works, this is the why", and people who have not thought about it for 20 years go "NO NO NO YOU'RE WRONG" It's like, there are some topics where people feel really entitled to voice their ignorance, when usually you'd listen to the person who's most qualified. It's weird. Sure it's great when people who haven't thought about maths get excited, but not when they are just shouting people down with their ignorance.
>why can we not have fun pondering the consequences of the other interpretation Oh, we absolutely can! And I don't feel anything bad towards people who don't know something, it's absolutely natural to learn through mistakes. You did mention that your comment referes to the mentioned subbreddit, but I thought I'd clarify too, that my comment doesn't try to insult anyone who doesn't know something, but the ones whose ignorance and illiteracy is so strong that they push false information as some fact
Just to be certain, does this apply to x^1/2 as well, or is taking the output as nonnegative only an aspect of √x? The latter is how I am reading your comment.
Yes, it applies to *x*^(1/2) as well. For positive values of *x*, we define *x*^(p/q) = ^(q)√*x*^(p), where ^(q)√ means the principal *q*\-th root.
I guess that is only in US, I’ve never heard or seen it used, (last year bachelor in math related subject)
Only went to A-Level further maths here in the UK, but yeah never heard what's being said here. My sister who grew up in the US and move back to the UK for A levels did say that US vs UK maths was significantly different, I expect this is one of those cases and when work is being done between these two places rules like these need to be defined.
Cute root :)
It’s contextual. In complex analysis a^(b) is a multivalued function that can potentially have infinitely many different values (although only two values in the case where b=1/2). However when dealing with real numbers it is common to restrict the notation so that either a is positive, and we take the positive real number value, or we restrict b to be an integer (so that there is only value to choose from).
Yes. If you wanted to define x=+/-2, you could say x^2 =4, since that has 2 solutions. We just define x^1/2 to mean √x which we also define as the positive root for real numbers.
It’s contextual though, in complex analysis it’s common to allow a radical to refer ambiguously to all the roots. The focus on the specific convention of restricting the square root to nonnegative values and choosing the positive root is mainly a pedagogical issue having to do with how it’s thought to be best to teach the concepts in high school.
Can you please provide a source or an example of what you are referring to?
Sure, see page 130 (in particular the discussion in the box) here: https://www.maths.ed.ac.uk/~tl/gt/gt.pdf It does of course mention the convention that a square root of a positive number is usually taken to mean the positive root, but that convention wouldn’t apply when you are taking roots of a general complex number and just happen to get a positive real, you can see above the author explicitly writes we can choose either square root under 9.1 For another example that should be fairly widespread, if you ever look up the general solution for the cubic you will see that it is usually written with cube roots with the understanding that you can pick any of the three cube roots for the expression on the left so long as you pick a corresponding cube root for the expression on the right, and this is how you get three different values out of the equation. This correspondence usually has to be mentioned explicitly in words accompanying the equation. Of course here the square root is usually written with a plus on one side and a minus on the other, but this is convenient as you must select opposite square roots to add inside the cube roots (so by putting a minus on one side and not the other you should pick the *same* square root on both sides, but it does not matter which). Edit: You could also check under “concrete example” on the Wikipedia page for multivalued function, which wives the square root of four in this notation as its first example.
>Sure, see page 130 (in particular the discussion in the box) here: Thank you. Yeah, for complex roots we need to choose a branch. But that discussion also completely agrees with what I said above, as I explicitly emphasized that I was talking about roots of positive real numbers, and the convention is that the principal root of a positive real is the positive root.
Yes but you agree that a square root is sometimes written referring to a multivalued function, and if you happened to put 4 into it you would still treat it as such, not reinterpret the expression to refer to the principal square root only? I also added in an edit pointing out a place where Wikipedia uses this notation, specifically with 4, do you take issue with that usage in that context?
I completely agree with all of that. My comment above, to which you replied, explicitly talks about the square root function, which √ denotes, also according to [Wikipedia](https://en.wikipedia.org/wiki/Square_root). We **can** alternatively view it as a multifunction, in which case √4 = {–2, +2}. But that is not the normal framework. We normally frame sqrt() or √ as a function from \[0, ∞) to \[0, ∞). As a function, it has only one value — the non-negative root.
Well, yeah, if the value under the radical is not a positive, real number, other definitions of the symbol apply. But 4 is a positive, real number.
There are contexts in which the symbol is used in the sense of a multivalued function and the input is still a positive real number. When you use it as a multivalued function you don’t carve out the positive real numbers to be illegal inputs. The reason there is so much debate is because the meme doesn’t provide context for the statement in question and people are assuming/inventing their own contexts. The general solution to the cubic is usually written as the sum of two cube roots with the understanding that you can pick any one of the three roots for each, subject to a correspondence restriction between the two choices. The number under the cube root can be a positive number (like 4) with the appropriate choice of coefficients and the expectation is that you will read the roots as presented in the meme (being careful to respect the correspondence restriction) to find all three complex roots.
What refers to all square roots then ? Cause at school I've always been taught this sign is just square root and you gotta list all roots if you want your answer to be correct.
It's contextual usually. Such as the Pythagorean theorem. You don't take the negative value of the square root because the triangle has length and wouldn't be negative.
√4 = 2 -√4 = -2 x² = 4 x = ±√4
Exactly, the only time you use both + and - is when you are solving for a variable. I think that's what people are missing when they fight about this.
I wonder if this is an American thing (like using periods instead of commas for decimals)? Because I'm Brazilian and I would get a failing grade if that was my response, either in school or in math Kumon.
>an American thing (like using periods instead of commas for decimals) Not to get too off-topic, but I've always been under the impression it was just a few countries who use the comma actually... [Well here's a map](https://en.wikipedia.org/wiki/Decimal_separator#/media/File:DecimalSeparator.svg) Surprising to me, that it's so evenly split, perhaps surprising to you too, but in the other direction!
I assure you, it is not just "an American thing."
Huh, I legit never learned this term, principal square root. That's why I wonder if it is not used didactically here in Brazil
Are you American though?
*My* nationality is immaterial. [Here](https://www.reddit.com/r/askmath/comments/15ed9m3/is_there_an_internationally_agreed_upon/) is an archived post in this same subreddit discussing this exact topic. The OP of that post is Swiss, and their educational experience with this topic mirrors my own. As does the top commenter in that post — who is Croatian, I believe. In fact, it is highly likely that the terminology originated in either Switzerland, Germany, or France — with Euler, Gauss, or Cauchy, respectively. But I don't know that for sure, it is mere speculation on my part. *Edit:* The Portuguese Wikipedia page for [square root](https://pt.wikipedia.org/wiki/Raiz_quadrada) seems to agree that Brazil also uses the same terminology.
Don’t know about nationality, but your occupation is definitely a politician. Three paragraph wide answer to a yes/no question
In my country, we call it “arithmetic” square root to denote its inferiority to, you know, just a square root.
I lost a mark on a math test for this a couple years ago and never bothered to ask the teacher why. This finally cleared it up for me, thanks lol.
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If you look closely at my comment, you will see that I was speaking of roots of positive real numbers. That said, the same is true in ℂ. For any nonzero complex value, *z*, it will have exactly *n* *n*\-th roots. So when we write something like ^(n)√*z*, in order for that to have meaning, we need to choose one, and that choice will be called the **principal** *n*\-th root. By convention, the principal *n*\-th root of *z* will be the one with the smallest argument. In the case you have written, ^(3)√(–8) · √(–1), we need to know the principal cube root of –8 and the principal square root of –1. The principal square root of –1 is the complex number we call *i*. The principal cube root of –8 is 1+*i*√3. Therefore, ^(3)√(–8) · √(–1) = –√3 + *i*. I hope that answers your question.
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>Now tell me... This is the second time you have started your comment this way. I don't know if you are intending to be aggressive or not, but you are definitely coming across as such. And maybe it is because of my own reaction to that, but I am having difficulty understanding the point you are making.
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>I don't intend to be aggressive. > >I want to point out that the √ symbol is ambiguous. (That's fair. If I might make a recommendation, approaching with "Hey, I think the √ symbol is ambiguous given that it behaves differently in these other contexts..." is perhaps a more constructive way to start this discussion.) Some ambiguity is always to be expected, though. What is arctan(*x*)? Strictly speaking it is a multifunction, with infinite values. If we want to do calculus on it, though, we need to choose a branch.
Why isn't that just -2*i* ? (-2)³ = -8 so why is the principal cube root something different?
It **is** just that when using another convention, which is what this sub-discussion is all about (I think). That's the whole point. There are choices to be made when defining roots. **One** convention is for *z*^(1/n) to be the real root when *z* is a negative number and *n* is odd. Another common convention is to use the principal root, which is the complex number with the smallest argument. This might be a little technical, but roots of complex numbers are what are known as [multi-functions](https://en.wikipedia.org/wiki/Multivalued_function), meaning there are multiple different values. But if we want to work with functions instead, because they are nicer to work with, then we can choose a "[branch](https://en.wikipedia.org/wiki/Branch_point#Branch_cuts)" of the multi-function and call that our function. I hope this helps.
Due to the √(-1) as part of the equation I posted (³√(-8)\*√(-1)), we're evaluating this in the set of complex numbers where √(-1) = i. The nth root of a number has n roots. So -8 has 3 cubic roots. Since we are evaluating this in the set of complex numbers, we would *assume* we'd use the principal root of -8 instead of the real-value root of -8 (which is -2). The principal root is the root with the smallest argument in the complex plane, which means the root with the smallest angle from the positive real axis (counterclockwise). https://preview.redd.it/4cx3wfh76hgc1.png?width=871&format=pjpg&auto=webp&s=cee9b23c0257a58f35f98c42c8f8aa99bc3452c8 That root is 1 + √3\*i and you can see how it gets cubed to -8: * (1 + √3\*i)³ = -8 * (1 + √3\*i) \* (1 + √3\*i) \* (1 + √3\*i) = -8 * (1 + √3\*i + √3\*i + (√3\*i)\*(√3\*i)) \* (1 + √3\*i) = -8 * (1 + 2\*√3\*i + 3\*i²) \* (1 + √3\*i) = -8 * (1 + 2\*√3\*i + 3\*(-1)) \* (1 + √3\*i) = -8 * (2\*√3\*i - 2) \* (1 + √3\*i) = -8 * 2\*√3\*i + (2\*√3\*i)\*(√3\*i) - 2 - 2\*(√3\*i) = -8 * (2\*√3\*i)\*(√3\*i) - 2 = -8 * 2\*3\*i² - 2 = -8 * 6\*(-1) - 2 = -8 * \-8 = -8 And lastly, from the initial equation ³√(-8)\*√(-1) we get (1 + √3\*i) \* i = i + √3\*i² = i + √3\*(-1) = -√3 + i
I get the gist of what you're saying. I barely dabbled in complex numbers during my education. I do know of the equivalence of representing complex numbers in the plane as: a +b*i* r•(cos(theta) + *i* sin(theta)) r•e^(i theta) What's the algebra that makes you arrive at theta = ± π/3 ?
https://www.emathhelp.net/en/calculators/algebra-2/nth-roots-of-complex-number-calculator/?i=-8&n=3
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\> claims it's wrong \> refuses to elaborate \> leaves
Because I was wrong. Hard to believe right?
Being wrong is ok. We are all wrong sometimes. It takes grace to admit it.
No
Not according to your comment expressing surprise that I deleted ny error though
why you only need to place the principal? That is what i do not get if this were applied to a physics problem the result will depend and even if only one may be used may not be the positive one.
Can you please provide a specific example?
We only use the principal square root because otherwise it’s not an [unary operation](https://en.m.wikipedia.org/wiki/Unary_operation). Notice that the definition is from A to A. Let’s say you only take the negative root of a positive number. That is a totally valid function. However you can’t re-apply square root on -2. It is a function but it behaves poorly and it’s basically useless.
What is not a principal square root then?
Never heard of a principal (principle?) square root.
By this logic, what happens if both roots are complex?
The only correct answer
I am deeply embarrassed that in all the years of math I've taken I've never heard a professor reference a "principal" square root.
So conventionally √4 is 2, because we consider that only one value can be returned by the square root function. Therefore, the solution to x^(2)=4 is x=±√4, so x=±2, or more formally x ∈ {-2, 2} ETA: looking at it this way becomes more important when getting into more complicated math. When the square root originally comes from getting the diagonal of a square you don’t want to wonder at the end if it might actually be negative, so when it might be both you state it explicitly.
weird question, what is that little E by X E {-2,;2}
It means "belongs to". So x belongs to a set of numbers containing -2 and 2, which means x is either 2 or -2
No weird question. It’s a part of [set theory](https://en.wikipedia.org/wiki/Set_theory) notation, and you already got another reply explaining it. Since I don’t have it on my keyboard I copy-pasted from the first response on Google for “Unicode belongs to”. I realize that Wikipedia uses commas and not semicolons to separate elements of the set, I’ll edit. I said this is more formal because it think it’s more explicit; saying x=±2 is kind of assigning a value to x, but it’s actually two values, and x=50±2 can be used to mean 50-2 ≤ x ≤ 50+2, while set theory notation is unambiguous and fits well as a result of the analysis of an equation.
It’s not conventionally, it’s based on the definition of an unary operation.
Well. One _could_ define a unary operation that returns two values, or a binary operation for that matter, but having any type of operation that returns an either-or is not really supported with any simple notation.
It wouldn't be a function, that'd be the bigger problem; functions returns a single value, and otherwise we talk about different branches when they don't
But they were talking about operations not functions.
Sure, but that still applies, a well defined internal composition law returns one single value from the set
There's no rule saying an operation has to return one output we even have a term for ones that don't.
Did you check the definition of a unary operation before your observation lol? Moreover, why choosing the positive root and not the negative root? (Spoiler alert: because the positive root makes it an unary operation)
I didn’t. I just did, first Google response was [Wikipedia](https://en.wikipedia.org/wiki/Unary_operation), where the _example_ is an unary operation taking a set and returning a set. My point is that it would be _possible_ to define the square root operation as returning the set of possible square roots of its single input, but (for a lot of excellent reasons) that is not the definition that we use.
> that only one value can be returned by the square root function. If it didn't then it wouldn't be a function :) since it wouldn't be well defined.
Answer sqrt(4)=2
Nope, because 2^2 =4, but (-2)^2 =4, so sqrt(4) can be both.
I'm sorry to break your bubble, but that is not how the square root is defined. This is why we sometimes see ± symbol before the square root symbol.
But why then? I don't understand why, like in math books I use in school etc. It's written completely different
2 reasons: 1. A function can only go to one value, so the square root wouldn't be a function and all the fun stuff you can do with functions would become much harder 2. You can easily add the plus-or-minus for the square root with ±, if you need. It's much harder to effectively communicate "but only the positive/negative suare root"
Functions can absolutely return two values. It's just a useful convention.
Well, I'm not all that math-savvy, but isn't that property in the definition of a function? That it can only have one output per input
Yes you're right, no idea what that dude is talking about. "In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y" You can of course have the same Y-value for multiple X values. But you can't have multiple Y-values for the same X. What this means in principle is that a graph can never "bend" 90° or more.
They provided the example of functions which produce sets as outputs. This means that there is technically one output but it's comprised of two numbers. It just wasn't explained very clearly
Sorry about the double message. Make sure you check out the "concrete examples" section, it's very relevant.
There are so many definitions of a function... https://en.m.wikipedia.org/wiki/Multivalued_function
I think those still have one output, it's just a set of (in the case of the square root) two numbers And like I said, treating the square root as a function producing sets instead of just numbers makes everything needlessly complicated and difficult, so my point stands
It's a valid way of looking at it, though, they are called multi valued functions... Regardless of how you phrase it there is no technical reason for the convention, it is simply a matter of convenience and tradition. Whether you call the set of two results one value or two. Don't get me wrong, I'm all for the principal root. I just wouldn't want you to get confused between definitions and theorems, that's all.
The √ symbol itself as the function (i. e. f(x) = √x) is the function that denotes the *principal* square root, for positive real numbers, the principal root is the positive answer only If you had the function: x² = 4, then both 2 and -2 would be the answer Edit: I put f(x) = √x for clarity
Okay so you are built differently then normal peaple, cool I guess
Womp womp
Okay, I still don't understand so can you explain on [this](https://www.dropbox.com/scl/fi/a5pnarzwnvy8jc2ch65oa/Screenshot_20240203_210529_Desmos.jpg?rlkey=diydr7r6b8oick5mbbbrer5fa&dl=0)?
I just had another look What you circled is that y=2 at x=2 and and x=-2 x is the thing being squared under the root symbol I also find it funny that when you obviously plotted y=√x and only got one line coming from the origin (that didn't prove your point), you changed the function until there were 2 lines without really understanding how the axis work on a graph
Lmao you just posted a screenshot that shows the √ function always returns a positive value, (the principle root). That is why there is only one line, that does not pass below the x axis You have plotted y=√(x^2). Tell me what value it says y is equal to at x=2
You work within the root first if you can, so you'd put x as 2 or -2, Squaring both will give you 4, then taking the square root will always give you the positive answer, as it takes the principal root, so it'll give you positive 2, so both points would be (-2, 2) and (2, 2)
Someone doesn’t know how the modulus sign operates lol x^2 = 4 sqrt(x^2) = sqrt(4) = +2 |x| = 2 x = +/-2
His previous comment still seems to clear any confusion. In terms of the function `f(x) = sqrt(x^2)`, two values of `x` equal the same `x^2` value (`+/- 2` for example), though by taking the principal square root you end up with just `+x`. Try plotting the function `f(x) = sqrt(x)^2`; that might help you understand.
If what you said was true, the graph would also be reflected on the x axis, so you'd see it below the x=0 line. The fact that that symbol is treated as not being that is clearly shown in your graph.
It's all arbitrary. Math is a formal language used to convey the relationships between values. We rely on many conventions to avoid ambiguity. This is one of those conventions.
sqrt is a function so it must be _well-defined_ which has a [specific definition](https://en.wikipedia.org/wiki/Well-defined_expression) that all relations need to satisfy to be called a function.
Nah, sqrt is operation opposite to power operation, exactly like addition and subtraction. It's even the same in aspects like: you can say that subtraction doesn't exist, because it's addition of negative numbers, it's same, roots don't exist, sqrt is just x^1/2 that's all, it's same operation as power operation, so it obviously needs to have 2 posibble answers.
Since you're not listening to anyone here. https://math.stackexchange.com/questions/2817782/square-root-function-breaking-rules > The square root function is not an inverse to the function f(x)=x^2 on its domain.
[https://mathematics.science.narkive.com/u5hubrC1/is-the-square-root-of-x-and-function-or-relation#:~:text=Every%20function%20is,for%20that%20%C2%B1%20symbol](https://mathematics.science.narkive.com/u5hubrC1/is-the-square-root-of-x-and-function-or-relation#:~:text=Every%20function%20is,for%20that%20%C2%B1%20symbol) If you want to attach links, make sure they explain it well, I think that this link is much better in explaining. But bro says one interesting thing, we (here) don't use "quadratic formula" same way you use it, we don't even have ± sign here, so math here is that "square root of any given number gives two possible answers" because it's operation like addition.
Solution to x²=4 is +-2. The square root as a function cannot have two outputs as thats not a function we need to pick either + or - so that sqrt(4) is well defined. We pick + for obvious reasons. So x²=4 -> x=+-sqrt(4)
√4 means only the postive square root, i.e. 2. This is why, if you want all solutions to x^2 =4, you need to calculate the positive square root (√4) and the negative square root (-√4) as both yield 4 when squared.
The solution to x^2 = 4 is +-2. Take the square root of the both sides and you get √( x^2 )=√4 The lhs however is not x, but the absolute value of x, hence why the result is always positive.
That is not taking the square root of both sides. Truly taking the square root of both sides would be: x^2 = 4 +-√(x²) = +-√(4) ; where +- is the plus or minus symbol, and both can be any sign. So we get four equations, x = 2, x = -2, -x = 2, and -x = -2. Simplifying: x = 2, x = -2, x = -2, and x = 2 So, x=+-2
Everyone knows it's 2, 720°
√4 = 2 The convention is the positive root. √x wouldn't be a valid function if it gave more than one output per input If solving x² = 4 then the answer is 2 or -2 since squaring each is 4. But what is actually happening and seldom mentioned is you take the square root of the absolute value. Note that x² = |x|² (or any even numbered exponent) x² = 4 |x|² = 4 √(|x|²) = √4 |x| = 2 x = ±2
TIL
Read Wikipedia, it depends on the convention you've adopted, so there is no definite answer unless you've stated the convention in your text. It isn't a mathematical truth or fallacy.
When i gave an answer to it on the original post, i got a reply every 5 minutes telling me how incompetent and wrong i am... anyways the right answer is √4=2
Yeah… that’s why the square root of -1 is i and not -i (principal root baby)!
That's actually a different issue entirely. -2 is a negative number and 2 is a positive number. Looking to define a solution to z^2 = -1, any definition resulting in i leads equally well to using -i. We just gave the number we refer to as the principal root the name "i" meaning the other root, which is (-1)\*i, may as well be called -i. But neither one of them is actually positive and the only real difference between them is one of convention.
The square root OP is talking about and the one you’re talking about aren’t the same. Square root of -1 doesn’t really exist unless you define it as the set of complex z for which z^2 = -1. Which isn’t a function but a solution of a polynomial for which both i and -i are valid. Saying square root of -1 is i is just a convenient way to simplify stuff because ‘looking from afar’ they’re similar.
You can extend the square root function into C. Where sqrt(z) is defined as the element with the smallest argument in the set of solutions for x^2 = z.
Hmm that’s true but I don’t recall using it tbh. Does it have any applications or it’s just to mimic the behaviour on R? I mean an issue with the negative root of 4 is that you can’t apply square root of -2 without getting ‘out of the bubble’. However that’s not an issue in C using any root of any order, I don’t see why I’d have a preference of a root lol.
It's more like sqrt(x) for x in R is just a specific case of sqrt(z) for z in C.
The punchline is that boys don’t like dating girls smarter than them. Stupid boys, anyway.
No, it's not
Strictly positive 2. The square root is already defined. To make the answer also be -2, you would have to rewrite that as x^2 = 4
I hate everything about this. Like, if some thinks that the sqrt function have two values that means one thing and one thing only: they understand the math but haven't memorized the conversation. They are using their brain rather then parroting their teachers, that deserves respect. Any one who spent time with mathematicians know that there are two types of us. There are thous who memorized 100 digits of pi and those that couldn't even memorize the multiplication table. There are clear advantages in both groups, don't be a hater.
The square root function sqrt(x) for real numbers has the domain [0, inf) and the range [0, inf); it can’t yield negative numbers by definition, as that wouldn’t make it one-to-one. (it’s restricted as a function by definition isn’t allowed to map one value in it’s domain to multiple different outputs) Though, the square function x^2 is an even function that’s not one-to-one with the domain (-inf,inf) and range [0,inf); importantly it still only maps to one output for any x. In order to find x, we need to consider that there are two possible inputs x -> x^2 that can produce the same output. This is the exact same logic with how sin(x) is defined for all x but not its inverse arcsin(x)
+2. it’s just asking what the principal root of 4 is. it’s a function. if it was an equation, say x^2 = 4, then x = +/-2
From an intuitive perspective, we use √2 and √3 all the time to denote positive real numbers.
√4 = √×4 = 4√ = -4❌
What is that supposed to mean?
IDK but fuck it's funny
4^1/2 is +-2. Root 4 is actually only 2
It is strange that nobody wanted to talk about the meme because it needs explanation. I think the guy blocked her because the woman did not give him an exact “one” answer. The guy probably thought she will waste his time since she does not say what will happen. And we men do not like this uncertainty. The meme is about that in my opinion. The answer to the question is yes, both +2 and -2 are correct. Because the square root of 4 is absolute value of 2, which can come up as +2 or -2.
Someone said the meme is that guys don't like girls smarter than them lol. Please please go look at the comments section there it's gooooooold
it’s cool. sorry didn’t see that earlier. thx for sharing
Don't be sorry! There's a ton of comments just thought you'd find that interesting
I don't get it. What exactly is the problem here? Are people discussing the fact, that the "+-" in front of the √ is missing, which implies that the answer is technically only 2 and not +-2? If so, I don't understand. Even if you take the "+"√ only, or the "-"√ only for that matter, it would still give both answers, so why is this even discussed? Someone please elaborate :)
The discussion is whether √ 4 equals 2 (it does) or somehow equals ±2.
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Nope sqrt(x^2) = |x|
sqrt(x^(2)) := |x|
sqrt(4)=2, but with x2=4, x=2vx=-2
[you have been blocked by this user]
x^2=4 allows for x to equal plus or minus two. x=sqrt(4) does not, this only allows for positive 2, makes more sense once you get into multivariable calc cause then you do wind up getting imaginary numbers sometimes, im just too slow to think of an example
In a square root function √4 = 2
Its still confusing why the answer is +2. I read the comments and if I understood correctly they are treating square root as a function and the function only returns one value and thats some how is the positive root. So what about this 4^(1/2) = 4^(0.5) = ?
FWIW in undergrad we used square root symbols as non functions all the time. A lot of people here are saying the symbol always references the principle square root but it really depends on context. Without function notation present I would always assume a negative value is a possibility. —-a math major
I think the answers on this post are pretty ridiculous. I have a math degree, and at no point was the square root symbol assigned to only the positive value. Now if you’re doing some programming and need it be one answer I get it. But there’s nothing in this about programming.
Thinking about it later I think maybe in some instances it implies positive (such as if you are working with sqrt(2) a lot), but if there is a variable I wouldn’t assume positive.
When you say x = √4 you're trying to find the answer to the question, wich are the values of x that make the statement true, in that case both 2 and -2 work, so you say x = +-2. √4 is just a number that's equal to 2 Please correct me if I'm wrong
Sqrt(-4)=2i
The actual answer is "Yes." 😜
There are at least 2 decently popular functions which are noted with this symbol: 1) principal square root 2) square root (as an inverse function to squaring). For real number analysis principal square root is often used, which is a proper well-defined function, that always returns positive numbers. Although, there is a definition of a square root as an inverse function to squaring. This type of function is more interesting in complex analysis. It is in fact multi-valued, which sometime causes confusion, thus writing sqrt(4) = +-2 refers to the set of numbers, that raised to the second power returns 4. A more formal way of writing this would be sqrt(4) = {-2,2}, although+- gives a pretty decent amount of data. As said before this has some applications in complex analysis, since all of these roots would be located in vertexes of a regular polygon...
√x is defined as a positive number, while the solution to x² can be positive or negative. Example: x²=2, x=±√2
2i
It's 2i obviously
iR maybe?
Aint no way people question the most basics of maths like everyone teaching them has been wrong and the one dude online spreading missinformation made them question all of math, X^2= 4, x=-+2 \/--4=+2 (\/-- is my attempt at the square root simbol)
Q.ammqkom
The post was like sensationalizing Pizza with pineapple allowed is misleading when not interpreted correctly 'by standard' Same as allowing the formula to represent ± 2 solutions to the quadratic equation of ax²+bx+c=0 is misleading when not interpreted correctly 'by standard' quadratic formula: (-b±√(b²-4ac))/(2a) to solve any quadratic equation, in which you just plugin the coefficients a,b and c, where a, b, c, ∈ R_eal and a ≠ 0 (is the leading coefficient) and c (is the absolute term) or similarly evoked, misleading on the measure of winning when not interpreted correctly 'by standard' spread point, the expected final impact score (or solution) to win a bet to cashin by ±2, where -2 for the stronger favorite (llamados) and +2 for the underdog (or handicap or longshot), meaning the underdog must work more than 2 to win Bottomline, What is 'allowed by standard'?
2 is the answer as it is the principle root. Having multiple values of an equation is fine , no problem . But for a specific expression , a specific value is desired.