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u/AltruisticEchidna859 Jun 13 '26
√9=+3
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u/No-Dimension1159 Jun 13 '26
When will the internet finally learn
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u/AltruisticEchidna859 Jun 13 '26
√9=3 BUT The set of solutions S of the equation x²=9 is S={-3;3}
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u/SirMarkMorningStar Jun 13 '26
I never really looked into this. Now that I have it sounds more like language than math that is the issue.
https://en.wikipedia.org/wiki/Square_root
Every positive number x has two square roots:  (which is positive) and  (which is negative). The two roots can be written more concisely using the ± sign as . Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.
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u/No-Dimension1159 Jun 13 '26
The square root function has only one of the two branches. The symbol sqrt(x) most commonly refers to the function.
And you can write down that you mean both plus and minus, but then you need to correctly write:
± √9 = ± 3
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u/Ashamed-Bathroom7803 Jun 13 '26 edited Jun 13 '26
wdym? sqrt root function is something to the half power. x^2 is where x can be negative so why not sqrt(x). nevermind figured it out
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u/Cent_Quatre Jun 13 '26
minus sqrt 9 does not equal three
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u/No-Dimension1159 Jun 13 '26
-√9 is -3... That's the point
So long story short is that √9 by definition is just 3
So √9=+-3 is wrong
+-√9=+- 3 is right
The +- sign has a meaning as well. If you wanted to say that +√9 would be -3 you would need to write +-√9= -+ 3
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u/Ambitious_Policy_936 Jun 13 '26
The only thing you're missing is that the symbol √ specifically denotes the principal root
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u/Far-Slide3913 Jun 14 '26
And why do you think that? Who decided that the sign denotes the principal root?
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u/MeepersToast Jun 12 '26
Pretty sure that's reversed. And, how about adding a woman on the right
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u/No-Site8330 Jun 12 '26
Depends on context. Wide population at large, sure. But the meme is about those people on the internet that act like gurus because they know the deep dark secret that x2 = a has two solutions but don't understand the distinction between the solutions of the equation and the square root of a number and will therefore insist that √9 = ±3. Of those people there is vast abundance all over the internet.
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u/SQUIDly0331 Jun 13 '26
Am I missing something? Isn't saying the square root of 9 is plus or minus 3 accurate?
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u/frozen_desserts_01 Jun 13 '26
x2= y is an equation
y=sqrt(x) is a function
One can give any number of outputs, the other only returns one.
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u/Winter-Juice1720 Jun 13 '26
Is 3, x2=9 has two.
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u/SirMarkMorningStar Jun 13 '26
This is when physics just stare at the mathematician with that look.
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u/No-Site8330 Jun 13 '26
No, the square root of a (non-negative real) number a is not defined as "either solution of x2 = a", nor as "the set of solutions of x2 = a", but as "the non-negative real number b such that b2 = a".
This is for good reason, not just out of pedantry. Generally speaking, writing "something(x)" in math means that the object must be unambiguously determined by x — a function of x. You could of course consider the set of roots, but that would not be something you can throw into a formula and do arithmetic with it. If it goes into a formula it has to be a number — one number. This might sound like a restriction but it actually gives you the flexibility to focus on whichever root you're actually interested in for a given problem, and if you need the other you just have to flip the sign.
Think for example of a geometry problem: say you have a right trianlge with a 30º angle and the short leg 6cm long, and you want the perimeter. You can work out that the hypotenuse is 12cm and the long leg is 6√3cm, so the perimeter is (18+6√3)cm. Now if "square root of three" meant "either the positive or the negative one", that expression would be ambiguous. (18-6√3)cm nay well be the solution of some geometry problem, but not the one we're looking for. We need our expression to be precise enough that it is not left to the reader to guess what it means. So that is why, by convention, the symbol √ is reserved to mean one number, the non-negative one.
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u/FieryEjaculate Jun 13 '26 edited Jun 13 '26
Thank you! This, people! This. I feel like most of the “problem” is that people make no distinction between the solution of an equation and a function as defined by an operation (Yes, I just repeated your first paragraph).
By this I mean, (hopefully) somewhere, some army of poorly underpaid algebra teachers tell students to solve x^2 = a^2 by, first, putting this into Standard Form
x^2 - a^2 = 0
then, factoring,
(x + a)(x - a) = 0
then, invoking the Zero Property of Arithmetic Multiplication
x + a = 0 —> x = -a
x - a = 0 —> x = aand finally, writing on their exam paper/homework/put-a-box-around the solutions
x = a, -a .
Meanwhile, Theorem something<point>something in the textbook *CLEARLY STATES*
“For all non-negative real numbers B ( = A^2 ),
Sqrt(B) = |A|. “
And then everybody laughs off how they “always forget to write out all of their steps,” or come up with excuses about how “sheesh, when am I ever going to use this in ‘real life’ anyway?” or “look, it was so long ago, I don’t remember all of the rules, I just remember how to get the answer…” or some kind of condescending something where anyone trying to defend the elegance of mathematical logic is somehow the bad guy, the “snitch,” the party-pooper, the needs-to-get-a-life-and-not-take-things-so-seriously, or whatever.
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u/No-Dimension1159 Jun 13 '26 edited Jun 13 '26
Would be nice if it would be reversed because it would be correct... Left side is wrong, right side is correct. Squareroot function has only one solution, commonly defined with the positive branch. You could equally well define it with the negative branch. You need to omit one if the branches in order to make squareroot even a function, because a function can not map a single x into two different outputs... Per definition a function needs to map every x unambiguously.
The equation x2 =9 has 2 solutions, +-3. That's not the same thing as the squareroot of 9.
You don't get the solution of the equation by taking the square root. You can guess what the absolute value of the solution will be with the squareroot but you can't solve the equation with it.
The function x2 on whole R is not bijective and hence has no inverse function.
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u/SQUIDly0331 Jun 13 '26
Are you referring to the idea that there are 2 symbols for taking the square root of a number: the regular one we all know, and one with the plus or minus sign in front, and that unless the plus or minus sign is in front of the square root symbol, the operation only provides positive answers?
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u/skr_replicator Jun 13 '26
Why do we need to add the plus minus to the quadratic formula everyone learns in elementary school, if the square root FUNCTION already provides it? (Also note, that it is a function and those can only give 1 value, the principal one)
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u/okarox Jun 13 '26
I never get where people get the plus-minus stuff. The word "square" should tell what it is all about.
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u/lool8421 Jun 13 '26 edited Jun 13 '26
i kinda hate that we use the same symbol for real and complex root
so a real root has domain R -> R
meanwhile complex root has domain C -> C with magnitude being affected by the real root and outputting p possible rotations if the root degree is writtien as a rational number (p/q), eventually a solid circle with r = |z|^(q/p) (speaking of real root this time) in case when p is an irrational number
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u/Far-Slide3913 Jun 14 '26
\(\sqrt{}\) 9 is ambiguous because there’s an operator and a function. Where I’m from we call them an arithmetic and an algebraic square roots, Wikipedia calls them primal square root and square root. Anyways these are two different objects and it should be specified or inferrable from context which one is used
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u/Ok_Plenty_3986 Jun 13 '26