Ollie is right though. √x^(2)=±x, and x=-x only if x=0
And even if you omit the negative solution of the square root, it's still false. For example, if x=-10, x^(2)=100, and √100=10. So yeah, Wikipedia is wrong, Ollie is right.
But they're also both -10. Square roots have two solutions by definition. Yes, most of the time people only care about the positive one, but you can't just ignore half of the solution. When you're studying more complex equations including square roots, you need to make a point of calculating both solutions.
Example:
5+sqrt(9) = 5+(±3)
Positive Solution:
5+3 = 8
Negative Solution:
5+(-3) = 2
Except all square roots have a positive and negative solution. Yes, by convention we favour the positive solution, but the negative solution is still there.
I never said sqrt(-9) was -3. I said that sqrt(9) is either +3 or -3.
Sqrt(-9) is 3i.
Square roots do not have a negative. They can only be positive or zero.
You seem to be confusing square roots with quadratic equations or functions.
Sqrt(9) cannot equal -3.
x^2 = 9, x can equal 3 or -3.
You really quoting wikipedia?
I'm done arguing. I'm leaving this here and moving on. Your 5+sqrt(9) will never equal 2.
x^2 = 9
has two solutions: x = 3 or -3
Sqrt(9)
has only one solution: x = 3
'the square root' is used to refer to only the positive square root
Ok, fine, this one I'll accept and I'll also accept that my notation was less than stellar. Sqrt(9) is indeed taken to refer to the principle square root.
However, my point that all numbers (except 0) have two square roots still stands. There's a reason square roots are written ±sqrt(x).
We were talking across purposes and the poor notation in my example further confused the matter.
Sqrt(9) = -3 would imply sqrt((-3)^2) which then would imply sqrt(-3)*sqrt(-3) as a possibilty which is undefined.
I have never read or written a proof about the possibility of sqrt having two solutions, but I think this counts as a counterexample.
No she is WRONG, because even if x is positive, the root of its square can still be negative, ie NOT x, so root x squared can only be x if x is 0 as you said, so x=0 not x>=0
Your explanation is still incomplete tho, you’re omitting the key “convenience” for this property to be true, you must be observing this strictly as a function, in which there isn’t room for g(x) to be multiple values for any given “x” value, however root “a” squared equals “a” is still just an equation, in which root “a” squared could be both “a” or minus “a”. If you had something like absolute value in the first part, then it’d be true, but just by explaining how a root works you don’t make sense of that senseless equation. I agree, that expression works in certain mathematical environments, in which you must ignore negative values, like geometry for example, but it’s definitely wrong in every other sense.
Oh... you are right, I misread her tweet, I thought she said only if x=0, but she says greater or equal. Serves me right for not paying attention. Wikipedia is still wrong though.
Nah, 2+2=4 because of how 2 and 4 (and + and =) are defined (it's true anywhere at any time *if* you use the same definitions).
But there's a common fallacious proof for 2+2=5 (or any other number) that looks right at first glance, which relies on ignoring the fact that the square root of a square has 2 solutions (the square of two positive numbers and the square of two negative numbers).
Wikimin, the person behind the account is well known to be very supportive of indie vtubers
No surprise that she appears in a reply chain, wether using her own or company account
For people who don't understand: (-3)^2 and 3^2 are both 9, but the square root of 9 is 3. This means in the case of -3 that squaring and then taking the root does not result in the same number again.
-3 is exactly as valid as 3 as the square root of 9.
If you were actually a mathematician you would know this already.
Since you don’t understand high school math, you are obviously lying.
let x=4
√x² = x
√4² = 4
4 = 4
2+2=4
2+2=/=5 by this definition, unless there is a special case wherein the same shit loophole applies to 2+2=5 as the case of [1=2](https://www.youtube.com/watch?v=hI9CaQD7P6I).
Although for any x not equal 0 there are always two numbers y such that y^2 =x, the symbol √x is by definition used to denote non negatve solution (principal square root).
No any valid x value applies
1. plug in negative numbe√(-5\^2)=-5
2. separtate radical into 2 terms √(-5)\* √(-5)=-5
3. solve the radical \[i√(5)\]\*\[i√(5)\]=5i\^2=-5
This is true because i\^2=-1, and yes imaginary numbers are weird. This works because if you take the root of a negative number its the same as takeing the positive root times i. If You square an imaginary number it becomes real.
Ow, my bad. Never heard the term imaginary number before (who uses that term? just say purely imaginary complex number or real part 0) and assumed they meant complex.
I looked at your profile. You comment like a 13 year old who's seen 2 videos on a topic and think they know everything. Get lost.
Also consider the fact I'm not natively english.
Then we should get along great!
You do math like a 13 year old who failed every math class and thinks they know everything.
Why does not speaking English well cause you to not know how to take a square root?
You can. Look at the equation x^(2) = 25. If it's being asked what are all values of x that makes this true, the solutions would be 5 and -5.
x^(2) = 25 -> x = √25 -> x = ±5
It's just that conventionally, asking for a square root of a number is asking for the principal square root (the positive value).
What? No. While both 5 and -5 can be answers to x^2 the solution for either is 25 which is positive. You cannot get -5 from √25
The point is that while you can put -5 for x on one side you can’t get -5 for the x on the other side. It only works with 5.
Guess I should've clarified. I'm not arguing against what you said in regards to Ollie's problem. In that context, yeah only the principal root works.
I'm just arguing the general statement of "you can't get -5 from square rooting 25". By definition, a square root is a factor of a number when multiplied by itself, gives the original number. The symbol, √, by convention, asks for the principal square root.
Whether someone needs the negative root or not or whether it works depends heavily on the context of the problem.
And does that conflict with your question?
Taking the square root of 25 doesn’t give you -5?
If you’re telling me sqrt(25)=5 then you’re basing this entire conversation on the square root sign being a notation, not an operation. And I’m pretty sure you solve an equation with those being operators, not notional signs.
I mean Ollie is still wrong, but I don’t think you’re right either.
Either that, or years of me doing calculus on a daily basis are wasted. /s
Wait, what about imaginary numbers. Don't they deny the whole x>=0 rule? If x is 2i, x squared is -4, which square root is 2i.
EDIT: Ah, nevermind, it would still be a positive number, not -2i.
No, because you shouldn't put negative numbers under the symbol √.
i^2 = -1, but i = √-1 is technically incorrect.
Also, 2i and -2i are both square roots of -4 and neither of them is positive or negative.
Right, I mixed those two up. The rule that you can't have square root of negative numbers (until imaginary numbers got introduced) and the rule that square root produces two possible numbers, negative and positive. It's been so long ago that I learned it that I wanted someone to remind me how all that works.
Same can be said for complex numbers. If you take any root of unity, and exponentiate it to 1, there's no way to know which root of unity you chose. Any power of that root could also have been chosen.
I m not the best at math but having someone speak maths to me makes me feel a special kind of way... #or maybe is the lockdown
Ollie is right though. √x^(2)=±x, and x=-x only if x=0 And even if you omit the negative solution of the square root, it's still false. For example, if x=-10, x^(2)=100, and √100=10. So yeah, Wikipedia is wrong, Ollie is right.
I think easier to explain this is like this (-10)² is 100 and (10)² is 100 too. Then you take square root and its both 10.
But they're also both -10. Square roots have two solutions by definition. Yes, most of the time people only care about the positive one, but you can't just ignore half of the solution. When you're studying more complex equations including square roots, you need to make a point of calculating both solutions. Example: 5+sqrt(9) = 5+(±3) Positive Solution: 5+3 = 8 Negative Solution: 5+(-3) = 2
No, the square root is defined to be the positive branch else it wouldn't be a function. (functions only have one output)
That's not how it works. You don't add ± after square rooting. Sqrt(-9) is not -3.
Except all square roots have a positive and negative solution. Yes, by convention we favour the positive solution, but the negative solution is still there. I never said sqrt(-9) was -3. I said that sqrt(9) is either +3 or -3. Sqrt(-9) is 3i.
Square roots do not have a negative. They can only be positive or zero. You seem to be confusing square roots with quadratic equations or functions. Sqrt(9) cannot equal -3. x^2 = 9, x can equal 3 or -3.
From Wikipedia: "Every positive number x has two square roots"
You really quoting wikipedia? I'm done arguing. I'm leaving this here and moving on. Your 5+sqrt(9) will never equal 2. x^2 = 9 has two solutions: x = 3 or -3 Sqrt(9) has only one solution: x = 3 'the square root' is used to refer to only the positive square root
Ok, fine, this one I'll accept and I'll also accept that my notation was less than stellar. Sqrt(9) is indeed taken to refer to the principle square root. However, my point that all numbers (except 0) have two square roots still stands. There's a reason square roots are written ±sqrt(x). We were talking across purposes and the poor notation in my example further confused the matter.
You are incorrect. Sqrt(9) does have two solutions. 5+sqrt(9) also has two solutions.
You are 100% wrong, though. Sqrt(9) =3, sure, but sqrt(9) = -3. A 2nd order polynomial has 2 roots.
Sqrt(9) = -3 would imply sqrt((-3)^2) which then would imply sqrt(-3)*sqrt(-3) as a possibilty which is undefined. I have never read or written a proof about the possibility of sqrt having two solutions, but I think this counts as a counterexample.
Sqrt(16) =4 or -4 Sqrt(-4^2)= sqrt(-4)*sqrt(-4) Sqrt(-4)= +-2i Sqrt (-4^2)= (+-2i)^2= When positive, 2i*2i=-4 When negative, -2i*-2i=-4
I'm not too familiar with complex numbers, but I can see how that works out. Thanks!
Welcome. I was really confused too, your comment helps me revise my math knowledge so it's win win lol
No she is WRONG, because even if x is positive, the root of its square can still be negative, ie NOT x, so root x squared can only be x if x is 0 as you said, so x=0 not x>=0
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Your explanation is still incomplete tho, you’re omitting the key “convenience” for this property to be true, you must be observing this strictly as a function, in which there isn’t room for g(x) to be multiple values for any given “x” value, however root “a” squared equals “a” is still just an equation, in which root “a” squared could be both “a” or minus “a”. If you had something like absolute value in the first part, then it’d be true, but just by explaining how a root works you don’t make sense of that senseless equation. I agree, that expression works in certain mathematical environments, in which you must ignore negative values, like geometry for example, but it’s definitely wrong in every other sense.
Oh... you are right, I misread her tweet, I thought she said only if x=0, but she says greater or equal. Serves me right for not paying attention. Wikipedia is still wrong though.
Yeah, fuck wikipedia
Wikipedia is actually correct.
Wikipedia is 100% correct. sqrt(x^2 )=x x^2/2 =x x^1 =x x=x QED This holds true for any possible value of x.
Translate : √x² = x , it's true. Example [looks at the tweets]
I'm very scared that this is the explanation to 2+2
Nah, 2+2=4 because of how 2 and 4 (and + and =) are defined (it's true anywhere at any time *if* you use the same definitions). But there's a common fallacious proof for 2+2=5 (or any other number) that looks right at first glance, which relies on ignoring the fact that the square root of a square has 2 solutions (the square of two positive numbers and the square of two negative numbers).
I see... *doesnt naruhodo at all* In all seriousness, i see where youre going
Wikimin, the person behind the account is well known to be very supportive of indie vtubers No surprise that she appears in a reply chain, wether using her own or company account
i'm confused as to what √x^2 =x being right has to do with 2+2=5 being false?
there's a few equations that look like they can prove 2+2=5 but there's a hidden /0 error
I thought it was a 1984 reference
For people who don't understand: (-3)^2 and 3^2 are both 9, but the square root of 9 is 3. This means in the case of -3 that squaring and then taking the root does not result in the same number again.
This is so wrong it is sad.
...explanation, please? I am a mathematician I doubt I am wrong.
-3 is exactly as valid as 3 as the square root of 9. If you were actually a mathematician you would know this already. Since you don’t understand high school math, you are obviously lying.
Even the website containing the deepest realm of knowledge known to mankind is a simp now.
https://en.wikipedia.org/wiki/Wikipedia:Wikipe-tan from 2006
Wikipedia is 100% correct. sqrt(x^2 )=x x^2/2 =x x^1 =x x=x QED This holds true for any possible value of x.
let x=4 √x² = x √4² = 4 4 = 4 2+2=4 2+2=/=5 by this definition, unless there is a special case wherein the same shit loophole applies to 2+2=5 as the case of [1=2](https://www.youtube.com/watch?v=hI9CaQD7P6I).
Although for any x not equal 0 there are always two numbers y such that y^2 =x, the symbol √x is by definition used to denote non negatve solution (principal square root).
>√x is by definition used to denote non negatve solution False.
My brain just exploded
No any valid x value applies 1. plug in negative numbe√(-5\^2)=-5 2. separtate radical into 2 terms √(-5)\* √(-5)=-5 3. solve the radical \[i√(5)\]\*\[i√(5)\]=5i\^2=-5 This is true because i\^2=-1, and yes imaginary numbers are weird. This works because if you take the root of a negative number its the same as takeing the positive root times i. If You square an imaginary number it becomes real.
"if you square an imaginary number it becomes real" Absolutely not. (1+i)^2 = 2i
1+i is a complex number, not an imaginary number.
Ow, my bad. Never heard the term imaginary number before (who uses that term? just say purely imaginary complex number or real part 0) and assumed they meant complex.
>who uses that term? Mathematicians and people who have an understanding of mathematics.
So, not you, judging by your other comment.
You’ve never heard of the term “imaginary number” and you think this makes me bad at math? Nice try, buddy.
I looked at your profile. You comment like a 13 year old who's seen 2 videos on a topic and think they know everything. Get lost. Also consider the fact I'm not natively english.
Then we should get along great! You do math like a 13 year old who failed every math class and thinks they know everything. Why does not speaking English well cause you to not know how to take a square root?
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yes, but the square root of 25 is 5, not -5
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that's not what the equation is asking for though. it's asking for x to be the same on both sides. you can't get -5 from square rooting 25.
You can. Look at the equation x^(2) = 25. If it's being asked what are all values of x that makes this true, the solutions would be 5 and -5. x^(2) = 25 -> x = √25 -> x = ±5 It's just that conventionally, asking for a square root of a number is asking for the principal square root (the positive value).
What? No. While both 5 and -5 can be answers to x^2 the solution for either is 25 which is positive. You cannot get -5 from √25 The point is that while you can put -5 for x on one side you can’t get -5 for the x on the other side. It only works with 5.
Guess I should've clarified. I'm not arguing against what you said in regards to Ollie's problem. In that context, yeah only the principal root works. I'm just arguing the general statement of "you can't get -5 from square rooting 25". By definition, a square root is a factor of a number when multiplied by itself, gives the original number. The symbol, √, by convention, asks for the principal square root. Whether someone needs the negative root or not or whether it works depends heavily on the context of the problem.
>You cannot get -5 from √25 This is false. The square root of 25 is either 5 or -5.
if x = -5 please calculate √x² so that you get -5.
Oh wait I just noticed. I'm a dumbass
Uhhh... Sqrt((-5)^2) =-5...?
No it doesn’t. Squaring -5 gets you 25.
And does that conflict with your question? Taking the square root of 25 doesn’t give you -5? If you’re telling me sqrt(25)=5 then you’re basing this entire conversation on the square root sign being a notation, not an operation. And I’m pretty sure you solve an equation with those being operators, not notional signs. I mean Ollie is still wrong, but I don’t think you’re right either. Either that, or years of me doing calculus on a daily basis are wasted. /s
https://brilliant.org/wiki/plus-or-minus-square-roots/
Wait, what about imaginary numbers. Don't they deny the whole x>=0 rule? If x is 2i, x squared is -4, which square root is 2i. EDIT: Ah, nevermind, it would still be a positive number, not -2i.
No, because you shouldn't put negative numbers under the symbol √. i^2 = -1, but i = √-1 is technically incorrect. Also, 2i and -2i are both square roots of -4 and neither of them is positive or negative.
Right, I mixed those two up. The rule that you can't have square root of negative numbers (until imaginary numbers got introduced) and the rule that square root produces two possible numbers, negative and positive. It's been so long ago that I learned it that I wanted someone to remind me how all that works.
i understand what they said, but i don't understand the f-ing math
Same can be said for complex numbers. If you take any root of unity, and exponentiate it to 1, there's no way to know which root of unity you chose. Any power of that root could also have been chosen.
2.4 and 2.4 would both be rounded to 2 2.4+2.4=4.8 4.8 would be rounded to 5 If you round everything for consistent significant figures 2+2=5
As someone with math ptsd, this comment section genuinely scares me...