Thank you! I have a PhD in math, algebraic number theory, and I'm glad to see videos addressing this topic correctly. I like to offer to my students: When you introduce the square root to an equation, that's where you add the "±", but if the √ symbol is already there then the selection of + or - has already been made. I also will ask them, if the ± is needed (e.g., √16 = ±4) then "Why does the quadratic equation have a '±' in the numerator?" That would be redundant if 3+√16 simplifies down to 3+(±4).
The clue for that kind of problem: where you have x with the power of 1 there is only one solusion ex. √x = 3 x = 9 where you have x with the power of 2 there is two solusions ex. x^2 = 9 x = 3 or x = -3
From line 4 you wrote +-x=4, and using line 2 you can write +-x=sqrt(16). 4 is a valid solution to x so therefore plugging it in reads +-4 = sqrt(16). I don't understand where I went wrong.
Yes, it is true that x² = 16 is equivalent to ±x = √16. But you need to understand that "±" hides that this is not one equation but a sort of system of two equations that are joined by an "OR": ±x = √16 really means +x = √16 OR -x = √16 You pointed out that 4 is a solution to the original equation x² = 16. In the system of equations this translates to +4 = √16 OR -4 = √16 Now keep in mind that it is not the case that both equations must simultaniously be true. This is because they are joined by an "OR": It suffices if one of them is true, then the whole system of equations evaluates to true. Now, because √16 is defined to be the value 4, the first equation is true and the second equation is false. Therefore, the whole system of equations evaluates to true which proves that 4 is indeed a solution to the equation. Note that -4 is also a solution. If you plug it into the system of equations, the first equation is now false while the second equation is true (as -(-4) = 4). This makes the whole system of equations true so that -4 is another solution to the original equation. Hope that helps 💪
@@complexcreations5309 Thanks for the response. It makes sense. But now the statement sqrt(16) = +-4 is actually sqrt(16) = +4 OR sqrt(16) = -4 which is a true statement even if sqrt(16) = +4 is only ever the true equivalence. I just point this out because the video claims it is wrong.
@@user-ze2yk7cd7g Well, this is a case of misusing notation. When you write ± you create a system of equations joined by OR. But this is not what people want to express by stating √16 = ±4. What it actually means is that if you create a system of equations by replacing ± with + or - respectively, at least one of the equations is true. But what people want to express by stating √16 = ±4 is different. What they mistakenly think is that √16 is equal to both 4 and -4 (i. e. creating a system of equations with AND). The issue here is that ±4 is not "the value 4 AND the value -4" but instead "the value 4 OR the value -4". So yes, technically √16 = ±4 is right under the aforementioned definition of ±, but because that is not how you typically read ± when it appears in an equation without a variable (which serves to clarify that you in fact deal with an equation which needs to be decomposed further) I think you should refrain from using ± in such a way as it will only lead to misunderstanding and confusion.
I don't fully comply with the video. He claims that the result of a principal square root only has one single value and is always an absolute number but he then mentions that the result of an absolute number still has two solutions. So due to transitivity it would indeed mean that in the end (an equation) of the square root has two solutions (while the outcome just of the square root still only provides a single value). Like after "x^2^1/2 = |x|" he further resolves to "|x| = |+/- x|". So "16^1/2" isn't just 4 its actually "|4|" and so it can be +4 as well as -4. If you have an equation like "a^2 = b^2" it resolves to "|a| = |b|" and so you have four possible solutions. |+a| = |+b| |+a| = |-b| |-a| = |+b| |-a| = |-b|
It's not "+4 = sqrt(16)" nor "-4 = sqrt(16)" it's "|4| = sqrt(16)". You can't alter the equation itself you can only show the possible solutions. So the complete equation would be "x = sqrt(16) = |4| = |+/- 4|", so there are two solutions with "x1 = +4" and "x2 = -4". Both are equally correct and if it would be a guessing game there is no way of knowing which of them is used. Basically such a behavior is used in cryptography where an input is put into an aquation which provides a single result like "-4^2" only becomes 16 but trying to reverse it gives you multiple solutions where you don't know what the original input was when you try to calculate it backwards. Like "x % 2" either resolves to 0 or 1 but where is no way in knowing what the original number was. You would only know if it was odd or even.
Thank you! I have a PhD in math, algebraic number theory, and I'm glad to see videos addressing this topic correctly.
I like to offer to my students: When you introduce the square root to an equation, that's where you add the "±", but if the √ symbol is already there then the selection of + or - has already been made.
I also will ask them, if the ± is needed (e.g., √16 = ±4) then "Why does the quadratic equation have a '±' in the numerator?" That would be redundant if 3+√16 simplifies down to 3+(±4).
My teacher was saying this to me all the time.
That it is only positive it is.
I was the one insisting on the negative's addition. On graphs too.
0:51 but why?
The clue for that kind of problem:
where you have x with the power of 1 there is only one solusion
ex. √x = 3
x = 9
where you have x with the power of 2 there is two solusions
ex. x^2 = 9
x = 3 or x = -3
From line 4 you wrote +-x=4, and using line 2 you can write +-x=sqrt(16). 4 is a valid solution to x so therefore plugging it in reads +-4 = sqrt(16). I don't understand where I went wrong.
Yes, it is true that x² = 16 is equivalent to ±x = √16.
But you need to understand that "±" hides that this is not one equation but a sort of system of two equations that are joined by an "OR":
±x = √16
really means
+x = √16 OR -x = √16
You pointed out that 4 is a solution to the original equation x² = 16.
In the system of equations this translates to
+4 = √16 OR -4 = √16
Now keep in mind that it is not the case that both equations must simultaniously be true. This is because they are joined by an "OR": It suffices if one of them is true, then the whole system of equations evaluates to true.
Now, because √16 is defined to be the value 4, the first equation is true and the second equation is false. Therefore, the whole system of equations evaluates to true which proves that 4 is indeed a solution to the equation.
Note that -4 is also a solution. If you plug it into the system of equations, the first equation is now false while the second equation is true (as -(-4) = 4). This makes the whole system of equations true so that -4 is another solution to the original equation.
Hope that helps 💪
@@complexcreations5309 Thanks for the response. It makes sense. But now the statement sqrt(16) = +-4 is actually sqrt(16) = +4 OR sqrt(16) = -4 which is a true statement even if sqrt(16) = +4 is only ever the true equivalence. I just point this out because the video claims it is wrong.
@@user-ze2yk7cd7g Well, this is a case of misusing notation. When you write ± you create a system of equations joined by OR. But this is not what people want to express by stating √16 = ±4. What it actually means is that if you create a system of equations by replacing ± with + or - respectively, at least one of the equations is true. But what people want to express by stating √16 = ±4 is different. What they mistakenly think is that √16 is equal to both 4 and -4 (i. e. creating a system of equations with AND).
The issue here is that ±4 is not "the value 4 AND the value -4" but instead "the value 4 OR the value -4".
So yes, technically √16 = ±4 is right under the aforementioned definition of ±, but because that is not how you typically read ± when it appears in an equation without a variable (which serves to clarify that you in fact deal with an equation which needs to be decomposed further) I think you should refrain from using ± in such a way as it will only lead to misunderstanding and confusion.
I don't fully comply with the video. He claims that the result of a principal square root only has one single value and is always an absolute number but he then mentions that the result of an absolute number still has two solutions. So due to transitivity it would indeed mean that in the end (an equation) of the square root has two solutions (while the outcome just of the square root still only provides a single value). Like after "x^2^1/2 = |x|" he further resolves to "|x| = |+/- x|". So "16^1/2" isn't just 4 its actually "|4|" and so it can be +4 as well as -4.
If you have an equation like "a^2 = b^2" it resolves to "|a| = |b|" and so you have four possible solutions.
|+a| = |+b|
|+a| = |-b|
|-a| = |+b|
|-a| = |-b|
It's not "+4 = sqrt(16)" nor "-4 = sqrt(16)" it's "|4| = sqrt(16)". You can't alter the equation itself you can only show the possible solutions. So the complete equation would be "x = sqrt(16) = |4| = |+/- 4|", so there are two solutions with "x1 = +4" and "x2 = -4". Both are equally correct and if it would be a guessing game there is no way of knowing which of them is used.
Basically such a behavior is used in cryptography where an input is put into an aquation which provides a single result like "-4^2" only becomes 16 but trying to reverse it gives you multiple solutions where you don't know what the original input was when you try to calculate it backwards. Like "x % 2" either resolves to 0 or 1 but where is no way in knowing what the original number was. You would only know if it was odd or even.
What if your teacher marks the right answer as wrong?
Then show him the second and third paragraph of the Wikipedia article on square roots