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# Common misconception Watch

1. Why is it that these have different solutions ?

x = √16
Why is it that the first one has 2 solutions and the second one had only one. My confusion arises since the second equation is just the first one square rooted.
2. The second equation defines x to be the (positive) square root of 16.

The first equation defines x to be any value which, when squared, will equal 16. The positive and negative root of 16 satisfies the first equation.
3. (Original post by MangoFreak)
The first equation defines x to be any value which, when squared, will equal 16.
.. But so does the second one..
I kind of understand you... but the second equation says that x is any number that when timed by itself equals 16, doesn't it? So that includes -4.
5. (Original post by Noble.)
I don't understand it...
6. It only makes sense if you do the principal square root.
7. (Original post by MathMeister)
I don't understand it...
Did it just pop into your head, or did it arise while problem solving? I ask because your problem is almost a non-problem, the only reason the second equation can be considered to only have one solution is because often the sign is taken to mean "take the principal square root", but it's by no means unambiguous.
8. (Original post by Noble.)
Did it just pop into your head, or did it arise while problem solving? I ask because your problem is almost a non-problem, the only reason the second equation can be considered to only have one solution is because often the sign is taken to mean "take the principal square root", but it's by no means unambiguous.
It is rather important to know the difference, as in an exam, if a question asks you to square root something and you need to find 2 values, you may get it wrong and only find one.
Thank you , as well. You have made me understand.
9. (Original post by MathMeister)
It is rather important to know the difference, as in an exam, if a question asks you to square root something and you need to find 2 values, you may get it wrong and only find one.
Obviously yes, however I think depending on the situation it is normally clear what is required because I personally can't think of a time (either from A-Level maths to undergrad) where I've not known whether they want the principal solution, or all solutions.
10. (Original post by Noble.)
Obviously yes, however I think depending on the situation it is normally clear what is required because I personally can't think of a time (either from A-Level maths to undergrad) where I've not known whether they want the principal solution, or all solutions.
Have you got examples?
11. (Original post by MathMeister)
Have you got examples?
If you have an equation that involves then there will be 2 solutions (perhaps repeated)

I am intrigued by your OP

is not an equation as such so it does not have solution(s)

It is a bit like asking how many solutions x = 5 has

Also it is not "the first one square rooted"

It you take the square root of both sides of the original equation you get

12. (Original post by TenOfThem)
If you have an equation that involves then there will be 2 solutions (perhaps repeated)

I am intrigued by your OP

is not an equation as such so it does not have solution(s)

It is a bit like asking how many solutions x = 5 has

Also it is not "the first one square rooted"

It you take the square root of both sides of the original equation you get

Thanks, it all makes perfect sense now.
13. (Original post by Noble.)
often the sign is taken to mean "take the principal square root", but it's by no means unambiguous.
Often? Isn't it always? Is there ever a scenario where is interpreted as anything other than ? (at least in the modern usage - maybe 100 years ago there was some ambiguity)
14. (Original post by TenOfThem)
is not an equation as such so it does not have solution(s)
I'd have to disagree: it's a complete standard equation, as it asserts that one quantity is equal to another quantity, and it most certainly does have a solution (for example, is not a solution since , but is a solution since ).

To put it another way, if you were asked to write down the solution set of that expression, what set would you write down?

It is a bit like asking how many solutions x = 5 has
That may or may not be a trivial question. In modular arithmetic, that equation can have an infinite number of solutions.
15. (Original post by atsruser)
I'd have to disagree: it's a complete standard equation, as it asserts that one quantity is equal to another quantity, and it most certainly does have a solution (for example, is not a solution since , but is a solution since ).

To put it another way, if you were asked to write down the solution set of that expression, what set would you write down?

That may or may not be a trivial question. In modular arithmetic, that equation can have an infinite number of solutions.
Hence my use of "as such"
16. (Original post by atsruser)
Often? Isn't it always? Is there ever a scenario where is interpreted as anything other than ? (at least in the modern usage - maybe 100 years ago there was some ambiguity)
It probably is, I know I've seen that convention used in at least one textbook but I don't actually know how universally accepted it is.
17. I did a mini-study on square roots/sqrt symbol etc. a few months ago and I was amazed by how many experienced maths teachers taught things like , which confuses students when they see referring to the principal square root in A-Level textbooks.
18. (Original post by Noble.)
It probably is, I know I've seen that convention used in at least one textbook but I don't actually know how universally accepted it is.
It's pretty much universal now that both and refer to the positive square root of x when x is a positive real number.

I suspect it has been the case for a significant period of time, otherwise the plus-or-minus symbol used in the standard quadratic solution formula would be redundant, and I learned that formula over 30 years ago!

It still doesn't stop textbooks being in error, though - I've seen examples where a student is asked to simplify something innocent-looking, like and the "answer" is given as without the required modulus sign around the x.
19. (Original post by atsruser)
Often? Isn't it always? Is there ever a scenario where is interpreted as anything other than ? (at least in the modern usage - maybe 100 years ago there was some ambiguity)
If x is a positive number I agree with you, but you might still see questions like "what is the Riemann surface of ?" in other contexts.
20. (Original post by RichE)
If x is a positive number I agree with you, but you might still see questions like "what is the Riemann surface of ?" in other contexts.
I'm afraid you're being too subtle for me here. Isn't a function single-valued on its Riemann surface, in which case I'm not sure what ambiguity can arise. Or have I forgotten too much complex analysis?

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