DanZ (![[info]](http://l-stat.livejournal.com/img/userinfo.gif){kind=small} fclbrokle) wrote,@ 2009-02-27 18:18:00 |
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XKCD, right again
It really happens!
I came across this handout on "Common Mistakes in Algebra." All well and good, except for #14, which is not wrong! Or, if it is, it's wrong for a very different reason than the solution gives.
#14 claims that it is false to say that sqrt(-x) * sqrt(-y) = sqrt(xy).
If both x and y are negative, then this is totally correct.
If one or more is not negative, then you are taking a square root of a negative number, so you really have no choice but to think about multiple square roots. Then this must be read as "the product of any square root of -x with any square root of -y gives a square root of xy," which is true! (One could quibble with the use of an equals sign here, but if you give it the generous reading then this is entirely correct.)
The solution key says that this is wrong, because sqrt(-x) = i * sqrt(x), and sqrt(-y) = i * sqrt(y), and so multiplying those together you get -sqrt(xy). But the negative of a square root is still a square root! And sqrt(-x) = i * sqrt(x) in a consistent fashion only if you let your square root take on multiple values!
So I wrote to the author of the worksheet to point out this (admittedly subtle) fact which she got from a published textbook. I spent quite some time crafting a nice e-mail and explaining the mathematics in detail so that this would be, as much as possible, a learning experience.
The e-mail bounced from her now-defunct e-mail address.
*sigh*
That is, I suppose, how it goes when you try to correct the Internet. The result lives on, and keeps its high page rank on Google, too.
(Incidentally, I'm sure that those of you who browse carefully will notice that yes, this is a teacher at the college level of other mathematics teachers, and this is someone who was studying to get a PhD in mathematics education. If you feel like launching into a rant about teacher educators, then I suggest you actually do something productive with yourself to help improve mathematics education instead. If you feel like launching into a rant about textbook publishers who don't get mathematically competent editors, then by all means, proceed. :))
It really happens!
I came across this handout on "Common Mistakes in Algebra." All well and good, except for #14, which is not wrong! Or, if it is, it's wrong for a very different reason than the solution gives.
#14 claims that it is false to say that sqrt(-x) * sqrt(-y) = sqrt(xy).
If both x and y are negative, then this is totally correct.
If one or more is not negative, then you are taking a square root of a negative number, so you really have no choice but to think about multiple square roots. Then this must be read as "the product of any square root of -x with any square root of -y gives a square root of xy," which is true! (One could quibble with the use of an equals sign here, but if you give it the generous reading then this is entirely correct.)
The solution key says that this is wrong, because sqrt(-x) = i * sqrt(x), and sqrt(-y) = i * sqrt(y), and so multiplying those together you get -sqrt(xy). But the negative of a square root is still a square root! And sqrt(-x) = i * sqrt(x) in a consistent fashion only if you let your square root take on multiple values!
So I wrote to the author of the worksheet to point out this (admittedly subtle) fact which she got from a published textbook. I spent quite some time crafting a nice e-mail and explaining the mathematics in detail so that this would be, as much as possible, a learning experience.
The e-mail bounced from her now-defunct e-mail address.
*sigh*
That is, I suppose, how it goes when you try to correct the Internet. The result lives on, and keeps its high page rank on Google, too.
(Incidentally, I'm sure that those of you who browse carefully will notice that yes, this is a teacher at the college level of other mathematics teachers, and this is someone who was studying to get a PhD in mathematics education. If you feel like launching into a rant about teacher educators, then I suggest you actually do something productive with yourself to help improve mathematics education instead. If you feel like launching into a rant about textbook publishers who don't get mathematically competent editors, then by all means, proceed. :))
