20130212, 15:46  #1 
Romulan Interpreter
"name field"
Jun 2011
Thailand
7·1,423 Posts 
square root equal to a negative
Now I have a big problem...
I have the next equation: Does it, or does it not have a (real/integer) solution? 
20130212, 16:00  #2 
"Nancy"
Aug 2002
Alexandria
2467_{10} Posts 
It does not. The two (Edit: complex) solutions are ±2i.
Last fiddled with by akruppa on 20130212 at 16:00 
20130212, 16:01  #3 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
No, its only solutions are complex (with non zero imaginary part). Anyone who claims a real solution forgets that squaring both sides introduces extra solutions which do not satisfy the original equation.
Edit: akruppa's solutions are not, I think he read the question too quickly. Last fiddled with by Dubslow on 20130212 at 16:05 
20130212, 16:04  #4 
Romulan Interpreter
"name field"
Jun 2011
Thailand
7×1,423 Posts 

20130212, 16:06  #5  
Romulan Interpreter
"name field"
Jun 2011
Thailand
7×1,423 Posts 
Quote:


20130212, 16:11  #6  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
Quote:
? You pose an interesting problem. Last fiddled with by Dubslow on 20130212 at 16:12 Reason: Clarification 

20130212, 16:15  #7 
Romulan Interpreter
"name field"
Jun 2011
Thailand
7×1,423 Posts 
Story is like that, this IS homework. My daughter's. She had to solve a system which came to this, after a lot of calculus. She said is no solution. I said the solution is 16. Indeed, sqrt(16), according with the math I know 40 years ago, is +/4. We argued, we looked in the book (answers section). They said the solution is the empty set. I got angry and looked into wiki, and wolfram alfa, they all agree with my daughter , and I am getting crazy... We live the days when the egg teaches the hen, you know?

20130212, 16:31  #8 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Hmmm... thinking of as a function, then it has no solutions (sqrt(16)=4). If you prefer to think of sqrt(x) as the set of all (complex) numbers whose square is x, then *maybe* you might say sqrt(16)={4,4}.
Edit: What was the original context of this equation? As in, what was the homework question? Last fiddled with by Dubslow on 20130212 at 16:35 
20130212, 16:33  #9 
"Ben"
Feb 2007
3617_{10} Posts 
x = 16*i^4

20130212, 16:41  #10  
"Nathan"
Jul 2008
Maryland, USA
1115_{10} Posts 
Quote:
For instance, in my first algebra textbook, I was taught that any positive real number x had two square roots, and to answer a question like "what is the square root of 36?" I should write . If I were solving an equation such as , I would perform the steps , hence , hence . Needless to say, it was a little embarrassing and cost me a few points on some homework assignments when I got into college, where the definition of the radical sign was taken to mean "the positive square root". But you can see now where your daughter's question might be entirely valid and have a solution in one setting, but not in another. If, as I suspect, her present textbook is making use of the "plusminus" definition of square roots, then one could say that, yes, if , then . On the other hand, if her textbook has taken on the more advanced, "principal"/positivesquarerootonly definition, then there would be an argument that there is no solution, as there is no possible way *under the understood definition of square root *as it is in this textbook** for the square root of a number to be negative. Hence the solution would be undefined. This, of course, is an excellent time to have a chat with your daughter (especially if she plans on going higher into math) about why this matter is an issue in the first place, and why we might like to fix "square root" to mean "only the positive square root" (e.g. to avoid a multivalued function, to be able to take limits in calculus, to not have a weird function that raises a positive number to a positive power and gives a negative answer...) Basically, this is one of her first lessons that what works well in algebra does not necessarily work well in calculus, and why certain changes to definitions and constructs have to be made in order to fix these issues. Last fiddled with by NBtarheel_33 on 20130212 at 16:44 

20130212, 17:14  #11 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 

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