The principal square root function under discussion is certainly not from R to R, since it sends negative inputs to “imaginary” outputs.
Which is vital in the example Senegoid gives which Saint Cad responds to. And this is “ill behavior” insofar as it is counter-intuitive that the normally reasonable (and desirable) pattern (√a)(√b) = √(ab) should go out the window. The reason we are discussing this example at all is precisely because it behaves “pathologically”.
You may consider it irrelevant dick measuring, but we are both responding to a post where Saint Cad desired to remark upon a natural sense in which we needn’t consider this pattern broken. Dick measuring or not, it seemed to me therefore quite relevant to back Saint Cad up with my agreement in perspective.
Well, the function as defined by Mathworld has a domain of x >=0, so I suppose it’s fair to say it’s not really R -> R. But still the point is that the radical symbol has a well-known conventional meaning, and if you’re going to generalize it to negative or complex inputs then you have to say so. As written in Saint’s post, it is correct to reject √(1) = -1 as fallacious.
[On edit: I wrote this before seeing the last post from friedo; this was not in response to that, but rather a continuation of my last post.]
Mathematics generally is not so interested in arbitrary, ugly definitions. Generally, it is more interested in clean definitions leading to nice patterns. For example, having first been told that x[sup]y[/sup] means the product of y many xes when y is a cardinal number, why then do we choose in generalizing this to say that x[sup]-1[/sup] should mean the reciprocal of x? Why not say “x[sup]y[/sup] is the product of y many xes when y is a cardinal number, and is 7 otherwise”? Well, because the choice of x[sup]-1[/sup] as the reciprocal of x allows us a clean pattern; it lets us continue to say that x[sup]a + b[/sup] = x[sup]a[/sup] * x[sup]b[/sup] and so on. This focus on maintaining clean patterns comes up over and over.
To re-examine a conventional definition, point out its warts, and consider what would be in many contexts a less ugly perspective instead is entirely in keeping with the manner in which mathematics is developed. Mathematics isn’t about definitions handed down on stone tablets from God to be received unquestioningly; it is about seeking the greatest clarity of understanding. In this example, I believe attaching undue significance to the arbitrarily defined principal square root function generally obscures more than it clarifies.
And I agree that the square root function from non-negative reals to non-negative reals is perfectly well-behaved, as I noted before. Note that this function continues to satisfy the pattern (√a)(√b) = √(ab). It does not furnish a counter-example to this rule.
Mathworld claims “The concept of principal square root cannot be extended to real negative numbers”. Others instead say “The principal square root is defined at all complex numbers; for negative inputs, it is chosen as a positive multiple of i, while for other inputs, it is chosen to have non-negative ‘real’ component”. I find this latter messy function not frequently useful or natural, and thus am more sympathetic to the perspective Mathworld takes.
My point was that I always considered the square root operator to be open to equivocation.
sqrt4) = 2 [principal root]
sqrt(4) = sqrt(-2 x -2) [so -2 is a square root of 4]
= sqrt(-2) x sqrt(-2)
= sqrt(2)i x sqrt(2)i [but now we only consider the prinicipal root?!]
= sqrt(2 x 2) x (i x i)
= 2 x -1
= -2
Therefore 2 = -2
To me Senegoid’s statement showed that ignoring that a square root really has two solutions leads to contradictions that don’t really exist.
I think the usefulness of defining the square-root-function as the principal square root is this:
It provides us with the vocabulary (that is, the notation) to specify whichever kind of square root(s) we want to deal with for the problem at hand.
If we want only the positive square root, then √x provides us the notation we need to say that.
If we wanted the negative square root, then we can write -√x to specify that.
If we want to talk about both of the square roots, then we can write ±√x to indicate that. This is used, for example, in the quadratic formula as friedo mentioned.
Sure. The notions of “positive” and “negative” square roots make natural sense when the input is positive. And, as I said, the square root function from (semi)positives to (semi)positives is perfectly well-behaved.
But when the input is negative, the notions of “positive” and “negative” square roots make rather less natural sense. There are few contexts in which one will want to distinguish one of the complex square roots of a negative number as more relevant than the other.
Thus, I don’t consider it generally valuable to distinguish principal square roots for negative inputs. [In some particular niche situations, there may be reason for this, though the appropriate way to do so will then depend on the niche]
This does mean understanding the appropriate interpretation of square roots is context-sensitive. But so it is, with everything, forever and always. That bothers some, but them’s the breaks; that’s the reality of things. [Do negatives even have square roots?: That depends on the context: should we consider only “real” values, or complex values as well, or even values of some other kind, such as square matrices? How many square roots does -1 have?: Over the reals, 0; over the complex numbers, 2; over quaternions or 4 by 4 integer matrices more generally, infinitely many. Does 2 have a square root?: Over the reals, yes; over the rationals, no. Everything is context-dependent, inevitably, whether we like it or not. But in very few contexts is there natural call to think of square roots of reals as principally defined as single complex numbers in general in such a way as that √1 is uniquely 1 and √(-1) is uniquely i, breaking the pattern (√a)(√b) = √(ab); though it is convenient for teachers to sidestep this sort of discussion, nonetheless, that particular notion of “principal square root” is rarely the clearest way to think of things.]
Come to think of it, the same method that was used to untie the Gordian knot can also be used to color any map with fewer than four colors, and Alexander made a pretty good attempt at that one, too.
