<p>But see, the problem for me is, I woulda said, no, that's not the right answer, and gone looking for something more esoteric!</p>
<p>(pats on back, got it fast)...but...garland technically I think you're right, -44 is less than the OTHER square root. But I guess you're reading too much into it??? Evidently one is to make the assumption 44 is THE square root of 2006....if so then what's -44?</p>
<p>The point is that 4 has two square roots. One of the square roots of 4 is 2; the other is -2. However, -1 has only one square root: i.</p>
<p>And coranged: square roots can indeed be negative (-2 is one of the roots of 4), BUT a negative number cannot have square roots in conventional math, which is why mathematicians came up with i, which is the square root of -1. </p>
<p>Then it turned out that the concept of an imaginary number (i) is incredibly useful is certain applications within engineering and science: <a href="http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH033.HTM%5B/url%5D">http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH033.HTM</a></p>
<p>"Right. So I'm still missing something about this question. Isn't the negative square root of 2006 still a square root of 2006? So it's not less than the square root of 2006; it's just less than one of them."</p>
<p>An imaginary number is the square root of a negative number, not a negative square root of a positive number (addressing some other thread). This is a very difficult question. The wording is slightly ambiguous, but they are correct. There is only truly one root of a postivie number: the principal root (positive root). The principal root of 49 is 7. Just because -7 squared equals 49 does not mean it's the square root of 49. It's merely the negative of the square root. The "logical" reasoning would be the instance of a cube root (or any odd exponent for that matter) of a number. So, for instance the cube root of 27 equals 3, but -3^3 does not equal 27. Difficult and tricky, I know. This is why I prefer history and literature.</p>
<p>So, technically the square root of 2006 is not -44.79, but -44.79^2 = 2006</p>
<p>...Excuse syntax errors</p>
<p>qwilde, I disagree with your logic. garland, you are correct. The problem is I didn't use the right wording for the problem. Here is the exact question, and you can see it is worded in a a way that avoids the "garland paradox" :)</p>
<ul>
<li>What is the only number less than (square root symbol)2006 whose square is 2006?</li>
</ul>
<p>no, wait. You still have the same problem if you read "sqare root 2006" as plus or minus. I guess I revert to the quote above:
[quote]
In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root.
[/quote]
</p>
<p>Thanks for naming a paradox after me :)! But I'm not sure it gets around "my" paradox. Okay, I get that there's this principle square root thing (first I heard of it, I'm embarrassed to say) but all my ten seconds of reading up on this make it clear that the square root can still be positive or negative; one's the principle square root, but they're both square roots. So it's still not less than the square root; it's one of 'em. That's my story and I'm sticking to it!</p>
<p>I'm with you Garland - stupidly worded question.</p>
<p>I think you have to not read too much into it. We had this problem on VaML (well I guess everyone has the same problem). And this month they let the preIB Alg 2/Trig kids take it, it's easier I guess. Honestly when I first looked at the problem, my initial reaction was that it was stupid and had no answer, but a bit later I realized the principle it was asking for. Stupidly worded maybe, but I don't see any other answer than -sqrt(2006). </p>
<p>I agree that if you analyze the problem, I suppose you could argue that phrased as sqrt(2006) one could assume it was negative already. But 1) I think that's giving the problem way too much credit, and 2) then there would be no answer to fit the second criteria for the problem which is that it be less than something. There is nothing less than -sqrt(2006) that will square to 2006, therefore I think one is safe in the assumption that it's the positive sqrt. </p>
<p>Basically I think if you look at it as a problem with two boundary conditions (sqrt(2006) and "less than"), the answer becomes fairly clear. I put down that answer last, at the end of test when I realized it. I also realized on the last problem it was asking the area of a semi circle and I wrote my answer down before I divided by 2.</p>
<p>Now that the original wording is in, it makes perfect sense. </p>
<p>[square root symbol]2006 has an implied positive sign in front of it in a much more definite way than the English expression "the square root of 2006". The way to indicate both square roots would be [+ or -][square root symbol]2006, as numerous math teachers have drilled into my head over the years. So written in English, explicitly stating that implied sign, the question would be,</p>
<p>What is the only number less than the positive square root of 2006 whose square is 2006?</p>
<p>And once you have that, it's pretty obvious.</p>