How do you compare square roots? Of course, the positive square root of 49 is greater than the positive square root of 36. However, what if you were to have $\pm\sqrt{49}$ ? $\pm\sqrt{36}$? Would it be $\gt$, $\lt$, or some other symbol. Also, what if you had $\pm\sqrt{16}$ ? 0?
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2$\begingroup$ Usually when people say square roots, we look at the positive number - i.e. when asked for the square root of $49$, we usually say $7$ and not $-7$. But if we are asked to compare the square roots of $a$ and $b$ where $a > b$, then $\sqrt{a} > \pm \sqrt{b}$ and $-\sqrt{a} < \pm \sqrt{b}$. $\endgroup$– 2012ssohnCommented Jun 4, 2015 at 0:53
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You can only compare two real numbers. $\pm \sqrt{x}$ may be interpreted as the unordered pair $\{\sqrt{x},-\sqrt{x}\}$, so it is usually meaningless to compare it with something else, in the same way that it is meaningless to compare $\{1,3\}$ and $2$ or $\{1,4\}$ and $\{2,3\}$. What you can say is $1 < 2 < 3 < 4$.
answered Jun 4, 2015 at 1:36
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$\begingroup$ Not true. You can compare strings (character-by-character), arrays, and other non-numeric items. It depends on what properties you want the comparison operator to have. You can even compare answers to problems. $\endgroup$ Commented Jun 4, 2015 at 2:15
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$\begingroup$ @martycohen: Of course I am aware of that. I just chose to focus on the problem that the asker had in understanding the meaning of $\pm$. $\endgroup$ Commented Jun 4, 2015 at 2:23
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1$\begingroup$ In fact, I did think of saying something like "If you want to make a comparison between real numbers, then you must compare real numbers.". $\endgroup$ Commented Jun 4, 2015 at 2:27
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$\begingroup$ And if you don't want to compare real numbers, then don't. $\endgroup$ Commented Jun 4, 2015 at 2:45