We all know that for every real and positive number $N$ we have

$$\sqrt{N} = \pm a$$

for example $\sqrt{25} = \pm 5$.

Now: why the plot of the function $\sqrt{x}$ has only a positive part in the Cartesian plane? I have two results, plus and minus a certain number, so why the plot takes into account only the positive results?

  • 6
    $\begingroup$ No, $\sqrt{25}$ is not $\pm 5$; it's $5$, by definition. $\endgroup$ – David Mitra Nov 26 '15 at 17:58
  • $\begingroup$ You could plot $f(x)=-\sqrt x$ to get the negative branch. $\endgroup$ – JB King Nov 26 '15 at 18:21

By definition $\sqrt{x}$ indicates the positive (or principal) square root of $x$. If we want indicate all the two root we have to write explicitly $\pm\sqrt{x}$.

  • 1
    $\begingroup$ Oh thank you!! That was a doubt I had since lots of time. Thank you! $\endgroup$ – Von Neumann Nov 26 '15 at 18:21

Actually, the function $f(x)=\sqrt{x}$ usually refers to the principal square root which is defined as the nonnegative solution to $r^2=x$ for $x\ge0$.

While it's true that in general there are two solutions to the equation $r^2=x$, we really like single valued functions.

  • $\begingroup$ Of course, a multi-valued function would be an oxymoron (i.e. not really a function). $\endgroup$ – John Joy Nov 27 '15 at 0:22
  • $\begingroup$ @JohnJoy Not true, this would still be a function, but on a different target set. $\endgroup$ – Did Nov 29 '15 at 9:03
  • $\begingroup$ @Did Of course, you are correct, if the elements of range of the function are themselves, sets (e.g. $\{-1,1\}\in S$). $\endgroup$ – John Joy Nov 29 '15 at 14:07

The √ symbol is used to denote the principal square root of a number, i.e. the positive one.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.