Why do we take the positive answer?

[Masua]

This is copied from Paul's online math notes

There is one final topic that we need to touch on before leaving this section. As we noted back in
the section on radicals even though √9=3 there are in fact two numbers that we can square to
get 9. We can square both 3 and -3.
The same will hold for square roots of negative numbers. As we saw earlier √-9=3i . As with
square roots of positive numbers in this case we are really asking what did we square to get -9?
Well it’s easy enough to check that 3i is correct.
(3i)2=9i2= -9
However, that is not the only possibility. Consider the following,
(-3i)2=(-3)2i2=9i2= -9
and so if we square -3i we will also get -9. So, when taking the square root of a negative number
there are really two numbers that we can square to get the number under the radical. However,
we will ALWAYS take the positive number for the value of the square root just as we do with the
square root of positive numbers.


Why do we do so with complex numbers and radicals?

 

[Mingfeng]

I just had a class in Analysis.
My professor said that √ is defined to get positive number.
√9=3 that is arithmetics.
when you have x^2=9, you get x = +or -√9 = +or-3 that is algebra.
Note: in both case √9=3

For the complex number
√-9= √- * √9 = i3

 

[Simon Bridge]

I would argue that Paul's notes are incorrect.
For instance - if you followed Paul's advice literally then you'd only ever find one root of a quadratic.

It is a convention of notation only. It is not always appropriate to take tho positive root.

Sometimes we will only need the positive or negative square root as the valid physical result (we know from the physics that we started out squaring a positive of negative number). Without the convention, we would have to write a +√ or a -√ for which one we mean, and √ would allow either.

What he's doing is continuing the convention of not writing the sign of positive numbers explicitly. So we get √ for the positive root, -√ when we mean the negative root, and ±√ when we mean either. This is more convenient since we almost always only need the positive root.

Thus the quadratic equation has an explicit ± in it.

 

[Simon Bridge]

Mingfing is making a reasonable point with the distinction between algebra and arithmetic.

I should also point out that for $y=x^2$ then $\sqrt{y}=x$ is fine, and we note that x can take positive or negative values. What we may be doing is measuring x and computing/deducing √y

However, to reverse it [we have measured y and we want to find x] we'd have to write $x=\pm\sqrt{y}$ because $\sqrt{y}$, by convention, only yields positive values.

It's a little annoying since the relationship is described by the former and, as a mathematical relation, is quite fine as $x=\sqrt{y}$ because, algebraically, it does not matter which order you write the terms around the = sign.

 

[BloodyFrozen]

Following what others have said,

we get $\pm$ when we solve equations (like Mingfeng's) because

$$x^2=9$$
$$x^2-9=0$$
$$(x-3)(x+3)=0$$
$$x=\pm3$$ which equals what we would have gotten if we just took the square root of both sides.

$$x=\pm\sqrt 9$$