# Notation for Principal Square Root?

1. Oct 21, 2009

### DocZaius

Hello,

I was just wondering if there is a special notation for a principal square root...

I suppose using absolute value would work..
$$\left|\sqrt{9}\right|=+3$$

But it doesn't seem as fitting as an actual special square root symbol. Maybe something like this?
$$\sqrt[+]{9}=+3$$

Also, $$+\sqrt{x}$$ could be ambiguous since it could mean $$+(\pm\sqrt{x})$$

Last edited: Oct 21, 2009
2. Oct 21, 2009

### pbandjay

I always thought it correct to denote the principal square root as

$$\sqrt{9}=3$$

whereas both roots would be denoted as

$$\pm\sqrt{9}=\pm{3}$$

3. Oct 21, 2009

### DocZaius

That's interesting...To me it would seem that when using the square root symbol in its purest form, it should denote the meaning of a square root in its purest form...and that would be to indicate that positive numbers have two roots, positive and negative.

In other words, it would make sense to me that the default interpretation of the symbol in its simplest form should be its meaning in its broadest sense (two roots).

But if that's the convention, that's the convention I suppose...

4. Oct 21, 2009

### rasmhop

$\sqrt{x}$ is not defined to be simply the square root of x, but the principal square root of x so there is nothing "pure" about $\sqrt{x}$. pbandjay is right.

Actually what would make sense was for the default interpretation to be the easiest one to work with and the one you usually need. In many contexts there is no need to discriminate between $\sqrt{x}$ and $-\sqrt{x}$ so we only need one. When there is a need we just use the symbols $\sqrt{x}$ and $-\sqrt{x}$. Attaching multiple values to $\sqrt{x}$ means that you can't attach a unique value to expressions such as $\sqrt{4} + 3$ and when actually working with square roots this is often desirable.