# Square Roots

1. Aug 29, 2009

### S_David

Hello,

My calculus book says that readers who are writting $$\sqrt{9}$$ as $$\pm3$$ must stop doing that, because it is incorrect. The question is: why is it incorrect?

Regards

2. Aug 29, 2009

### arildno

Because the square root of a number A is DEFINED to be the unique, non-negative number whose square equals A.

3. Aug 29, 2009

### S_David

I didn't understand. For any real number a there are two square roots: a positive square root, and a negative square root. How is the square root is unique?

4. Aug 29, 2009

### fleem

When you encounter a square root in an equation, use the +/- thing. If somebody asks you, "What is the square root of four", say "two". Its just what mathematicians have decided we will mean, when "square root" is used in each of those contexts.

5. Aug 29, 2009

### arildno

Incorrect.

For any non-negative number "a", the equation:
$$x^{2}=a$$
has two SOLUTIONS:
$$x_{1}=\sqrt{a},x_{2}=-\sqrt{a}$$

The $\sqrt{a}$ is a non-negative number.

6. Aug 29, 2009

### HallsofIvy

Staff Emeritus
To expand on arildno's point: S David, would you say that the solution to $x^2= 5$ is $\sqrt{5}$ or $\pm \sqrt{5}$? I suspect you will say the latter and the point is that the whole reason we need the "$\pm$" is because $\sqrt{5}$ itself only gives one of them: the positive root.

7. Aug 29, 2009

### S_David

Referreing to the book whose name is: Calculus (7th ed), for Anton, Bivens, and Davis, Appendix B at the bottom of the page, it says the following:

After this review, it says the thing I started with. Is this differ from what I said in post #3 in this thread?

8. Aug 29, 2009

### arildno

The SYMBOL $\sqrt{a}$ always signifies a non-negative number.

Therefore, $-\sqrt{a}$ is always a non-positive number.

Colloquially, we call this "the negative square root of a", whereas if we want to be über-precise, we ought to call it "the (additive) negative OF the square root of a"

(alternatively, "minus square root of a", in complete agreement of calling -2 for "minus two")

9. Aug 29, 2009

### slider142

The symbol $\sqrt{9}$ is shorthand for "the principal square root of 9" (not simply "a square root of 9") where the principal square root is a function. A function has only a single output for each input, therefore equating it to the symbols $\pm 9$ which is shorthand for the set {9, -9} is an error.