## Friday, February 28, 2014

### 43: What is a Root?

We all know that $\sqrt{4*4} = 4$ and $\sqrt{3*3} = 3$. We know that $\sqrt{12.4*12.4} =12.4$.

What's $\sqrt{-4*-4} = ?$

Which is it ... or both?
$\sqrt{4} = 2$ or $\sqrt{4} = -2$ or $\sqrt{4} = \pm 2$ ???

From Gabriel Rosenberg via email:

Which is it ... or both?
$\sqrt{4} = 2$ or $\sqrt{4} = -2$ or $\sqrt{4} = \pm 2$ ???

1. Just the bald square root is positive. If you want to find solutions to x^2 = 16 for example, square root both sides the square root of x^2 and the square root of 16 yield abs(x) = 4. Both are positive. The values that are true for x are then x=4 or x=-4.

2. I was told in school that i is the square root of -1. Does that mean i is positive or was I lied to in school?

3. Galois: i is *a* square root of -1, but it isn't *the* square root of i. I think if you'll look carefully at what textbooks say, you won't see "sqrt(-1) = i". Rather, you'll see i^2 = -1 and you'll be told to replace i^2 with -1. so that (for example) i^3, which is (i^2)(i), becomes (-1)(i) = -i, etc.

4. Dave, in better textbooks you'll see that, but I've seen a few books that do say sqrt(-1)=i. I have one that says this, although it's at the beginning and when it comes time to construct complex numbers and be a little more rigorous, they're more careful. Still, it's there before the reader is asked to prove that the complex numbers are not ordered.

Still, it's very common to see sqrt(-1)=i used as a definition, especially in high school level books. Galois raises a good point (as he always does when he explains what he was told in school) about extending the sqrt function to negative numbers and what this means in terms of the previously learned rule of taking the nonnegative root.

5. Some guy is correct that many textbooks do day "i=sqrt(-1)" (except using the standard symbol for square root which I'm having difficulty writing in comments.) See, for example: Precalculus: 2nd ed. by Faires and DeFranza p. 173. Prentice Hall Brief Review for Algebra 2/Trigonometry: NY 2014 by Primiani and Caroscio p. 168, Holt's Algebra 2 NY 2008 p.350. What's worse standardized state exams in NY have often asked a problem like find "sqrt(-16)" and expect 4i as "the answer". Guides to the exam expect students to use invalid properties like sqrt(-16)=sqrt(16*-1)=sqrt(16)*sqrt(-1)=4i

1. Galois, Some guy: After I wrote my comment I began thinking exactly what you two said, but I didn't go and look at any recent high school books to see what I'd find. This reminds me of an incorrect method many textbooks use to show that y = +/-(b/a)x are horizontal asymptotes to the hyperbola x^2/a^2 - y^2/b^2 = 1, which I explained in the following Math Forum AP-calculus post back in November 2010:

http://mathforum.org/kb/message.jspa?messageID=7258230

[This reminds me that I should probably take out of the mothballs the article I said (in the above AP-calculus post) that I was writing and do something with it.]