Common Mistakes: Square Roots

A look at the precise definition of the square root function and problems arising from misuse of square roots.

Category:

Education

Tags:

• square root
• mathematics

[mascoteponto]

I agree with your arguments, but the whole thing gets shattered when we get to complex numbers, so that's what I would have liked to see you discuss. Thanks, anyway.

[DrKevinHouston]

@mascoteponto Thanks for the comment. I'm not sure what you mean by shattered when we get to complex numbers. One of my reasons for doing the real case carefully is that students often don't fully understand the subtleties of the complex case. The video helps lay the groundwork for dealing with those problems (eg, multi-valued functions such as square root or log). I'm teaching Complex Analysis again this coming semester so if I have time I may post a video about it. (That's a big 'if'!)

[mascoteponto]

@DrKevinHouston What I mean is that it is not clear (at least to me) which one is THE square root of a complex number, and it gets even worse for roots of higher indexes.

[DrKevinHouston]

@mascoteponto I would say that there is no unique square root and so we can't talk about THE square root of a number. The point is that when we move to complex numbers then we have to introduce multi-valued functions or make branch cuts in the complex plane to define roots and logs. By first being clear about what is a function and what are solutions of equations in the real case we can then (hopefully) make the complex case clear.

[mascoteponto]

@DrKevinHouston Yes, that is the point. The concept of a multivalued function is flimsy to me, as the very definition of a function requires it not to be multivalued. Moving to complex numbers, then, sort of demotes the concept of roots to the broader notion of a correspondence. We define, for instance, square roots as "x is a square root of y iff x^2 = y". By this definition, however, the number -2 is a square root of 4, contrary to what you exposed in this video.

[DrKevinHouston]

@mascoteponto Actually I think it's fine to make the definition "x is square root of y iff x^2=y" (we're mathematicians so we can make whatever definition we like!). So -2 is a square root of 4 as you say. And if we want to talk about THE square root of y, then we could take the complex number with the smallest argument. (This also works for higher degrees as well.) All well and good. The trouble is too many students miss the difference between "the" and "a". And confuse function and set.

[DrKevinHouston]

@DrKevinHouston Now that I think about it, I may try to use this approach to clarify these ideas in my upcoming course. Thanks for the comments - they keep me thinking!

[Scp1966]

You are a true genius. You made me realise something that might be useful to me in the future. Thanks a lot!

[DrKevinHouston]

@esper109 As for the material at 7:07, what is meant is that at least one of "x=1", "x=-2" is true. Think of it like the statement "David has a dog or Alice has a dog" being true. We know that at least one of them has a dog (possibly both!) but it does not mean that both definitely have a dog. So the equation sqrt(x+3)=x+1 true implies that x=1 or x=-2 means that at least one of "x=1", "x=-2" is true. Now, we know from earlier that one of them is not true.

Hope that explains that!