Square Root Scandal

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This is still in very early draft form. It's at the stage where I take more extreme views, then sleep on it, then look at it and see how much I still agree with.


So as you read, bear in mind that it's intended for my own internal discussion, and will change.

You don't need to read TheTrapeziumConundrum before reading this post, but it might be interesting and help to set the scene, as it were.
So, following on from my previous post in which I talked about definitions, conventions, and mathematical taste, I want to move on to another discussion from Twitter.

Again, let me emphasise that I sympathise with the problems teachers face. And it's not just having to maintain discipline, or do enormous amounts of paperwork, or cover a curriculum. It's also the problem of catering to such a wide range of ability, interest, and eventual career destinations. Some students are destined for service industries, some are destined for academic positions, and still others will be managers or workers. We can argue endlessly about the role education should be playing in preparing students for jobs which haven't yet been created, using technology that hasn't yet been invented, but I don't want to go down that route yet. Instead, I want to look at one very specific topic in one very specific subject.

When teaching maths in particular, the question is how to make it relevant when there is such a huge range of possible interests. What's relevant to one student will be totally irrelevant to another.

And so on.

So what follows isn't a criticism, it's a plea for awareness, both by others of the problems teachers face, and by teachers of the points of view of others.

So what on Earth am I on about?

Square roots.

Now, some of you will have just sighed, rolled your eyes, and possibly switched off, I know. It's an issue that for some of you has been done to death, and you're going to get #mathsrage unless I'm careful. I appreciate that. I'll try to be careful.

The place where this started was Twitter, when @JamesGrime tweeted me and asked: What's the value of $\sqrt{9}$ ? I was a little taken aback, but it turned out that he'd been spoken to most unpleasantly by someone who'd even gone so far as to say he should never be allowed near children again.

His most heinous crime was that he'd suggested that the square root of 9 has the value $\pm{3}.$ His interlocutor was most adamant that the answer is 3, definitely 3, positively 3, unequivocally positive 3. Further, he claimed that to say anything else was, without question, going to damage children's maths education permanently.

Some of you will agree with that, and some of you will sit open-mouthed and wonder how anyone can possibly be so wrong. Let me put both sides of the argument as best I can. For the avoidance of doubt, I do have a specific opinion on the matter, and I'll come to that later.

So first, let me put the argument as best I can that $\sqrt{9}$ should take the single value 3, and not the other possible choice of $\pm{3}.$ (Yes, I know that the simple -3 is another choice, but I'm going to ignore that.)

Part of the problem is that if we allow the value of $\sqrt{9}$ to be either plus or minus 3 as we choose, then we can no longer say that $\sqrt{9}\times\sqrt{9}=9$ because one of those square roots might be the positive one, and one of them night be the negative one. That means that when you multiply $\sqrt{9}$ by itself, the answer isn't necessarily 9. That opens several cans of worms for the students, who like to have single, definite answers, and expect maths to have clear definitions, and clear definite answers.

And for those students who are not going on to study physics or engineering or maths or astronomy or statistics or architecture or economics, this very well might be doing them a favour.

But the vast majority of students won't be going on to advanced study in a maths-heavy subject, so surely the right thing to do is to say to your students that the square root is always the positive square root, that's that, and if you find you deal with square roots later, you can worry about it then.

Well, that's one point of view. You can probably tell from my tone that it's not one I actually agree with, but to be completely honest, I can understand why teachers will take this point of view, even if they themselves don't entirely agree with it. After all, most of teaching is dramatic over-simplification. We're first told that atoms are small, indivisible units. Then we're told that that's wrong, and in fact they are made up of an indivisible nucleus with electrons orbiting it a bit like planets around the Sun.

Then we're told that that's wrong, and in fact the nucleus has indivisible bits called protons and neutrons, and that the orbits aren't arbitrary. Then we're told that the indivisible bits in the nucleus aren't indivisible after all, and the electrons aren't actually in orbit at all. Then we're told that electrons aren't really particles, but are sort of particles and sort of waves. Most recently we've even been told that electrons are no longer fundamental particles, but can actually be split [1].

What will we be told tomorrow?

It's a succession of simplifications, each one more sophisticated than the last, and no matter what stage you're at when you stop learning chemistry (or physics or astronomy or biology or whatever) you are left with an image (or model) that's suitable for your level of understanding.

And I have some sympathy with that progression. After all, much in the world can be explained to a ten year old using the idea that stuff is made up of indivisible atoms that combine together in different ways. You won't get very far if you have to start with quantum mechanics.

But having said that, I also think there's value in at least hinting that there's more to come, and that the picture is incomplete. Surely it's worth baiting the hook and laying the foundation that things are actually more complicated, and this is a simplification that will do "for now."

This is disputed by some teachers, who get very angry when I suggest it. "After all," they say, "nothing goes wrong when you always use the positive square root. We then always have that $\sqrt{9}\times\sqrt{9}=9,$ and everything seems to work. Why confuse the students to no good purpose."

Well, the point is firstly that it doesn't always work. And the reason it doesn't always work is because it's a lie, and sometimes you'll get caught out. Here's an example:

Start with any two numbers a and b, and take their sum: $s=a+b.$ Let t be half of s, so now we have $s=2t.$

$a+b=2t$ t is half s
$a=2t-b$ Subtracting b from both sides
$a^2=2at-ab$ Multiplying each side by a
$a^2-2at=-ab$ Subtracting 2at from each side
$a^2-2at+t^2=-ab+t^2$ Adding $t^2$ to each side
$a^2-2at+t^2=-(2t-b)b+t^2$ Substituting 2t-b for a
$a^2-2at+t^2=-2bt+b^2+t^2$ Expanding the LHS
$a^2-2at+t^2=b^2-2tb+t^2$ Rearranging the LHS
$(a-t)^2=(b-t)^2$ Factoring both sides
$a-t=b-t$ Square root both sides
$a=b$ Add t to both sides

Using the simplification of always taking the positive square root, without questioning why, appears to have led to a proof that every number equals every other number.

It's easy to see where this goes wrong, but if you've taught students that there is always just the one true square root, it's hard to explain this result. All you've done is lay the foundation for them to believe that maths is just an random collection of arbitrary rules, with neither rhyme nor reason, with no order, and with no purpose.

I've had this discussion with teachers. One point of view that was put forward is that students who are confused by this will never "get maths" anyway, so it's of no value to try to be more complete or more precise. But if they won't "get it" regardless, why not do the right thing by those who will "get it" and at least tell them that it's complicated, and that sometimes you need to use a negative square root?

See, the point is that it is complicated. Glossing over it without even a hint is short-changing them. At least give them a glimpse of the fun to be had with Riemann Surfaces, complex numbers, and finite fields. Tell them that mostly it works with just the positive square root, but that sometimes there's no such thing, sometimes you need to have a mixture, sometimes you need to be careful, and that it's a part of a bigger scheme of things that they can explore later if they want to.

To look for a moment at some of the fun we can have with this, let's think about "nice properties" for functions.

Another nice property is differentiability, but we're not going to go there.
Functions should always give a single answer for any input. This is one reason that teachers and students both want to have a single, definitive value for "the square root." It's also nice when functions are continuous, so that varying the input by just a little moves the output by just a little. It's nice when feeding in similar values to a function gives similar answers.

So let's look at the complex numbers, and let's suppose we insist that the square root of 9 is +3. It would be nice if varying the input slightly only varies the output by a little, so it would be nice if the square root of 9+0.6i is roughly 3 (and in fact it is very, very close to 3+0.1i, since $(3+0.1i)^2=8.99+0.6i$ ).

So here's an idea. Starting with 9 and 3, move 9 by a little, then find a number close to 3 which, when squared, gives the new number. As we gently and slowly move away from 9, the square root moves gently and slowly away from 3. Here's a table of the values as we move around a circle of radius 9 in the complex plane. In each case I've started from the previous square root and found the nearest complex number that squares to give 9. We can do that numerically just by looking at nearby numbers, squaring them, taking the one that gives the answer closest to 9, then repeating.

Moving from 9 Moving from 3
9.00 + 0.00i 3.00 + 0.00i
8.46 + 3.08i 2.95 + 0.52i
6.89 + 5.79i 2.82 + 1.03i
4.50 + 7.79i 2.60 + 1.50i
1.56 + 8.86i 2.30 + 1.93i
-1.56 + 8.86i 1.93 + 2.30i
-4.50 + 7.79i 1.50 + 2.60i
-6.89 + 5.79i 1.03 + 2.82i
-8.46 + 3.08i 0.52 + 2.95i
-9.00 + 0.00i 0.00 + 3.00i

Doesn't seem so bad. We get that the square root of -9 is 3i, which is as expected.

Let's keep going:

Moving from 9 Moving from 3
-9.00 + 0.00i 0.00 + 3.00i
-8.46 - 3.08i -0.52 + 2.95i
-6.89 - 5.79i -1.03 + 2.82i
-4.50 - 7.79i -1.50 + 2.60i
-1.56 - 8.86i -1.93 + 2.30i
1.56 - 8.86i -2.30 + 1.93i
4.50 - 7.79i -2.60 + 1.50i
6.89 - 5.79i -2.82 + 1.03i
8.46 - 3.08i -2.95 + 0.52i
9.00 - 0.00i -3.00 + 0.00i

Now there's a problem - we have that the square root of 9 should be -3, not +3. By using our idea of how a nice function should behave we've derived behaviour that's not nice.

This means that the square root as a function from the complex numbers to the complex numbers is either not a function (because it must take multiple values) or it's not continuous (because it has to jump). Mathematicians get around this by having two copies of the complex plane and cross-connecting them. When you run around the origin once you end up on the other copy, and then you run around again to end up back where you start. This is a Riemann Surface.
So if you insist that the square root of 9 is +3, you have to give up all sorts of other nice things. In particular, you have to decide when the above process "goes wrong" - when do we have to use a number that's not close to the previous value for the square root.

This harks back to the question of whether 0 is even or odd, or even a whole number at all. Yes, we can simply declare that 0 isn't a whole number, but we lose all sorts of benefits.

In the interests of full disclosure, there are some mathematicians who say that generally they would give the answer +3, and then in practice never take the square root without explicitly putting in the $\pm$ sign. That really muddies the waters.
And similarly, yes, we can declare that the square root of 9 is +3 and only +3, but we lose all sorts of benefits. And that's why mathematicians (in general) have chosen the convention that the square root of a number is always the $\pm$ version. It's not arbitrary, it's a part of a structure that's been worked on for the last 5000 years, and it continues to grow as we find out more, and find more uses for the techniques and ideas.

Maths, in the end, is a bit like a diamond. Yes, it's hard, but it can also be useful. More, it can sometimes be genuinely beautiful.

OK, so it's plausible that I've upset some teachers here. It's not only possible, but likely that I haven't understood completely the concerns they have, or the reasons why some of them teach, emphatically, that $sqrt{9}$ always means the positive square root.

If I haven't understood then please, email me. Contact details are at the bottom of this page. I have temporarily configured my spam filter to pass automatically any email that contains the string "SquareRoot" with that exact capitalisation and no spaces.

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