Best way to explain the thinking steps from x² = 9 to x=±3

I'm a huge fan of using the zero factor property: $x^2 - 9 = 0$, which factors as $(x+3)(x-3) = 0$. It's not the most straightforward method, but it gives my students a tool for avoiding losing solutions. For instance, in solving $x^2 = x$, it's common to lose $x = 0$ as a solution, but rewriting as $x^2 - x = 0$ and factoring gives $x(x-1) = 0$. The zero factor property gives us both solutions this way.

You might consider having mom draw a plot of $y=x^2$, and then have her turn it on its side.

“Well, if you start with x2 = 9, you know that x can be either 3 or -3 to make it true. If you think of this as an algebra step, it would be taking the square root of both sides. But you would want to keep both the answers, 3 and -3.

To me, it seems you are saying (or, at least the child will understand) that $\sqrt{x^2}=x$, which is wrong, can be easily explained under the previous agreement that the square root is always positive and explains everything:

• Since $3^2=9$, we have $\sqrt{3^2}=\sqrt{9}=3$.
• Since $(-3)^2=9$, we also have $\sqrt{(-3)^2}=\sqrt{9}=3$.
• In general, $\sqrt{x^2}=|x|$.
• Thus, taking the square root of both sides: $|x|=3$, that is, $x=\pm3$.

The original text does not clarify why on one side we get only one value ($x$) and on the other we get two values ​​($\pm3$). Since it is not a matter of definition, "want to" is not a reasonable mathematical explanation.

I'm not sure I'm qualified as a "good teacher" (most likely not, but, like with everything else, it depends on what you compare to), but I find this conversation as written more confusing than revealing. It mixes at least four things, each of which requires a separate discussion:

1. The history of the introduction of negative numbers (which I don't know really well, so I'll abstain from further comments here)

2. The definition of the function $x\mapsto \sqrt x:[0,+\infty)\to[0,+\infty)$ (that is about where you say that there cannot be two square roots of the same number, i.e., a function (to deserve being called "a function") should assign exactly one value to each argument.

3. The fact that $\sqrt x$ is only a right inverse to $x\mapsto x^2$ as a function from $(-\infty,+\infty)\to[0,+\infty)$ but not the left one and one needs to use the left inverse in passing from $f(x)=y$ to $x=f^{-1}(y)$ (that "weird algebra").

4. The imprecision of English language and the necessity of having a clear definition, which has to be consulted in all questions of the type "Is X an A, or isn't it?" in mathematics or in the real life.

Althea is confused for a reason - (her?) Mom in mixing things - exactly:

Geometry vs. algebra/calculus - "...long time ago people..." have been asking about length of side of a square not a square root - thus they focused on positive numbers => one answer.

But we are asking about equation or function and here I would go with @Pedro answer - explaining that square root and square are inverse operations and as result we get absolute value $|x-0|=3$ which is a question about numbers in distance of 3 from 0 - or I would go with arms of function but this path requires additional knowledge (what function is, why there are no gaps between natural numbers for which we draw/check it values etc.).

Though I don’t know for sure that I am a “good teacher”, I have taught mathematics to the age group of your 12 year old protagonist Althea. And I apologize for the longwindedness of this, and it’s rambling disorganization. To borrow from Pascal, “I am sorry I wrote such a long response, I didn’t have time to write a short one.”

First, I see you are based in the USA, as am I, so my answers are going to be based on my understanding of “what 12 year old American students know”; having been a middle and high school math teacher (who’s background was a biologist that happened have an undergraduate math degree – I am no mathematician!). Also I assume, since this is a young adult novel and the protagonist is about 12, that 12 or younger is the is the age of your intended audience. (Otherwise you are violating one of the Rules of YA books – the hero is just older enough for the reader wants to grow up to be!)

So some things to keep in mind; at age 12, which is approximately 7th grade, neither the protagonist Althea or the reader is likely to have had any formal introduction to functions per se. They will have, for instance, graphed an equation like y = 3x – 2, but any functional notation such as f(x) = 3x-2 will be a newish and abstract concept, and you’d have to spend much time unpacking that. Since the protagonist/reader won’t really know about functions, then the idea of an inverse function will be even further from their understanding. Formal introductions to inverses may be as early as Algebra I (usually taken in 8th or 9th grade, so when students are 13 or 14) but often not until Algebra II (usually taken in 9th, 10th, or 11th grade – so maybe as old as 16).

Finally, I argue that the real intellectual problem with square root notation is not the idea of inverse functions, or more generally, multiple correct answers/results, but with the annoying existence of the highly unintuitive irrational numbers. More on that in a moment.

Students at that age have no problem with there being more than one solution to a problem, though perhaps a review for them is appropriate. A non math example would be the mappings between individual female gymnasts in the 2024 Paris Olympics and the teams they are on. If you ask what gymnastic team Simone Biles is on, there is one and only one correct response – The US team. Simone Biles maps to the US Team, not the Russian, Chinese, or Uzbekistan Teams. Similarly, Jordan Chiles, Jade Carey also competed for – and only for – the US Team, while Kaylia Nemour only competed for Algeria.

At that 12 year age both your protagonist Althea and your reader will get that asking the related and sort of “opposite” question of “who was on the 2024 Olympics women’s US gymnastic team” has multiple correct answers; Simone Biles is A correct answer, but not THE ONLY correct answer; Jordan Chiles, Jade Carey, and a host of others are also correct answers. And that sometimes a single answer is ok, but often you need to know ALL the correct answers. So the full answer to who is on the US team is the whole team roster, not just one or two of the members.

And of course one can do the same with number questions; 8 x 3 = 24, but the question “what two numbers multiply to give you 24?” has many answers. Obviously 8 x 3, but also 6 x 4, or 12 x 2, and so forth. And you can even get into issues of restricted domains now. Do I allow negative numbers, that is is my domain whole numbers or integers? How about fractions, that is rational numbers? (and at 12 they may have only recently been introduced formally to the words “integers” , “rational numbers” and maybe even “irrational numbers”)

The student/reader gets the above arguments immediately, but may be confused at their age if you use formal function definitions and the concept of inverse functions.

So I will argue that – depending on all the goals of your book – you might need to decide if getting fully into anything that hints of formal definitions of functions and inverses will require too much unpacking, as it may be a few years away in their formal math education.

So back to why I blame the problems of understanding that the equation x^2 = y has two solutions that we label +sqrt(y) and -sqrt(y) on the existence of irrationals rather than the need to have deep (or any/much) understanding of inverse functions. It has to do with why we even have the square root symbol at all. I argue that reason is because there are an infinitude of irrationals; “most” numbers are irrational, but we’ve only given symbolic labels to a few; pi, e, phi, and so on. We have never ever solved that problem – and never will, at least not until we become infinitely intelligent beings whose short term memories can simultaneously hold an infinitude of items rather than a measly 7 plus or minus 5!

But for the subset of irrationals that are square (or nth) roots of numbers, someone did come up with a clever annotation for them – namely stick the squared number (or cubed or nth powered) number inside that weird Vans shoes V symbol. (Aside – I’ve never met a middle school class where at learning the square root symbol, at least one person doesn’t make the joke that the Vans shoe logo is “the square root of the ans-wer”).

Because the number, x, when squared, that is equal to 2 is not 1.4 or 1.41, or even 17/12 or 577/408 – though lovely approximations, all we can do is come up with some way of symbolizing it. And we do that by sticking the square number 2 inside that wacky square root symbol.

And that (ignoring signs) doing the step x^2 = 9, so x = sqrt(9) and sqrt(9) = 3 is often taught but in some ways confuses the issue. The answer to X^2 = 9 is directly x=3. One really only needs to use the square root sign for those equations without a perfect square. It’s a place holder symbol for a number that you know exists (say the length of a diagonal of a square), but cannot enumerate exactly.

OK, so we need both the postitive roots (easy conceptually) and the negative roots. And I argue (perhaps poorly) that any formalisim of functional notation and inverses isn’t really necessary. One only needs the reader/student/child’s intuitive understanding of the above “who is on the US women’s Olympic team” having more than a single correct answer.

Oh another thing the student will know, has known for a while, but was perhaps only formally introduced around 11 years, maybe 12 (ie 6th ort 7th grade) is that “a postitve times a positive is a postive, a negative times a negative is a positive, a negative times a postive is a negative, and a positive times a negative is a negative.” And they know it because in 6th grade or so, they learn to chant it so it really sticks! And they also know that “zero times zero is zero”, and that “zero times any other number is zero”. This may be useful. Aside: the above rule/chant is an example of a generalization and abstraction that often slowly occurs. Sure they have been multiply positive numbers and getting positive numbers since 2nd or 3rd grade, and added negative numbers to their math toolbox a year or so later, so they already work with the concept. But though they’ve seen that result each and every multiplication they have done, it might not have been until 6th or 7th grade it is explicitly stated as a general abstract rule that is always true, not just the for the small set of multiplications they have personally done.

Getting to why the two roots to x^2 = y has two roots, namely positive sqrt(y) and negative sqrt(y). They will get the positive root exists without any problem, especially if we are talking square roots of perfect squares. And with proper motivation they will get/remember/understand that all those annoying unperfect numbers also have square roots, and the best we can do to represent them is use that clever square root symbol over the number in question.

The negative root becomes easy; they can easily be intuitively convinced (indeed they already know, but perhaps never had stated explicitly) that if (a) * (a) = b, then (-a) * (-a) is also positive and also equals b. Or in fancier math terms, if x is the square root of some number y, then so is x’s additive inverse.

So I don’t think I’ve solved any questions you had about the pedagogy of your explanation, but I hope I’ve – if not clarified anything (I probably confused things!) – at leas added food for thought.

So now on to a meta-question. If you are interested in explaining the imaginary number i, where i^2 = -1, what is your pedagogical purpose for the foray into negative square roots? A negative square root of a postivie number is a whole different kettle of fish than a square root of a negative number!

Your reader/hero already knows that “a positive times a positive is a positive” and that “a negative times a negative is a positive”, so it’s only a tiny intellectual leap to “all squared numbers are positive”. Or zero, if the original number is zero. So the wild thing about imaginary numbers, i, is that we are proposing a number that clearly can’t exist (at least on the real number line)! All squares are positive, so there can’t be a square that is negative! But the historical thing was that pretending i existed helped solve all sorts of tricky problems, especially (originally) finding all the roots of cubic equations.

Oh – fun fact – historically, my understanding is that imaginary numbers were grumpily accepted as useful decades to centuries before negative numbers were completely accepted! Go figure! And that imaginary numbers were invented far enough back that Christopher Columbus could have known of them before he “sailed the Ocean Blue” back in 1492! Crazy, right? But I digress…

So the intellectual difficulty is why are imaginary numbers so useful, when they clearly “can’t exist”? With my students I would present that this way. I get them to see that i cannot be a positive number (since all squared positives are positive) or a negative number, or zero (since zero times zero is zero). So I can't put i on the (real) number line. I usually say something like “so if we want to treat I as a number, which is useful, it just has to float here in space, above the number line, haunting and taunting it.”

But notice here, the issue of negative square roots (as opposed to square roots of negative numbers) never really has to come up. And the fundamental concept of i isn’t the square root of -1, it’s proposing that a negative squared number can exist.

And after introducing Argand Diagrams, formal definitions of complex addition and multiplication, I usually say something like this to my students:

“So you see, it’s easy to have a squared number that is negative! All you have to do is let me change what we mean by “number” (from a a point on a one-dimensional number line to a point in the two-dimensional complex plane) and what I mean by “multiplication” (from a stretching in one dimension to a stretching AND a rotation in two dimensional space) and we are good! Easy peasy, right kids?”

At which point they groan, and tell me I’m a horrible person who goes out of my way to break their brains. But they say it with smiles on their faces, so I think they mean it in a good way!

I apoligize for the rambling longwindedness of this. I hope it wasn’t completely useless as a response! And I also apologize to all the real mathematicians groaning and gnashing their teeth at the terribly innaccurate and likely incorrect things I have said.

Good luck with your books!

Discussion of the Problem

I think that the key to good teaching is anticipating the kinds of mistakes that learners are likely to make. A common (slightly incorrect) way of thinking about a problem like this is as follows:

If I have an equation, I can do the same thing to both sides of the equation, and the truth-value of that equation remains the same. For example, if $5x=30$, then I can divide both sides of the equation by $5$, giving $x=6$. If I apply the same process to $x^2 = 9$, I can take the square root on both sides and get $$x=\sqrt{x^2}=\sqrt{9}=3.$$ But wait... I know that $(-3)^2 = 9$, so $-3$ is also a solution to the original equation. I should have gotten $x=\pm 3$. The process I came up with didn't find that solution, so I must have done something wrong. It must be the case that $\sqrt{9}$ is not $3$, but rather that $\sqrt{9}=\pm 3$. So, more generally, $\sqrt{a^2} = \pm a$.

The fundamental mistake here is the idea that if something is done to both sides of an equation, this precisely preserves the truth-value of that equation. There is an implicit assumption here (in more technical terms) that all functions are one-to-one. This heuristic works in a lot of the cases shown in elementary algebra—for example

• $a = b$ if and only if $a+c = b+c$ (the function $x \mapsto x+c$ is bijective);
• $a = b$ if and only if $ac = bc$ and $c\ne 0$ (the function $x \mapsto cx$ is bijective for $c \ne 0$; if $c=0$ there is a bit of a problem, but most students have it hammered into their heads that division by zero is "not allowed", so there is rarely a problem here);
• $a = b$ if and only if $\mathrm{e}^a = \mathrm{e}^b$, from which it follows that $x = b^y$ if and only if $\log_b(x) = y$ (for "admissible" values of $b$);
• an so on.

Because students being exposed to this kind of algebra for the first time are not usually familiar with the idea of a function (let alone an injective function), the reason that the algebra works out in these examples is a bit mysterious, and students over generalize.

A more accurate statement is something like

If $a = b$ and $f$ is a function, then $f(a) = f(b)$.

In comparison to the previous statements, this is a conditional statement, rather than a biconditional statement. $a=b$ implies that $f(a)=f(b)$, but the converse need not be true (in order for the converse to be true, $f$ would need to be one-to-one).

But the function $x\mapsto x^2$ is not an injective function, so it is not true that $$ x = 3 \iff x^2 = 9.$$ It is true that $x=3$ implies that $x^2 = 9$, but the converse is not true. Oops... this causes problems when trying to solve the equation $x^2 = 9$.

Alternative Solution 1

"I know that if $x^2 = 9$, I can take the square root on both sides, and get $x=3$."

"Great. But do you know that this is the only solution to the equation?"

Althea stared at the page for a couple of seconds, then picked up her pencil and started writing again. "So I can rewrite $x^2=9$ as $x^2-9 = 0$. I know that the graph of $y=x^2$ is a parabola, and subtracting $9$ moves that parabola down $9$ units," she said, as she sketched out a quick parabola.

[insert picture here]

"The solutions to the equation are the places where the parabola crosses the $x$-axis. I've already figured out that one of those points is $x=3$, but it looks like there's another!"

"Oh?" Mom asked. "And do you see anything in your picture which might help?"

"Well, it looks like it is symmetric—if I fold the picture in half along the $y$-axis, it matches up. Because one solution is at $x=3$, the other solution must be it's mirror image—$x=-3$!"

"Exactly! Parabola's a symmetric, so equations involving parabolas typically have solutions which come in pairs. So if you imagine that the equation $x^2 = -1$ has a solution—for example, suppose that $\ddot\smile$ has the property that $$ (\ddot\smile)^2 = -1, $$ then we would expect there to be a second, symmetric solution, $-(\ddot\smile)$."

Alternative Solution 2

"If I take the square root on both sides, I get $x = 3$, but that doesn't seem to get all of the possible solutions, since I know that $x=-3$ is also a solution. So how do I get that other solution?" Althea asked, almost rhetorically, furrowing her brow.

"Why don't you try rearranging the equation a bit. Maybe something will pop out..." mom said.

"I can rearrange $x=3$ to $x-3=0$, and I can rearrange $x^2 = 9$ to $x^2-9=0$. I know that $x$ times $x$ is $x^2$, so maybe I can multiply $x-3$ by something to get $x^2-9$...?"

Althea thought for a bit, then started writing down some notation.

"If I were working with just numbers, the question is easy—for example, if $3$ times something is equal to $15$, I can write that as $$ 3y = 15. $$ I can solve that by dividing each side by $3$ to get $y=5$. But these $x$'s and $x^2$'s are making things hard!"

"So pretend that they are 'just' numbers."

"Well, if they were just numbers, the equation would be $(x-3)y = x^2 - 9.$ If I wanted to solve this, I would write $$ y = (x^2 - 9) \div (x-3). $$ Is there a way of doing that?"

"Do you know how to do long division?" mom asked.

"Sure... oh, I get it! When I do long division, I make a guess about how many times the divisor goes into the dividend, then multiply it out and work with the remainder. I can do the same thing here. $x$ goes into $x^2$ exactly $x$ times, so I can write

 x ---------- x-3 ) x^2 - 9 x^2 - 3x 

"After subtracting, the remainder is $3x-9$, and $x$ goes into $3x-9$ exactly $3$ times. So...

 x + 3 ---------- x-3 ) x^2 - 9 x^2 - 3x --------- 3x - 9 3x - 9 ------ 0 

"So it looks like $x-3$ goes into $x^2-9$ precisely $x+3$ times, with a remainder of $0$! In other words $$ x^2 - 9 = (x+3)(x-3). $$ But I want to know when $x^2 - 9 = 0$, and the only way that a product can be zero is if one of its factors is zero. So either $$ x+3 = 0 \qquad\text{or}\qquad x-3=0. $$ I already figured out the second one, but the first one gives $x=-3$!"

Mom beamed. "Exactly! You can solve these kinds of problems by thinking about factoring. The question to ask is, `Can something like $x^2 - a$ always be factored, or are there exceptions?' That's where imaginary numbers start to come into the picture."

This is an attempt to answer your question:

I'd like to know what good teachers think of the implied pedagogy in the following conversation between 'Mom' and Althea.

It doesn't answer your other questions, because I don't know what you're really asking. For example:

What is the best way (are the best ways) to explain the thinking steps that get us from $x^2 = 9$ to x = $\pm 3$?

That depends on what your thinking steps are to get from $x^2=9$ to $x=\pm 3$. You don't say. It's presumed that Althea already knows it anyway. Instead, your passage seems to be trying to address your other question:

"No, I was just thinking about your dilemma of whether square roots should have two answers."

This is an entirely different question which you also don't answer. You raise some of the issues with either convention, but you don't clearly give any definitive answer on which convention to pick or why.

There is also an implied question here: "What are the best thinking steps for getting us from $x^2 = 9$ to x = $\pm 3$?" Other people have tried to answer that one. Factorisation of $x^2-9$ is one way. Going in the other direction and introducing complex numbers first is another. Explaining why the answer is $\pm 3$ rather than just $3$ becomes trivial when you know about complex numbers. It's because of there are two angles that when 'doubled' give any given angle. You can teach that to a child by getting them to spin around on the spot. Complex numbers are a long way round to make that point, but if you're going to be talking about complex numbers anyway, then it makes more sense to talk about why there are two roots after rather than before.

So you need to think about what it is you are trying to do here.

• Are you trying to explain why there are two roots? How to create an intuitive understanding of its necessity?

• Are you trying to explain the terminology and notational conventions regarding the use of the square root sign, the $\pm$ sign (which is also used to express errors with an entirely different meaning), principal values, functions, and so on?

• Are you debating the meaning of the term "square root" and whether it is applicable to negative numbers - given its geometric etymology as the positive length of the side of a square of a given positive area?

• Are you asking how to explain what the meaning of $\pm 3$ is? Is it either value, or both values at once, or one of these values but you don't know which?

• Are you talking about the pedagogy of discussing these sorts of conflicts and puzzles as a way of opening a student's mind to new ways of thinking, that break with what they have been taught before?

• Are you trying to teach the process/algorithm that starts with the number $9$ and ends up somehow with $\pm 3$?

• Are you asking for such a process?

• Is Mom trying to argue that because $9$ has two square roots, it's therefore reasonable by analogy for $-1$ to have two square roots too, and you are asking us whether that argument is persuasive?

• Are you asking about the pedagogical wisdom of being asked a question about $\sqrt{-1}$ and instead of directly answering the question, going on a long and rambling diversion about whether positive numbers have two square roots?

It seems to me that Althea already knows that both $3$ and $-3$ square to $9$, so she's right to roll her eyes and complain that Mom is not answering the question!

If you want to explain imaginary numbers to a student, then this debate about $\pm$ is an irrelevant distraction. Construct the complex numbers (first concretely, then pictorially, then abstractly), define how to multiply them, show that there are two of them that square to $-1$. Whether you then want to call them both "the square root of minus one" or only one of them doesn't really matter.

This is a great piece of dialogue.

But there is one detail I find confusing. In the last paragraphs, Mom discusses "plus or minus three", with the plus-or-minus sign.

And then she justifies it with this:

“We read that as x equals plus or minus the square root of 9. It’s an unusual algebra step and people often forget the plus or minus part. [...] I was just thinking about your dilemma of whether square roots should have two answers. If we say that square root means just the positive value, then we have to do that weird algebra step.”

I find these last paragraphs confusing, and the most obvious way to interpret them seems to be:

Square root is a function. Functions have only one answer at every point. But there are two square roots of nine. To solve this dilemma, we do a weird algebra step of merging three and minus three into the new number plus-or-minus-three and we say that the square root of nine is the number plus-or-minus-three.

This is probably not what you wanted to say!

"x equals plus or minus three" is nothing more than shorthand for the disjunction of equations "x equals three OR x equals minus three". There is no new weird number involved.

Start with the idea that a negative times a negative is a positive. A good way to understand this is a negative is an opposite, and the opposite of an opposite is the original.

For instance, when we turn a coin upside down, and then do it again, it's back to where it was as the start.

Then we can look at how something like how $(-3)\times(-3)=9$.

So if we asked this question: What can we put in for $x$ to work for $x^2=9$? Is there only one answer?

Maybe Althea needs an identical twin sister. They do many things together, and have fun with the "power of two." Most strangers do not even know there are two sisters. For them, $\sqrt{\hbox{Althea}^2}=\hbox{Althea}$. But her mother, teacher, and friends know about the twins. They both have individual identities. The tool that these informed people use to distinguish them is the mirror image of $\sqrt{\phantom x}$ next to $\sqrt{\phantom x}$

¯\(ツ)/¯

We need to talk a bit first about "What are numbers?"

We use numbers to represent various things in the real world, and how they behave when we combine them.

The first sort of number we come across is counting numbers. We count the number of sheep in our flock, or the number of marbles in a bag, or the number of pages in a book. We observe that it doesn't matter whether we're talking about sheep or marbles or pages, if we add three of anything to two of anything we get five of that same thing. So we can work out the rule once, that three plus two is five, and immediately apply it to anything that can be counted. Five sheep, five marbles, five kittens, five whales - they all follow the same rules.

But some things in the real world don't quite act the same way. For example, let's take steps forwards and backwards. Three steps forwards plus two steps forwards is the same as five steps forwards, so that much is the same. But two steps forwards plus three steps backwards gives us one step backwards. That doesn't work with sheep. If we start with two sheep, we can't take three of them away. So we now have a bigger set of "numbers" that includes the counting numbers but also includes some new numbers that we didn't know about before: the negative whole numbers, and possibly zero. (Depending on whether we thought "zero sheep" was a thing.)

Now let's consider measuring things like lengths, areas, and weights. Lengths, areas, and weights are always non-negative but don't have to be whole numbers; they can vary continuously. So they include the counting numbers, but not the negative numbers.

And for another example, let's consider rotations, which we measure with angles. For small turns, the same rules seem to apply. Two ticks clockwise plus three ticks clockwise is the same as five ticks clockwise. But now we have a really weird thing that when we turn all the way around the circle, we end up back where we started! A whole turn is the same as two turns is the same as no turns. Angles are a very strange sort of number!

So when we get a question like "What is the square root of $-1$?" we first have to ask what sort of numbers we are talking about. There isn't an answer for any of the sorts of numbers we know, but maybe there is for some other new sort of number we don't yet know about, describing the way some other sort of thing in the real world behaves? It turns out there is.

We are going to talk about the scale-rotations. This is where we take all the points on a flat plane and do a combination of scaling and rotating about the origin. We chunk them together as one thing. So an example of a scale-rotation is the transformation that scales everything up by a factor of two and rotates a quarter turn anticlockwise. We can break every scale rotation into a combination of a pure scaling (by a non-negative number) and a pure rotation through an (anticlockwise, just to be unambiguous) angle.

We can also represent each scale rotation by specifying where the point $(1,0)$ (or the point $(0,1)$) is transformed to. This makes the arithmetic much easier, but is maybe conceptually not as clear, because it's not actually a point on a plane - the point is just being used to show its effect.

We invent two rules for adding and multiplying them. The one for adding them is geometrically a little strange (if the first maps $(1,0)$ to $(a,b)$ and the second maps $(1,0)$ to $(c,d)$ then their sum maps $(1,0)$ to $(a+c,b+d)$), but we don't need to worry about that for answering our question about square roots. The rule for multiplying them is easy: just do one transformation after the other!

These rules turn out to work very nicely. The sum or product of any two scale-rotations is another scale rotation. Any scale rotation apart from the zero scaling can be reversed by another scale rotation. And the pure scalings act exactly like the numbers we already know, so our existing system of numbers is included. (We mentioned earlier that we could choose the scale factor to be non-negative, but we can easily get negative numbers by adding a $180^\circ$ rotation.) At the same time, the pure rotations behave like the angle "numbers" we considered previously, so we have both sorts of number incorporated into the same system!

Now that we can multiply them, we can square. Just multiply a scale-rotation by itself. Or in other words, do the scaling twice, and the rotation twice. Say we pick a pure scaling by a factor of $3$. We do it twice, and get a pure scaling by a factor of $9$. So that works exactly like we expect squaring to work. We can do the same with $-3$ too. Take a rotation by $180^\circ$ combined with our scale factor $3$ and do it twice, and we find that we get the same answer. The scale factors multiply, the angles add, and $180^\circ +180^\circ =360^\circ =0^\circ$.

We can also reverse this process to find the square root. We want a scale-rotation that when done twice gives us the target scale-rotation. We find the scaling part of the answer using the ordinary square root, and we find the rotation part by halving the angle. It turns out this is ambiguous, because there are two distinct angles that when applied twice give the same answer. Since every angle is unchanged by a $360^\circ$ rotation, we find $a=a+360^\circ$ and so in a way we could say $a/2=a/2+180^\circ$. (Warning: that last equation is properly messed up. Angles can't be multiplied and divided like that. They're the wrong sort of numbers! I'm just saying that to make it sound plausible to you quickly without going into details about what it really means.)

So it turns out that the reason we have two solutions to $x^2=9$ is that there is a hidden rotational part to the numbers that is ambiguous when we try to find a scale-rotation that when done twice gives us a pure scaling. If we only ever looked at the pure scalings part, we would never have known geometrically why the ambiguity was there!

And finally, we can now easily answer our oringinal question. The scaling number $-1$ is a pure scaling by a factor $1$ and a rotation of $180^\circ$. It a simple half-turn. What operation, done twice, gives us a half-turn? Why, a quarter-turn, of course! A three-quarters turn will work too.

We can make everything easier to work with if we happen to be familiar with $2\times 2$ matrices. The scale rotations are matrices of the form:

$\pmatrix{a & -b \\ b & a}$

This sends the point $(1,0)$ to $(a,b)$ and the point $(0,1)$ to the point $(-b,a)$. These are the two columns of the matrix.

The pure scalings are of the form:

$\pmatrix{a & 0 \\ 0 & a}$

We can split any scale rotation into a linear combination:

$a \pmatrix{1 & 0 \\ 0 & 1} + b \pmatrix{0 & -1 \\ 1 & 0}$

We can write the multiples of the first matrix as if they were the ordinary numbers we already knew about, and drop the identity matrix. If we give the second matrix here a name: $i$, we can write:

$\pmatrix{a & -b \\ b & a} = a+bi$

This is called a "complex number". (The word "complex" just means it is made up of several parts, not that it is complicated.) It has two parts, the "real" part which is one of the numbers we already knew about, and an "imaginary" part (it's just a name - it's not any less real in the everyday sense of the word) that is a multiple of $i$. We can treat this like ordinary algebra, with the additional rule that $i^2=-1$.

So the complex numbers can be considered to be a special subset of $2\times 2$ matrices, representing the scale-rotations. We just use ordinary matrix addition and multiplication on them.

And if we consider how the matrix acts on the vector $(1,0)$, we can represent every complex number with a point in the plane.

There is absolutely nothing "imaginary" or unreal or mysterious about them. Rotations and scalings are as "real" as the translations we do when we step forwards or backwards along the number line. They're all pieces of the geometry of space - of the universe we live in. The numbers for representing space must include not just translations and scaling but also rotation (and reflections and shears and stretches and other things), all combined together into one system. It was purely a historical accident that we initially missed them, because we were looking at only a part of the geometric picture when we first invented "numbers".