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I'm not sure how the concept of $i$ is taught in other places, but in our district the curriculum defines $i = \sqrt{-1}$, which is how it has been traditionally taught (for a while now) and also how our state standards align the concept of imaginary numbers. This definition is, in my opinion and the opinion of Math StackExchange (here and here), a poor definition for many reasons.

This definition excludes the completely legitimate possibility of $-i = \sqrt{-1}$, it causes difficulty when combining radicals through multiplication, and it is inconsistent with the definition of square root for positive numbers, such as $\sqrt{4} = \pm 2$. It should instead be taught that $i^2 = -1$, which is a much better definition because it avoids all of these problems.

However this does not align with any of the curricula or standards that I am expected to teach to. Any suggestions on how to get around this unfortunate situation?

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    $\begingroup$ I agree with everything except $\sqrt{4} = \pm 2$, because by the definition of square root $\sqrt{4} = 2$. $\endgroup$ – Mark Fantini Jan 9 '15 at 14:35
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    $\begingroup$ My guess would be that when introducing a new symbol, it is probably easier for students to grasp one that is just to the first power. I'm sure there are other reasons though. As far as teaching around this, that would depend on how much you think it would change the curriculum. Do you have a proposed teaching plan that involves starting from $i^2 = -1$ ? If so, how much (if at all) does it overlap with $ i = \sqrt{-1}$? Can you combine them somehow (ie "some books/teachers define $i=\sqrt{-1}$, but I prefer $i^2 = -1$ and here's the difference." $\endgroup$ – PawnInGameOfLife Jan 9 '15 at 14:37
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    $\begingroup$ How is "$i^2=-1$" necessarily better? It leaves ambiguity by a sign, which is merely a transmuted version of the original ambiguity. Note that the natural-language version, "Let $i$ be a square root of $-1$... so that $-i$ is the other square root of $-1$", skirts "rules" about "the square root function", which are artifactual, in any case. $\endgroup$ – paul garrett Jan 9 '15 at 16:57
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    $\begingroup$ @paulgarrett starting with $i =\sqrt{-1}$ seems to imply that you have a square root function (defined on what domain?) prior to having $i$. I think it is more honest to say "We invent a new number $i$ which obeys all the usual rules of arithmetic, but for which $i^2=-1$". This is essentially the construction $\mathbb{C} = \mathbb{R}[x]/(x^2+1)$. Perhaps more motivated is to define complex numbers as rotations and dilations of the plane, but this misses their algebraic properties initially. $\endgroup$ – Steven Gubkin Jan 9 '15 at 17:01
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    $\begingroup$ @StevenGubkin, well, ok, if everyone's already committed to believing there's a square root function, ... but if that equality can be read as "let $i$ be a square root of $-1$", then there's not the same issue. I am as fond as anyone of "the square root function", but am less fond than many of attempts to legislate-away the natural issues that arise. To my mind, this "problem" is akin to the "problem" of assigning values to inverse trig functions, which textbooks seem to want to do. $\endgroup$ – paul garrett Jan 9 '15 at 17:23
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A bit of personal opinion: Any time I find myself debating over a bit of pedantry while preparing course material, I ask myself, "Is this something that is really important to a student taking this course, or is it just interesting to me because I'm a mathematician?"

I personally would put the question about how to define $i$ in the "only important to me" category, at least if I was teaching a high school algebra or calculus (or similar level course). And, as a result, I would teach $i = \sqrt{-1}$. So I guess my answer to the question in the post is: I don't think the situation is unfortunate!

I don't think this causes any difficulty, at least not any more than any other new concept. Yes, they will have to learn that $\sqrt{(-1)(-1)} \neq \sqrt{-1}\sqrt{-1}$, but I think they could be convinced pretty easily that this is no big deal! After all, this thing $\sqrt{-1}$ is new and weird. It's not one of the usual numbers we know and love, so why should we expect it to behave like one!

It sounds also like any other student taking this course in your district will likely be taught that $i = \sqrt{-1}$, so I would stick with this for the benefit of your students being able to communicate with other students.

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    $\begingroup$ Indeed, as you say, it's just not really important to or for the students in such a course. The distinction would be lost on them, and time taken will be wasted, and any ranting about such things is usually perceived as further evidence for the random fussiness of mathematics-as-rule-system. :) $\endgroup$ – paul garrett Jan 9 '15 at 19:55
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There are issues with how to define $i$ no matter which you choose, as the answers to the linked questions attest.

To begin with, it is not actually true that $\sqrt 4 = \pm 2$. The square root function is defined (by mutual consent of the maths community) to only produce non-negative outputs, so $\sqrt 4 = 2$.

There are two reasons for this. Firstly, a function can't produce two outputs simultaneously or it wouldn't be a function. Secondly, if it did produce two outputs then what on earth would be meant by the notation $\sqrt 2$? If $\sqrt 2$ was two numbers then so would $-\sqrt 2$ and indeed they would have to be the same two numbers so $\sqrt 2 = -\sqrt 2$ which would imply that $\sqrt 2 = 0$.

So $\sqrt 4$ is in fact just 2. If you want to point out that there are two numbers that square to give 4 then you have to say something like "The solutions to $x^2 = 4$ are $x = \pm 2$".

So in that sense, saying that $\sqrt{-1}=i$ does not at all preclude us from saying that the solutions to $x^2=-1$ are $x=\pm i$.

What it does do is implicitly assume that there is a mathematically consistent way to choose which of the two possible solutions is the one meant when you do the square root of a nonpositive number. For example, if you can unambiguously say $\sqrt{-1} =i$, then there ought to be an unambiguous way to define which of the two possible solutions to $x^2 = 1-i$ is the single one meant by $\sqrt{1-i}$. There is a way, but it is not obvious until you talk about polar form, which comes much later!

Defining $i$ using $i^2 = -1$ is problematic too, because if $i$ was a variable then that equation has two solutions, so which of them is $i$? In fact either of them could be $i$ and there is no problem whichever one you choose, as long as you choose one and carry through with it for the rest of the maths you do. If each individual is consistent and doesn't switch between them willy-nilly it will be fine, and it won't matter if your $i$ is different to mine because switching all the $i$'s for $-i$ produces the same set of complex numbers and doesn't interfere with the operations on them. This is the idea of conjugation!

(The resolution of all of this is the representation of complex numbers with points on a 2D plane. Mathematicians agree to choose $i$ to be the point anticlockwise from 1 rather than clockwise.)

I think it would be remiss of you as a teacher not to mention some of this, but on the other hand it would be a bad idea to go into full detail straight up because that would confuse the students. Also, if locally the definition is $i = \sqrt{-1}$ then they need to know this to get the marks in their exam locally.

I would choose the approach of using the definition you have, but pointing out that there are some technical issues you're glossing over by using this definition. I would also tell them that it's possible to define it as a number such that $i^2=-1$, which also has some technical issues. You can possibly get back to these issues later if they are interested.

Of course, if a task they are doing doesn't require them to explicitly mention the definition, then there is absolutely nothing wrong with them using the fact that $i^2 =-1$, which I find is easier than using square roots anyway. Indeed, many students I have taught don't realise it's ok to do this and end up having extra lines in their working and make more errors.

One final thought: whenever there are multiple ways that people define things, it's probably a good idea to let your students know this, and which definition is the preferred one locally. (Another common example is the definition of the number $e$.)

This tells students that it is possible for there to be differences in definition. It also means that they are less likely to be confused by resources they find online which use a different definition.

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    $\begingroup$ The business about making square roots of positive reals positive real is just a convention, not a fact-of-nature. There's no enforcement mechanism! :) $\endgroup$ – paul garrett Jan 9 '15 at 16:58
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    $\begingroup$ I've added a parenthesis to point out that it's by agreement rather than nature. But still to be a function you have to choose one, and it would be extremely odd to suddenly define it to be the negative one when it's been the positive one in geometrical situations for so long. $\endgroup$ – DavidButlerUofA Jan 9 '15 at 20:35
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    $\begingroup$ I guess in "geometric" situations the uniqueness of square roots is manifest, since measurements are non-negative... but/and that's a somewhat different context. I do think it's reasonable to convey to students the point that context matters, in terms of our use of technical language. This as opposed to some "universal conventions" (as with the bad idea of everything being a global variable in compute programming...). $\endgroup$ – paul garrett Jan 9 '15 at 20:43
  • $\begingroup$ Thanks @BenjaminDickman. Fixed. $\endgroup$ – DavidButlerUofA Aug 17 '15 at 20:27
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Another way to get to the definition of $i$ is by starting with that the Complex numbers are an ordered pair of Real numbers $(a,b)$. Let $x=(a,b)$ and $y=(c,d)$. Then addition and multiplication is defined as \begin{align} x+y &= (a+c,b+d)\\ xy &= (ac-bd,ad+bc) \end{align} From the definitions above, the Complex numbers are Field which you can very with the students or just say so. Now we define $i=(0,1)$. $$ i^2=(0,1)(0,1)=(-1,0)=-1 $$ Now we can show $(a,b)=a+bi$ to achieve the more common definition of the Complex numbers. \begin{align} a+bi &= (a,0)+(0,b)(0,1)\\ &= (a,0) + (0,b)\\ &= (a,b) \end{align}

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  • $\begingroup$ This doesn’t address the question: how should $\sqrt{-1}$ be defined? $\endgroup$ – Incnis Mrsi Aug 17 '15 at 10:39
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    $\begingroup$ it gives you a concrete definition of $i$ hence a natural context in which to understand how $z^2=-1$ can have a solution. $\endgroup$ – James S. Cook Aug 17 '15 at 18:43
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When Ī̲ was taught about this stuff, my teachers explained that there are two definitions of the square root function:

  • arithmetic (that produces non-negative results and is undefined on negative numbers)  and
  • algebraic (that is defined on the whole ℂ and is 2-valued).

From this perspective, $\sqrt{-1}$ is arithmetically undefined and algebraically $\pm i$. One may attempt to inject something in-between, namely to extend definition of the arithmetic square root in some way to obtain exactly $\sqrt{-1} = i$, but this won’t clarify the question. All the truth about $z^2 + 1 = 0$ is that the equation has exactly two roots in the extended field, and, unlike square roots of 1, both $\pm\sqrt{-1}$ are equivalent (formally speaking, switching between them generates a field automorphism). So we have to “label” one of these indistinguishable twins with $i$ to be able to write extended field’s elements explicitly.

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Celeriko,

I'll directly respond to your exact question first then rant on about my real feelings after :)

You are exactly right that this alleged definition which your standards is using is to say the least... unappealing... for all the reasons you mentioned. If you teach your students $i^2=-1$ then you can automatically show them the case with the square root. It is highly unlikely that on some state test that they would ask for something like "Define $i$," but if they did... your students would be giving a correct definition or would know to choose $i=\sqrt{-1}$ after working with you on this. Teach them the more logical thing, then point out how to pass the ridiculous test.

On an even more important note, really... complex numbers are points. It makes more geometric sense and the actual use of these numbers. They aren't "imaginary," and their uses are necessary for the modern world.

To see a geometric, more logical approach that is completely within the realm of high schoolers, watch this wonderful mathematics professor's videos: Complex Numbers (Wildberger) and his lecture on the history of complex numbers and algebra (more advanced but good info to have!)

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    $\begingroup$ I will point out that while Wildberger is currently one of the most popular sources of "higher level math" youtube videos, but he has some pretty unconventional view (including a distrust of the real number system, and I believe, Cantor's diagonalization argument). $\endgroup$ – Steven Gubkin Jan 9 '15 at 16:57
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    $\begingroup$ @StevenGubkin Wildberger does have some unconventional views, but for the most part they do not affect the quality of his videos. I have previously watched the linked video and found it accurate and accessible. $\endgroup$ – Richard Jan 10 '15 at 13:20
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    $\begingroup$ @Richard I agree that some of his videos are okay, this one in particular. Since this forum is populated with educators who might recommend his videos to impressionable young students, I think it is worth conveying some of the problems with them. $\endgroup$ – Steven Gubkin Jan 10 '15 at 14:13
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    $\begingroup$ @StevenGubkin "impressionable young students"... Should videos based on constructionist mathematical philosophy be marked R16 Warning: contains Adult Themes... $\endgroup$ – Richard Jan 12 '15 at 10:06
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    $\begingroup$ @Richard constructive mathematics (of which I am a big fan!) accepts Cantor's diagonalization argument. In fact, it is a constructive proof. Most students have been trained to accept things on the authority of the teacher, so that an authority with very nonstandard beliefs could cause lasting damage (potentially). I would just not recommend Wildberger as your one stop "higher math fix". I think it is worthwhile recommending his videos with a grain of salt. They could also be a source of fruitful discussions "what parts of this are good, and what parts are not". $\endgroup$ – Steven Gubkin Jan 12 '15 at 13:17
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The choice of i versus -i is correlated with the geometric orientation of the the plane. One choice, i, correponds to counterclockwise orientation which matches the righthand rule for orientation.

Conjugation, which is reflection across the x-axis, reverses orientation and exchanges the two choices.

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