The only point I made is that there are better ways to present i than saying "let's invent a number i such that i*i = -1", and that the above presentation is pedagogically horrible.
The amount of confusion surrounding complex numbers should make it clear that there is a problem. We should try to alleviate this problem, instead of sweeping it under the rug . "Deal with it, this is how math is" is not a solution, especially when much better ways of presenting the subject do exist.
And so you would propose to say, "let's invent a number i so that multiplying by i is a rotation!"
A priori, why should anyone have a reason to think of numbers geometrically? The truth is they're both arbitrary. What I'm trying to do is not dictate facts but show the process of exploring an idea. That is correct pedagogy, and this way one can be surprised by the geometric perspective arising from what I think is a more natural question (why can't negative numbers have square roots?). Indeed, I don't think anyone has ever naturally asked "why can't some numbers be thought of as rotations?" (that is, without already knowing about complex numbers, and extrapolating to symmetry groups of various geometric shapes).
Never did I ever say "deal with it, this is how math is." I say, "What if this idea weren't so crazy, then this is what would have to follow. Look at the nice patterns that come up, we must be scratching at the surface of something really interesting!"
Or at least I try to say this. It seems like you didn't get that message.
No, I don't propose we say that. I don't propose we start with "let's invent... now let's 'explore' if this makes sense". This is bad pedagogy: It is confusing, it gives no motivation for what is being invented, and ultimately it produces bad results and demotivated students.
What I do propose is that we start with an interpretation of multiplication as a geometric transformation of of the real line by scaling. Then think about equations such as 1xx=-1, and think what solving them means in terms of our new interpretation: is there a geometric transformation x that we can apply to 1 such that doing it twice results in -1? Yes there is: rotation by pi/2. Now we call this transformation i, and work out the rest of the details.
There is no "let's invent", no "believe me, it works", and no leaps of faith required. A simple problem is presented, and its solution naturally leads to to the definition of i. There is no mystery, there is no need to somehow force ourselves to accept this new symbol into our number systems without knowing why we are doing it or what it is, and most importantly: there are far less confused students.
Many non-geometric presentations are also good, but it should never start with "let's invent". It should start with a problem that people can relate to (just as it started historically), and the definition should arise from our attempt to solve this problem. More generally, it should come out of something we understand, not dropped on our heads by the teacher who "knows" math. Geometry is often slightly better because our intuition works better there (this is why VCA is so successful), but it is not necessary.
Exploring ideas is good. Presenting some hocus pocus definition out of the blue and expecting young students to then "explore" it because somebody said it's interesting - this is bad, and it's definitely not how math is done.
You "invent" when you extend the interpretation of "geometric transformation by scaling" to "geometric transformation in general." You're saying "believe me it works" when you do that. It's just as arbitrary, but you just don't see it that way because somehow geometry on a line and geometry in the plane are more difficult to distinguish than square roots of positive and negative numbers.
Why can't people relate to the question of negative square roots? Students ask this question all the time, along with why we can't divide by zero and what happens when you do 0^0. Rather than address their questions, you're telling then, "Now you have to think about multiplication as a geometric scaling (why? just do it, it's 'natural'), and now we get to add rotations because... it's interesting, believe me!" There is more hocus pocus in that than you're willing to admit.
The problem here is: can we have square roots of negative numbers, and the natural definition arising from an attempt to answer it in the positive is i. It's way more obvious to just declare something to be the square root of a negative number than to try to rack your brain for clever geometric transformations. We do this in mathematics all the time: if we don't know what the roots of a polynomial are, we just denote them by a letter and see what we can deduce about them. If you don't know whether something exists, you suppose it does and either arrive at a contradiction or don't.
Plenty of math is invented without motivation. I see it happen all the time. In fact, people will often give a problem or definition motivation after the fact to try to sell it better! I also see this happen regularly, and it all starts with "let's invent." Your gripe seems to be that you do always want the teacher to have the answer, and that every interesting pattern must follow from a non-arbitrary motivation. This is just not how mathematics works.
I "invent" it in the same kind of way that I "invent" that a certain cloud has the shape of a rabbit. Your "invention", on the other hand, is like inventing a new animal just so that it will fit the cloud. But it's not only the invention that matters - it is where it starts and what are its foundations. Your "invention" starts at the end and its foundations are hidden. You keep drawing those general analogies between things as if they mean anything, but they are so general that the only thing they do is hide the difference, instead of exposing the similarities.
By the same kind of reasoning we can justify an extremely large amount of bad expository articles on mathematics. There are clear conceptual differences between making people "see" multiplication as scaling, then asking a question about the possibility of extending it, out of which i arises naturally and it's clear to everybody how it fits in and in what sense i^2=-1 despite the seeming contradiction with everything we know about numbers and multiplication up to that point, and between your way of inventing this "number" because "if you think about it long enough, there are ways to make it work without contradictions". People don't understand it just as they don't understand what is the square root of an apple: how can you even take the square root of a fruit without extending the definition of what it means to take a square root? We are not dealing with mathematicians who have experience in abstract algebra where multiplication has this abstract meaning and we can just add new stuff to our set and see if it works. We are talking about people who's experience with multiplication and numbers comes from calculating the acceleration of a ball and counting coins. It shouldn't come as a surprise that one approach to teaching something works better than the other: Tim Gowers presents normal subgroups and quotient groups in the much better way [1] than the traditional approach, even though the same very broad analogies can be drawn between the two. VCA presents certain ideas about complex analysis in a much more intuitive and clear way than the traditional approach, even though all it does is place a much higher emphasis on geometry.
Why don't you grab a bunch of students with little prior knowledge of complex numbers, present them the complex numbers using those two ways, then ask them what seems more obvious and natural.
I don't care about what seems natural to you. I care about what produces better results for the majority of people. Your way of explaining complex numbers produces bad results.
You can either accept this, or you can keep ignoring it because you know better. There is no shortage of smart mathematicians with the wrong ideas about how to teach mathematics, and they all have plenty of excuses.
The amount of confusion surrounding complex numbers should make it clear that there is a problem. We should try to alleviate this problem, instead of sweeping it under the rug . "Deal with it, this is how math is" is not a solution, especially when much better ways of presenting the subject do exist.