Ok that’s what I was getting at for pedagogical reasons. Would you say that when you’re teaching, presumably somewhere around grade 9?
ETA: say “real” solutions. I would with polynomials obviously, but generally wouldn’t with triangles.
The OP might find this article from BetterExplained.com and some of the other articles useful.
It’s easy for folks who know this stuff to get deep quickly. Grokking why this stuff works might be closer to what the OP is looking for.
I might say “real solutions” with high schoolers, but I expect most of them wouldn’t grok that I meant that in a technical sense. But I strongly dislike telling students things that are, ultimately, wrong, just because they’re simpler than the truth.
Come to think of it, there are also “impossible triangles” for SSS (for instance, 3,4,10 ), and those also yield complex solutions via the Law of Cosines.
And why does nobody ever talk about the Law of Tangents? For any triangle ABC, tan(A)*(b^2 + c^2 - a^2) = tan(B)*(a^2 + c^2 - b^2) = tan( C)*(a^2 + b^2 - c^2).
Just don’t do this:
Complex solutions or no complex solutions, I would also be scratching my head and wondering exactly where and how this triangle with sides 3, 4, 10 is supposed to be constructed.
Something seems backwards there. It seems like you would have to introduce a lot of abstraction and technical gymnastics in order to develop calculus/differential geometry without ever talking about concrete things like sets and functions. It is probably clearer to say, we first start with the notion of Cartesian spaces and smooth functions between them, then from that we can quickly define and work with arbitrary smooth manifolds, including the synthetic approach which enables the use of infinitesimals without the classical use of limit processes.
I probably wasn’t very clear, but effectively that’s what I meant. Trying to do Calculus without functions gets unwieldy fast, and you’d have to introduce different abstractions to make progress past a certain point. But functions are pretty easily understood by high schoolers, and encompass lots of the stuff we care about, like polynomials, trigonometry, and log/exp. You can introduce the (f(x+h)-f(x))/h definition of the derivative and go from there. It limits you only a little but simplifies a lot.
I’d suggest there’s some value in “playing around” as per my example above, but mostly just as a way of gaining some perspective on the process.
You don’t have to construct it using an unmarked ruler and a compass you know. Just grab your complex protractor and make the angles pi+6.8i, 7.9i and 8.4i. Easy peasy!
LOL. But where can I construct it? On one hand you somehow have distances between points, but on the other hand it obviously can’t be any metric space.
I think you could do it in a Riemannian spacetime. Which is, as you correctly note, only pseudo-metric, not metric.
So let’s ask @Riemann.