I think (?) it makes perfect sense to consider Diophantine equations, like x² + y² = z² or x² + y² = z³ over number fields and not just rational numbers. So, sure, some of the integers may be imaginary.
On the other subject, people can and do talk about “pseudo-Euclidean spaces”; why not? 
There is definitely some sort of geometry there.
Incorrect.
http://mathonweb.com/help_ebook/html/functions_5.htm#sqrt
Definition: The square root function is defined to take any positive number y as input and return the positive number x which would have to be squared (i.e. multiplied by itself), to obtain y .
-5 squared is 25, but -5 is not a valid output of the expression √25. Functions which use the √ symbol exclude the negative root. If you want to get into negative roots, complex numbers, etc then use +/- √ or exponentiation to 1/2 to denote exactly what you mean.
dalej42 didn’t use the square root function and was not wrong to say that the solution to c^2 = 25 is c = ±5
The definition of the square root symbol is irrelevant to this question.
In fact, consider your Pseudo-Euclidean space; it is not a metric space any more. But the quadratic form allows you to define a “squared length” q(v) of vectors, except this might be negative or zero. Still, the analogue of the Pythagorean theorem holds, so that v and w are orthogonal if and only if q(v + w) = q(v) + q(w), essentially by definition.
As iamthewalrus_3 noted earlier in the thread, this gives you a one-dimensional vector, where the (negative) sign indicates the direction. This is called directed distance, and is fairly standard. But, by definition, the length (or norm) of any vector, one-dimensional or otherwise, is a nonnegative real number.
He used the Pythagorean theorem which is defined using a square root function specifically because physical objects cannot have negative lengths. The only way to create the problem in the OP is to misunderstand some very basic 7th grade math concepts.
The Pythagorean Theorem is that the identity a^2 + b^2 = c^2 holds for the sides of a right triangle. There’s no square root function there and Pythagoras wouldn’t have known what one was. Of course Pythagoras would have also said “WTF is a negative number?” and we could say that the fact that the theorem is about lengths make the negative root obviously inappropriate.
But what we can’t do is to pick a derived representation, the one using the root sign, and say that proves anything. Mathematically that is missing a ±, which has been left out due to the nature of the problem.
It is in fact an interesting question, and sometimes those have mathematical answers that those who know 7th grade math glosses over a lot can learn about and sometimes even discover have uses, even if that doesn’t apply in this one case.
I’d like to see one geometry source - whether it’s a middle school textbook or a paper in a cutting-edge journal published yesterday - that defines the Pythagorean theorem without reference to either the square root function (which cannot have a negative value) or a triangle (which cannot have negative side lengths). Whatever you think you have if you end up with a negative number, it’s not a square root function or a triangle. Hand-waving things to mean something other than what they mean and then observing that this creates problems isn’t as interesting as you think it is.
The point is that answering questions like “can a triangle have negative lengths?”, which is what the OP did, is inherently interesting, you can tell by how some very smart and math savvy people have replied to it earlier in the thread. Your attempt at dismissing it by referencing the square root function was irrelevant and misguided.
Diophantus is supposed to have said equations of the type 4x + 20 = 4 were absurd. Turns out he was wrong.
It doesn’t seem there’s much use for negative side lengths, but that is not because there’s a root sign in middle school math books.
But it’s only interesting if you redefine what a “triangle” is. The definition of a triangle involves positive side lengths and several other necessary features. If you use the actual meaning of the words instead of asking “what if triangle means fribajab in this thread only” then the answer is a very uninteresting “no.”
You can plot “complex” triangles in the argand plane (one in which one axis is the rteal part and the other is the imaginary part), but they would look exactly like completely “real” triangles, so I don’t think it’s very interesting or exciting.
As for triangles with two real sides and one imaginary side, if you’re looking for solutions to that using the Pytgagorean theorem (regardless of how it plots out), you should realize that the solutions are basically the same as those for a Pythagorean triangle Instead of x^2 + y^2 = z^2, you’ll have (ix)^2 + y^2 = z^2, or y^2 = x^2 + z^2. It’s just that now y acts like the hypoteneuse. The same thing happens with two imaginary legs and one real leg.
And that was the OP’s question. “Does really advanced math allow for crazy geometric figures with negative lengths?”
The answer to that still isn’t “no, because of the square root function” or “no, because that’s not how triangles are defined in elementary school”. It is, as was posted earlier, “Let’s see what ways we could define negative lengths and see what happens.”
As it is, nothing much interesting happens, but your approach would have taken the question “what if we make a triangle where the angles sum up to more than 180 degrees” and dead-ended it with “Ms. Crabtree told me in fifth grade you can’t do that” instead of the much more interesting “that’s non-Euclidian geometry! Let me tell you about that.”
But you can have a non-Euclidean triangle in spherical geometry etc. You’re confusing artificial limits (“let’s just worry about planes for the time being for pedagogical purposes”) with limits that exist throughout all levels of mathematical language (the definition of a square root and triangle) or even limits in physical reality.
Can you find me any source, anywhere, that includes the requirement of positive side lengths as part of the definition of a triangle?
Underlining mine.
I’m not sure whether you’re just writing imprecisely or I’m reading wrongly, but it sure sounds like you’re maintaining that in general the square root operation applied to any number for any reason never gives a negative result.
If I’m reading you accurately you seem to have really fallen for what you caution about:
Hand-waving things to mean something other than what they mean and then observing …
the spouting of nonsense.
If I’ve misunderstood you please try again.
That’s correct. The square root function √ is defined as returning the real positive square root of the expression under the function. The fact that negative, complex, and other kinds of roots exist does not change the definition of the square root function. If you need to know why this convention exists in all mathematical sources, recall the definition of a “function.”
A function must return a single output, yes. But there are functions for which the output is a single set, containing multiple elements.
The length of the side of a triangle on a Euclidean plane is the Euclidean distance between two vertices, which is always non-negative.
My take is that the former position represents a dogmatic assertion that the simplified understanding taught in high school is the truth, not a truth only true subject to certain other often unstated conditions.
I just spent 20 minutes hunting for older threads on the “lies to children” issue in pedagogy. I know we’ve had a couple good ones but I can’t find them.
You’re incorrect. The fact that in many applications it is very important to specify that the negative and complex roots will produce nonsensical/wrong answers is exactly why there is a difference between √ and exponentiation to the 1/2 or other notations. That the proof of the Pythagorean theorem requires positive lengths is an inherent property of the notions of length (and even of the actual universe if you use the fluid-in-squares demonstration of why the theorem is true in physical reality). That is why it is defined in reference to the √ function and not to all possible roots in arbitrary domains.
Precisely defining terms and trying to move from carefully articulated premises to conclusions in an impeccably logical way is the essence of mathematics - playing word games and going “what if zerb means blork because I say so” to achieve exceptions that you believe are “technically correct” when in fact a precise application of technicality makes them nonsensical isn’t anything.