Basic math teaches us that with a right triangle with legs of 3 meters and 4 meters and an unknown hypotenuse that we can plug it into the Pythagorean theorem. 3 squared plus 4 squared = C squared. C squared is 25 and the answer is plus or minus 5. We throw out the -5 and the answer is +5 meters.
Do we always have to throw out the -5? Does really advanced math allow for crazy geometric figures with negative lengths?
I’m picturing a negative right triangle as a regular triangle, just in the bottom left quadrant of the graph but keep in mind, IANAMathologist.
Not quite what you’re asking but …
The “space-time interval” is a measurement of a type of “distance” in space-time which has the useful property that it is invariant for any inertial observer, unlike, say, the ordinary Euclidean distance between two points which varies depending on the speed of the observer. The formula, in terms of the differences between each of the four coordinates is sqrt(Δx^2 + Δy^2 + Δz^2 - Δt^2). As you can see, it almost looks like the normal Euclidean distance formula in 4 dimensions, except the Δt component is subtracted rather than added. Some writers prefer to treat the time dimension as imaginary (multiplied by sqrt(-1)), so that when squared the result is negative, and then the formula uses a plus sign instead of minus, making it identical to the Euclidean distance formula.
You might have to come up with a creative definition for a what a negative length is. I can’t think of one that’s sensible.
In some mathematic concepts, negation indicates direction, treating the number as a vector. But “length” is a scalar. There’s no directional component to it.
You can’t have a triangle with negative side length any more than someone standing on their head would say they are negative six feet tall or you could have negative three dollars in your wallet, even though concepts like “six feet below sea level” and “in debt 3 dollars” are both reasonable things that can be represented with negative numbers in some contexts.
There’s a trick you can do to calculate the area of a triangle by adding the determinants of the matrices created from pairs of the coordinates of the three points. To do this, you need to treat the intermediate areas as positive if the second point is counterclockwise from the first point and negative if clockwise (or the other way around - I forget). Clockwise and counterclockwise relative to an arbitrarily chosen origin, btw.
Of course there are negative triangles, with the suggestions already mentioned, treating the length as being in the opposite direction. But what about imaginary values? (3i, 4i, 5i) is a valid triple but seems less interesting than something like (3i, 5, 4). Does that sort of right triangle exist?
Not totally sure I followed that. What are the intermediate areas? Presumably this process yields a positive number if you go around the points in one direction and a negative number if you go the other way, but the area is just the absolute value?
That said: I can see how area might end up having a positive or negative value depending on some right hand rule where the area implies a vector pointing in some direction. So maybe there’s a way that negative length is meaningful that I’m not thinking of.
It’s been a while since I tried this, and my description above is wrong in some details.
Consider a triangle with points (1,1), (4, 1), (1, 4). By inspection the area is 4.5.
By the (corrected) algorithm I describe the area is: 1/2 * det (1 4;1 1) + 1/2 det (4 1;1 4) + 1/2 det (1 1; 4 1) = 1/2(1-4) + 1/2 (16-1) + 1/2 (1-4) = -1.5 + 7.5 -1.5 = 4.5 - and notice that two of the intermediate areas in the calculation are negative.
Pythagoras? That’s advanced math. Let’s try an even simpler formula, like the area of a square: There are two very different squares, both of which have an area of 25 square meters. For one of them, each side of the square is 5 meters long. For the other, each side of the square is -5 meters long.
I don’t think that was very helpful, but maybe someone can improve on it. Good luck!
Ah, I see.
I’d have to delve into it but I expect what’s happening is that the large term is double-counting something and the negations are subtracting it to get back to the correct answer. This is negative area the same way that, say, you can find the area of the parts of a square with a circle inscribed in it that aren’t in the circle with 4r^2 - pi*r^2, or that if you sell three acres of land there’s a negative three acres term somewhere. But the circle doesn’t have a negative area.
You’re right. Each of the determinants is calculating the area of a triangle defined by the two points and the origin, and the negative determinants cancel out the extra areas if necessary to leave only the triangle you care about. There’s a 3-d equivalent, too.
The very unexciting answer is that ‘length’ is non-negative by definition.
Forgive me for being naïve here but I consider mathematics to model reality, not to be reality. Such that, like any other model, if it doesn’t match reality, the model is wrong…its not reality that is incorrect.
So I attribute the possibility of -5 as a reasonable answer to be a failure to correctly model reality. Reality drives the model, not the model drives reality.
This is probably more apt. For instance, there’s a lot of algebraic problems whose normal solution methods will give you solutions that don’t actually work, so-called “extraneous solutions”. The existence of these simply means that our algorithms for solving problems generate a possible list of solutions, and not the absolute list of solutions. Our model of the problem via the algorithm is close to perfect, but not quite if you don’t include checking to make sure that the supposed solutions actually work.
(This reminds me of when I was in math grad school taking a differential equations class, and one of the great things about it was that you could always check your solutions to differential equations. I ended up on one problem type being completely unable to see why the answer I was generating, by way of what I thought the algorithm was, was not working, and I ended up on medical leave before I figured it out.)
Here’s an article that also points out that the algorithm I describe will determine the area of any polygon no matter how many points https://www.baeldung.com/cs/2d-polygon-area
In mathematics the generalization of the notion of length is called the ‘norm’ and it is closely related to the genralzation of distance known as the ‘metric’. There are many further generalizations of the norm and metric and you might be able to find someone somewhere who has generalized it to include negative numbers. The basic problem though with allowing distance/length to be negative is that it doesn’t behave enough like distance/length to be interesting.
An interesting example has been mentioned above which is the Lorentzian norm, used in relativity. This is not actually a norm as strictly defined, but a generalization of a norm as it can be zero even when the vector is not a zero vector and the square norm can be negative. That’s not the same as allowing the norm to be negative though and it still has enough in common with the notion that it can be thought of as a type of length.
That said, I believe ‘negative norms’ are used in quantum physics as a calculational trick.
You can have negative area, at the very least. It happens all the time with integrals. However, that’s because your graph is on a plane that includes negative and positive values on its axes.
I’m sure you can come up with a reason to defeine a negative distance (or displacement) in a triangle. But I would not expect it to be defining a triangle in the normal, real world sense.
Supringly it is actually very difficult to find a good reason to define negative distances. A metric space is a set with a function, obeying certain axioms, that can be thought of defining a distance between memebrs of the set. The axioms of a metric space imply that the distance is non-negative. There are many extensions of the concept of a metric space, which alter the axioms, but nearly all use a distance function, which is by definition non-negative.
One fo the most obvious examples of an extension of the metric which doesn’t use a non-negative distance function is pseudo-Riemannian metrics used in relativitistic physics but even this doesn’t use negative distances, though the square of a “pseudo-Riemannian distance” can be negative.
The basic problem with negative distances isn’t that you can’t contrive a way to allow negative distances, it is more that finding interesting/useful concepts of negative distance isn’t easy.
You can define negative distances easily enough. Just say that the distance between two points on a number line is y - x , instead of |y - x| . And it really doesn’t break much if you do that. It just also isn’t particularly useful, either. And yes, one can probably, if one digs deep enough, find a few things that it does break, and a few things that it’s useful for. But usually it makes no difference.