Why aren't triangles where the side lengths [math]a + b < c[/math] considered valid triangles, Veeky Forums?
I know why you can't physically draw them, like a lot of abstract things in math, but you can still get an area out of them with the meme imaginary numbers, for example a triangle with side lengths 1, 2, and 5 has an area of:
[math]2i\sqrt{6}[/math]
define triangle
You could just start to use rational trigonometry.
You can have a triangle with quadrances 1,2 and 5 there.
Things that would be considered dumb useless shit in the past are serious research now
Look back on infinite sets and set theory, category theory, automatic sequences and many more
In 200 years someone might take this idea seriously and develop a new area of research about retarded imaginary triangles, and the formula for imaginary area will bear his name
this is why i go to Veeky Forums
So the area of this triangle is [math]i[/math]? How does that make sense?
so without the triangle inequality, "triangles" can attain complex volumes or complex angles. that's a fine observation.
cool
now what can you use them for and how are they useful?
>muh usefulness
Found the brainlet engineer
Have you though about how something like that could be represented?
Standard linear algebra with a norm (or just a metric) obviously does not work.
Mathematical usefulness is something to consider too.
>0 dimension
>imaginary area
It all makes perfect sense.
also this one
Triangles have to have three angles.
Getting an imaginary area from feeding some unplottable side lengths into a formula is one thing, but unless you can get some definitive angle sizes, they're not triangles.
Thinking about it a bit more, it could be possible on the surface of a sphere (or indeed a cylinder) if the longest side goes the long way round.
I don't know enough about curved surface trigonometry to know if they'd still count as triangles, but I do know you can't just assume your flat surface formulae will work on curved surfaces.
>what are the analytic continuations of sine and cosine
While we're at it -- let's talk about vectors with negative magnitude.
For example, if you have a vector v that has angle θ and magnitude r = -1, then what do you get when you convert that vector to rectangular (x,y) coordinates? Would x and y be complex in that case?
this made me laugh so hard
but isn't the area 0
magnitude, by definition, is positive
there is no way to calculate it that allows for a positive number
you obviously can have r = -1
but the magnitude of r, in the case that r = -1, is still 1, a positive number
Would the angles of these "triangles" still add up to 180?
Vectors with negative magnitude are the same as vectors with a positive magnitude in the opposite direction.
The most commonly encountered example is tangential acceleration (change of speed). When something is decelerating, the tangential acceleration is negative. But tangential acceleration is a form of acceleration, which is a vector; its direction is whatever direction the object is moving in at the time. And negative magnitude in the tangential direction is exactly the same as a positive magnitude in the opposite direction.
There's no such thing as "negative magnitude" in vectors you fag.
Negative direction, but the magnitude is always positive.
That's not really the area of any triangle, you are just extending the definition of the area of the triangle outside its valid domain. It's called the "imaginary" unit for a reason.
>you are just extending the definition of the area of the triangle outside its valid domain
Sure, but the point is why isn't it a valid domain?
What's wrong with these imaginary triangles?
>mfw number theory once was considered a joke, yet created the RSA
>mfw in a few decades ITT (imaginary triangle theorem) will porvide a fast and easy way to find factorials.
If it's imaginary how can it have an area?
Even if the mathematical construct has use, it would still misleading to call it a "triangle".
It is not an area. There is no geometric analogy for complex numbers. It is not the job of math to be applicable or to have a ready analogy for an explanation that is any way related to the real world (that is not math but rather science and engineering). It does turn out that many properties of imaginary numbers are useful so we study them.
>It's called the "imaginary" unit for a reason.
no, it's not. it was used solely to make them look ridiculous, because people thought they were useless back then.
>it would still misleading to call it a "triangle".
so call it "generalized triangle".
So you're saying any set of 3 numbers is triangle?
at this point it is semantics. Define whatever you want with whatever word seems most fit. There is no concept of a physical triangle with such lengths that give an imaginary area, thus you could call it a triangle if it makes you feel warm and comfy but you can call it hotdog, or ice cream or England, chips, etc.. it really makes no difference.
These numbers based on previous assumed rules that behave in a certain way; whatever name you give it bears no importance at all.
Well the problem with that is that the word triangle does in actuality have a definite meaning, and to use that word with a different meaning you are not really speaking English. I don't speak ching-chong, so bye now.
where the fuck did I say that ?
Good question. Do you know the answer? I've not encountered the sines and cosines of complex numbers before.
Can anyone who does know about them calculate the angles of OP's abstract triangle (side lengths 1, 2 and 5)? Preferably by multiple methods, as if it's a triangle the results should concur.
There is such a thing a negative magnitude in vectors (see my example). But because it's the same as positive magnitude in the opposite direction, there's rarely any reason to use it.
However I don't think there's such a thing as negative direction in vectors. BICBW on that.
use those formulas and you can do it yourself if you don't know anything about complex analysis. too lazy to do it myself.
Why not so it like this instead of using sides with 0
it just doesnt work that way
I'd say any set of 3 numbers where [math]a + b \neq c[/math] because if [math]a + b = c[/math] the area of the "triangle" wouldn't be unique, it will always be the same, 0.