Understanding uniform motion: are radians really necessary? | WildTrig

[@culater]

Just came across your series on Trig. Love what I'm seeing. Thanks for this video treasure !

[@OldSloGuy]

While the length of a circular arc is a common problem with a good approximation available, the length of an ellipse is another matter. There are no simple formulas for this length either for the whole curve or a portion. The circle is an ellipse without flattening, so the shape is fixed and the formula is simple because only the size varies and that is scalable. The flattening of an ellipse changes the shape, so an infinite number of formulas would be needed. The lengths of non circular arcs can be calculated, but this is avoided as much as possible. Most engineers don't know how to do it, or they have forgotten how they once did it.

[@theodosiosgouzios2918]

YOU ARE THE BEST. !!!!!!!!

[@brendawilliams8062]

Thankyou

[@meimoscoso]

I think i found why these kind of videos to me seem like a bit of a conspiracy theory. These videos are made by someone who doesnt believe in the infinite in math. This is extremely weird, since most mathematicians accept infinity and study it extensively. For example set theory we have a whole system of infinite sets, and from these we can get and construct basically everything in mathematics, using only set theory. There are other theorys, like category theory, homotopy type theory which i havent studied at all. These theories are quite complex and require a lot of time. From set theory, using the most accepted axioms, including the axiom of infinity (one of the ones Wildberger oposes, i presume) we can get that: the natural number exist, and their properties, similar to the integers and the rationals. For the real numbers, we have two routes, using Dedekind cuts, or using cauchy sequences. Either way you get that the real numbers are unique (up to isomorphism). The reason why we do calculus over the reals, an extremely natural extension of the rationals, is because calculus breaks over the rationals, there are gaps that just break. For example if you want to find the root of the polynomial x²-2 over the rationals, there's none. Even though it makes sense there exists at least one given that if evaluate it at 1, you get -1, and if you evaluate it at 2 you get 2. It switched signs, thus for just common sense, there must be a number where the polynomial equals 0 (since polynomials are continuous). I would say that it is obvious the necesity of the real numbers. The reason why i talked about the real numbers because they have a really nice property, that every nonempty set that is bounded above, then it has a least upper bound. This is vital since it is the fact thar enables us to do any calculus, including taking limits of sequences, functional limits, derivatives, and integrals in an exact way, no approximation. There's a point that probably will get thrown around, that what about the infinite digits? My answer is simple, if we dont care about application and we dont want approximations because they are not formal enough (btw, anyone who has taken a course in a Numerical Analysis, will tell you that you just cant half ass it) then why would we bother with them? I mean, does it reallt matter that the 6th digit of pi after the decimal is 2 or 7, from a theoretical point of view? Or is it easier to think as a constant, and not even be uncombered by the digits themselves? The important part is the ideas themselves not the numbers those change with basis. I could just as easily say that the digits of pi are 1, since i am working in base pi. Now lets actually talk about lengths of curves. These is quite well studied, even me a lowly undergrad is working on a generalization of it for higher dimensions. The way we can talk about lenghts of curves is by approximating them, just like the applied mathematician, and get all the approximations in a set. If this set is unbounded, then the curve cannot have a length since any number we assign we can find a "better" aproximations that grows beyond that said number, and if it is bounded we define the length of the curve as the least upper bound of the set. How do we calculate it for general curves? We don't, because the numbers in 99.99% of the cases are not important, but the ideas are. If the curve is nice enough, then we can calculate it using the integral in the video. I just wanna emphasize that saying that the length of a curve is not defined it is simply either feigning ignorance of the years of research, that it has been poured in to analysis, or just not understanding the importance of infinity in mathematics. Just to defend the Maclaurin expansion of sin and cos, they are used soo often in other courses. Having this definition is a bit weird, but soo freeing to do weird stuff like the cosine of a matrix (i think it is soo cool we can do that, i have no idea if it is useful tho). It can help you approximate integrals, or even define new functions. The reason we use series is because no matter the level of precision for your result you will clear it eventually. If you think this applied sipping through your precious pure mathematics, i have news for you. No one besides the mathematicians even bothered to consider what did it mean to converge, a notion that is in all places of math. If you dont want series because of the infinite sum (which is a limit in the space of functions). No worries we can define them using differential equations, and get the exact same Macluarin series soo basically nothing. For what is worth i think that trig in general, is just a bag of formulas that are an extension to calculus. I think of calculus like the general form of geometry since we can get all of the results of geometry using calc 1 to 3. Either way i just wanted to explain in detail why the way he is explaing everything he said is really "non standard" and really throwing under a bus a branch of maths that is extremely powerful. All because calculus relies on infinte digits, that 99.99% of mathematicians, just walk around them.