3 ms·

If you allow negative side lengths, as you should, much of geometry is unified and simplified, such as this situation. Negative (and imaginary) measurements ma

by lupire 3y ago

If you allow negative side lengths, as you should, much of geometry is unified and simplified, such as this situation. Negative (and imaginary) measurements make math better-behaved. (See also: quantum mechanics.) Rectangular coordinates are suboptimal because they are arbitrary in a way that hides some of the symmetry of mathematics. Your vector argument looks like a lot of algebra noise "without a diagram" because it relies on... triangles... for intuitive justification.

tromp 3y ago
> Instead of adding two separate angles α and β, we’ll use θ and −θ. It's not clear that the geometric proof based on pictures with positive angles α and β also applies to a negative β. I think one should at least provide a separate picture of that case...
Tainnor 3y ago
> If you allow negative side lengths, as you should [...] I mean, the blog post was about a pedagogically better suited construction of trig functions. If we allow negative side lengths, it's not really going to be intuitive anymore. At this point, you might just as well use the power series definition (or the one based on the complex exp function) since it makes things easy to prove.
- lupire 3y ago
  Imaginary lengths may be unintuitive but negative is pretty simple. The idea of negative measurements (meaning "opposite direction along a line") as is well understood before children study geometry. Kindergarteners learn "left" vs "right" and "forward" vs "backward". I agree with other poster about the value of showing negative and positive lengths visually in the same diagram.
  - Tainnor 3y ago
    Magnitudes are intuitionally directionless (hence why norms and measures are nonnegative reals). Of course, you can extend notions suitably and make it work, but I really don't think that this is terribly intuitive. (Of course, different people will find different things intuitive.)
jvanderbot 3y ago
The rotating angle suggests immediately the domain and definition of Sin(x), Cos(x) using simple projections. And, huuuuge bonus, that's a right triangle. I just think lesson zero in the above article should have been these projections, and a simple "From this all angles can be defined as right triangles in your preferred coordinate system", and you're off to the races in any old direction. This is not how I was taught it, and I'm retrospectively upset.

by lupire 3y ago

tromp 3y ago

> Instead of adding two separate angles α and β, we’ll use θ and −θ. It's not clear that the geometric proof based on pictures with positive angles α and β also applies to a negative β. I think one should at least provide a separate picture of that case...

Tainnor 3y ago

> If you allow negative side lengths, as you should [...] I mean, the blog post was about a pedagogically better suited construction of trig functions. If we allow negative side lengths, it's not really going to be intuitive anymore. At this point, you might just as well use the power series definition (or the one based on the complex exp function) since it makes things easy to prove.

lupire 3y ago
Imaginary lengths may be unintuitive but negative is pretty simple. The idea of negative measurements (meaning "opposite direction along a line") as is well understood before children study geometry. Kindergarteners learn "left" vs "right" and "forward" vs "backward". I agree with other poster about the value of showing negative and positive lengths visually in the same diagram.
- Tainnor 3y ago
  Magnitudes are intuitionally directionless (hence why norms and measures are nonnegative reals). Of course, you can extend notions suitably and make it work, but I really don't think that this is terribly intuitive. (Of course, different people will find different things intuitive.)

jvanderbot 3y ago

The rotating angle suggests immediately the domain and definition of Sin(x), Cos(x) using simple projections. And, huuuuge bonus, that's a right triangle. I just think lesson zero in the above article should have been these projections, and a simple "From this all angles can be defined as right triangles in your preferred coordinate system", and you're off to the races in any old direction. This is not how I was taught it, and I'm retrospectively upset.