Approximating Conics with Polynomial Curves | Algebraic Calculus One | Wild Egg

We introduce two important families of polynumbers: the cosine and sine polynumbers associated to the unit circle with equation x^2+y^2=1, and the cosh and sinh polynumbers associated to the relativistic hyperbola with equation x^2-y^2=1. These work together to give us circular polynumber curves, which are approximations to the unit circle, and (relativistic) hyperbolic polynumber curves, which are approximations to the relativistic hyperbola.

Both families enjoy pleasant differential properties, and approximate the corresponding conics in a purely algebraic, or modular, way.

Video Contents:
0:00 Intro
0:13 Approximating Conics with Polynomial Curves
6:33 Approximating the unit circle
10:23 Properties
13:49 Approximating the relativistic hyperbola
15:19A (relativistic) hyperbolic polynumber curve
17:51 Question

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[JoelSjogren]

Nice conceptualization. One might add that c_{2n} = c_{2m} mod α^(2n+2) whenever n≤m.

[dnaviap]

Dear professor Wildberger, what is the relationship between those approximations and the rational parametrization? How can we deduce them from the algebraic framework?

[txikitofandango]

I don't ever recall seeing the sine and cosine polynumbers anywhere else in your videos... do you ever make a connection between the sine polynumber and spread?

[postbodzapism]

I think you may want to discuss factorials in the course first ?

[billh]

At 13:42, how do you know or show that gamma(m) approximates the unit circle?

[postbodzapism]

Can you approximate pi using integration on these PPC?

[elkartoubi]

you use s for the signed area and for the summation and for the sine polynumber its a little bit confusing