idempotent

This account is for doing math just for the sake of doing math.

[idempotent @op_idempotent]

A similar trick can be done along the imaginary component of a complex s to extend our calculation over all of C.

[idempotent @op_idempotent]

It can be computed to extremely incredible precision with just a few terms of the power series over a very wide range of inputs using the following trick:

[idempotent @op_idempotent]

A short thread on the reciprocal of the gamma function

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[idempotent @op_idempotent]

We can transform between the two domains using:

[idempotent @op_idempotent]

Have you ever played around with the hyperbolic numbers?

[idempotent @op_idempotent]

While we can add, subtract, multiply, and exponentiate just fine, we cannot generally divide. For instance,

(1+h)(1-h) = 1 - h^2 = 0

which implies the existence of zero divisors and hence, it is not a field, unlike the complex numbers.

[idempotent @op_idempotent]

Basic stuff, i.k., but IMO this is quite pretty:

[idempotent @op_idempotent]

Did you know you can derive most important trig identities using only complex numbers and basic algebra?

Here are two examples:

[idempotent @op_idempotent]

The algebra is quite cumbersome in the reals but drastically simplifies in the complex numbers.

Algebraically, it is structurally far simpler to treat each pair of identities at once in one equation over C rather than splitting it up into two equations over R.

[idempotent @op_idempotent]

Addendum: slightly improved wording:

[idempotent @op_idempotent]

Once in a while I like to review basic but truly fundamental ideas in mathematics.

Here's a very short thread on how integral path independence leads to Cauchy which leads to power series:

⬇️

[idempotent @op_idempotent]

Addendum: I think I significantly improved the exponential rule argument without adding anything unnecessary:

[idempotent @op_idempotent]

Cauchy's foundations for calculus using integral path-independence is more ontologically fundamental, but pedagogically it is also useful to approach this from the opposite direction using more elementary tools.

I compiled a short thread of some of the most important basics: ⬇️