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Comment by nick__m | Hacker News Reader

And my favorite equation is ℇ^(ⅈπ)+1=0 !

It contains Euler, the imaginary unit, the unit, the zero and some hidden trigonometry.

P.S. does anyone know why the unicode symbol for the Euler constant render as a weird E when it is usually represented as a slightly italicized e ?

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One of the things I really like about the tau manifesto (the proposal to use tau == 2 * pi instead of pi in many situations) was their explanation of how tau made this equation all the more interesting (IMHO) by making it "almost like a tautology":

https://tauday.com/tau-manifesto#sec-euler_s_identity

Indeed. It shows that the pi formulation is actually somewhat ugly because it lacks symmetry.

The weird E is Euler's _constant_, and the slightly italicized e is Euler's _number_.

https://en.wikipedia.org/wiki/Euler%27s_constant https://en.wikipedia.org/wiki/E_(mathematical_constant)

But that is the Euler–Mascheroni constant which is normally denoted by gamma.

There is a footnote on https://en.wikipedia.org/wiki/Letterlike_Symbols that says:

> It's unknown which constant this is supposed to be. Xerox standard XCCS 353/046 just says 'Euler's'.

See also this discussion on math stackexchange: https://math.stackexchange.com/a/3123704

I think they're aware, but replying to when the parent asked "anyone know the symbol for Euler's constant" when they really neeeded the symbol for Euler's number.

This follows from e^(ix) = cos(x) + i sin(x)! I'm currently reading the Qiskit quantum computing textbook, and the appendix on linear algebra has a demonstration: https://qiskit.org/textbook/ch-appendix/linear_algebra.html

And that follows from De Moivre's formula, (cos(x)+ i sin(x))^n = cos(nx) + i sin(nx).

https://en.wikipedia.org/wiki/De_Moivre%27s_formula

Also an amazing way are Taylor polynomials, this article explains the process of thought really well:

https://betterexplained.com/articles/taylor-series/

Yeah, in the link I posted the formula is derived via Taylor (or Maclaurin) series, but the explanation in your link is great. Thanks for sharing!

Also π and the three most important operations: addition, multiplication, exponentiation.

> addition, multiplication, exponentiation.

Which are the hyperoperations of rank 1, 2 and 3:

> In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3).

https://en.wikipedia.org/wiki/Hyperoperation

Which only slightly obfuscates the fact that e^(iπ) = -1. Bamboozles the rubes!

e^(iπ) = -1 is basically the unit circle in the complex plane:

https://en.wikipedia.org/wiki/Circle_group

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