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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
by nick__m 5y ago
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 ?
- mkl 5y agoAlso π and the three most important operations: addition, multiplication, exponentiation.
- 323 5y ago> 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 https://en.wikipedia.org/wiki/Hyperoperation
- adunk 5y agoOne 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 https://tauday.com/tau-manifesto#sec-euler_s_identity
- janto 5y agoIndeed. It shows that the pi formulation is actually somewhat ugly because it lacks symmetry.
- Scarblac 5y agoThe 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/Euler%27s_constant https://en.wikipedia.org/wiki/E_(mathematical_constant) https://en.wikipedia.org/wiki/E_(mathematical_constant)
- poizan42 5y agoBut that is the Euler–Mascheroni constant which is normally denoted by gamma. There is a footnote on https://en.wikipedia.org/wiki/Letterlike_Symbols 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 https://math.stackexchange.com/a/3123704
- 0xdeadb00f 5y agoI 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.
- f00zz 5y agoThis 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 https://qiskit.org/textbook/ch-appendix/linear_algebra.html
- sidpatil 5y agoAnd 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 https://en.wikipedia.org/wiki/De_Moivre%27s_formula
- Rompect 5y agoAlso an amazing way are Taylor polynomials, this article explains the process of thought really well: https://betterexplained.com/articles/taylor-series/ https://betterexplained.com/articles/taylor-series/
- f00zz 5y agoYeah, 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!
- wrycoder 5y agoWhich only slightly obfuscates the fact that e^(iπ) = -1. Bamboozles the rubes!
- 323 5y agoe^(iπ) = -1 is basically the unit circle in the complex plane: https://en.wikipedia.org/wiki/Circle_group https://en.wikipedia.org/wiki/Circle_group