5 ms·
In my opinion, the truly beautiful concept involved here is Euler's formula: e^(ix) = cos x + i sin x. It unifies algebra, trigonometry, complex numbers, an
by susam 5y ago
In my opinion, the truly beautiful concept involved here is Euler's formula:
e^(ix) = cos x + i sin x.
It unifies algebra, trigonometry, complex numbers, and calculus. Euler's identity is only a special case of Euler's formula, i.e., Euler's formula with x = π gives us Euler's identity:
e^(iπ) = -1.
This is cute but Euler's formula is truly beautiful. In fact with x = τ = 2π, we get another cute result:
e^(iτ) = 1.
From Chapter 22 of The Feynman Lectures on Physics, Volume I:
"We summarize with this, the most remarkable formula in mathematics: e^(iθ) = cos θ + i sin θ. This is our jewel. We may relate the geometry to the algebra by representing complex numbers in a plane; the horizontal position of a point is x, the vertical position of a point is y. We represent every complex number, x + iy. Then if the radial distance to this point is called r and the angle is called θ, the algebraic law is that x + iy is written in the form re^(iθ), where the geometrical relationships between x, y, r, and θ are as shown. This, then, is the unification of algebra and geometry."
See the bottom of the page at https://www.feynmanlectures.caltech.edu/I_22.html https://www.feynmanlectures.caltech.edu/I_22.html for the above excerpt.
- SuchAnonMuchWow 5y agoe^(iτ) = 1, not 0
- susam 5y agoOops! I meant to write e^(iτ) - 1 = 0. I was going back and forth between the various formulations of the identities: e^(iπ) + 1 = 0, e^(iτ) - 1 = 0, e^(iπ) = -1, e^(iτ) = 1. The one involving pi looks more elegant when written in the first manner. The one involving tau looks more elegant when written in the fourth manner. I screwed up this one while going back and forth between these options. I have fixed it now. Thanks for noticing and commenting!
- BlueTemplar 5y agoI don't "get" the first two. e^(iτ/2) = -1 can be translated as "half a full rotation is equivalent to a symmetry" (And a full rotation to identity (for usual objects), but this is kind of obvious...)
- brandmeyer 5y agoFor those who are unfamiliar with this use of tau (a constant equal to 2*pi), see also https://tauday.com/tau-manifesto https://tauday.com/tau-manifesto
- rnhmjoj 5y agoAgreed, the full formula is much more intersting and deep: it shows there is a mapping between the reals and the circle group that preserves the structure (turns addition into multiplication). This concept leads to the very general exponential map of a Lie algebra to a Lie group and Pontryagin duality, which is the essence of Fourier transform.
- vishnugupta 5y ago> It unifies algebra, trigonometry, complex numbers, and calculus. I vividly remember my maths teacher impressing us upon what you stated above. By far the best math teacher I had. He didn't work out the proof on the board but made us work it out, took us on a journey, him as a guide, so to speak.
- cuspycode 5y agoIn a way it's analogous to how Einstein's E = mc^2 is a special case of how the norm of the four-momentum is defined in special relativity, which is mc = √((E/c)^2-p^2). For the special case of a stationary object we have p = 0, so E = mc^2 follows automatically. But the special case is somehow more memorable and more famous, and I believe something similar has happened with Euler's identity.
- User23 5y agoVisual Complex Analysis is the text that introduced me to this concept. The functions cos and sin suddenly look a lot like accessor functions for a two member object.
- deleted 5y ago[deleted]
- formerly_proven 5y agoThat's the essence of I/Q modulation (which is pretty much the backbone of all modern RF standards).
- lumost 5y agoIt's a unification of geometry and algebra in 2 dimensions, but generalizations beyond that are scarce. Effectively we're left with a solution where we can represent any space in dimensions modulo 2. While quaternions and some higher dimensional complex numbers exist, is there a unified formula expressing Euler's formula for arbitrary numbers of dimensions? is there one for an infinite dimension space?
- duckfang 5y ago> In my opinion, the truly beautiful concept involved here is Euler's formula: > e^(ix) = cos x + i sin x. And that right there, is the mathematical basis for a quadrature mixer... And all of software defined radios. https://hackaday.com/2017/05/16/if-the-i-and-q-of-software-defined-radio-are-your-nemesis-read-on/ https://hackaday.com/2017/05/16/if-the-i-and-q-of-software-d...
- ahmedalsudani 5y agoIt’s hard to think of anything more beautiful in all of mathematics.
- plasticchris 5y agoI've always thought ℵ₀ (more the series of proofs leading up to it) as just as beautiful and interesting.
- rsj_hn 5y agoAll of complex analysis is filled with magical results like this. For example, Cauchy's formula and its generalizations. Max/Min results, Loiville's theorem, Nevanlinna theory, Picard's Great and Little theorems. It's all absolutely bonkers. Just think: "Every complex function with an essential singularity achieves all but one value infinitely often in every neighborhood of the singularity" or "Every holomorphic function can be approximated arbitrarily well by a region of the Riemann Zeta function" There is this joke about how proofs start in different branches of math. So in real analysis you start with "Let ε > 0". And in Abstract Algebra, it's "Suppose not." But in Complex Analysis: "Nothing up my left sleeve, nothing up my right sleeve.."
- deleted 5y ago[deleted]
- dnautics 5y agocomparing the pi and tau formulas, though, the advantage of using the pi formula is you can pose the question (to someone who has never seen it) as such: Given the formula, e^(ix) = -1, what would you suppose x is? If you tried to use the tau formula, it falls flat: Given the formula, e^(ix) = 1, what would you suppose x is?
- zeroonetwothree 5y agoIt’s also just implied by squaring the pi formula even if that’s all you know so it’s somewhat less interesting in that respect.
- dnautics 5y agoi think you missed the point. In formula #2 there is a trivial solution 0.