Same here. I'm with Heisenberg, "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."
While not really a mathematical "equation", the Drake equation, simply for the debate it causes and the ramifications of parameter changes on what we think of as a race.
This equation was the first serious attempt to quantify the number of civilizations in our universe. As we explore Europa and others, we will get some serious data points to improve our understanding of the origin of life in the universe.
euler's identity e^ipi+1=0 of course. it connects all the dots (e, pi, sqrt(-1), 1, 0), is fun to prove, and just plain cool. i have no idea of its practical applications though.
In the form you wrote it, there are no real applications. But, written as $e^{i\theta} = \cos \theta + i \sin \theta$, it's frequently used in complex analysis and other areas.
A math professor of mine once said of this (paraphrased): yes it is cool, but really it is just a trivial consequence of how we defined our extension of the real exponential map to complex numbers. The really cool part, is that this extension is the _only_ possible holomorphic (differentiable for complex numbers) extension. All that to say that the really nifty part of this identity is the uniqueness of analytic continuations (on the whole complex plane).
Well, pi in there is a special case. The general case for any angle $\theta$ is $e^{\i\theta}=\cos(\theta)+\i\sin(\theta)$ It lets you transform between rectangular and polar representations of complex numbers. That allows for things like phasors [1] which are used quite a lot in electrical engineering. It's also used in Fourier analysis, IE all sorts of digital signal processing. Basically anywhere you get complex numbers, sines, or cosines, Euler's formula shows up. That's a lot of places in math/physics/engineering.
V - E + F = 2 - 2g, the defining equation for the genus/Euler characteristic of a surface. It has applications all throughout topology and graph theory.
Euler's equation is cool...
Written properly it contains addition, negation, multiplication, exponentiation, the imaginary unit, the additive identity, the multiplicative identity, the natural anti-derivative identity, and two irrational transcendental numbers in a most compact form with holomorphic continuity.
Your link is slightly misleading - it shows ways of approximating solutions to the equation. The equation itself is a partial differential equation [1].
The first time I saw it I was in high school and I was watching Andrew Ng's machine learning class for the first time.
My Math background was not the best and I didn't fully understand it until a couple of years later when I took MIT's free online course on linear algebra (OCW 18.06)
At some point in my past I became biased towards inequalities over equalities; they somehow seemed more fundamental (I have no intention to defend this sentiment... its probably insubstantial). So there are lots of fun inequalities, but probably my favorite is the bound on the support of eigenvalues of a matrix given by the Gershgorin circle theorem (https://en.wikipedia.org/wiki/Gershgorin_circle_theorem).
It looks like an equation, but it's really almost a Y combinator, almost in Haskell:
y f = f (y f)
Though this won't exactly type check, it's a pithy reminder of what I found so powerful about functional programming, which is that _any_ repetition in the code ought to be something you could factor out. The Y combinator shows you that _even the idea of a function calling itself_ can be factored out. It's hard to imagine you would really ever be forced to repeat yourself if this can be factored out.
Of course that isn't the real point of the Y combinator, but this thought, this "equation" completely changed the way I think about programming.
This works in Haskell just fine, thank you very much!
ghci> y f = f (y f)
ghci> factorial = y (\f x -> if x == 0 then 1 else x * f (x-1))
ghci> factorial 5
120
However, the Y combinator as defined in a non-recursive form is tougher to implement in Haskell [1][2]. It looks a lot better in Racket (and this is coming from someone who doesn't particularly like Racket).
Gauss's law for magnetism is so simultaneously simple and profound at the same time. It says that there is no such thing as magnetic charge (monopole).
Euler's formula. This one formula is the most practical formula I've ever encountered. Once you learn it, you will never stop using it. The Schrodinger equation is a lot of fun, but you will likely stop using it if you stop doing quantum mechanics. Maxwell's equations are beautifully simple and tend to appear in many other contexts other than classical electrodynamics. For example, in the absence of charge, Maxwell's equations describe free waves -- which can appear in many contexts other than electromagnetic fields.
Just to clarify, mass-energy equivalence implies nothing about mass-energy conversion. We've never witnessed such a conversion nor do we know if it's possible. The implication is that mass difference of an atom after energy absorption is directly proportional to the energy absorbed. Energy is still energy in that case, no conversion has taken place.
This means that sin(x) is an eigenfunction of the second derivative operator in the infinite dimensional vector space of functions.
This little fact makes numerically solving second-order differential equations using a sin(x) basis really cheap, just need to pay the upfront FFT cost... (O(n log n))
69 comments
[ 2.2 ms ] story [ 124 ms ] threadhttps://en.wikipedia.org/wiki/Mandelbrot_set
For equation, probably, Maxwell's equations, because I use them in my research.
N = R* • fp • ne • fl • fi • fc • L
https://en.wikipedia.org/wiki/Drake_equation
An excellent use of rough statistics to shatter long held beliefs. Even if life is incredibly rare, it most certainly exists somewhere else.
[1] https://en.wikipedia.org/wiki/Phasor
I immediately guessed Euler's identity and wrote "$e^{i\pi}+1=0$"
That was when we knew it was serious.
I got lucky (in more ways than one) because she'd nearly gone with her other choice, the continuum hypothesis
Oh so many reasons to love it and hate it
[1] http://www.espenhaug.com/black_scholes.html
used to find the line of best fit.
The first time I saw it I was in high school and I was watching Andrew Ng's machine learning class for the first time. My Math background was not the best and I didn't fully understand it until a couple of years later when I took MIT's free online course on linear algebra (OCW 18.06)
Proof: Since `B` is a minimum, the derivative of the expression minimized wrt `B` must be zero (the "derivative" is taken in the matrix sense here).
Solving for `B` you get the result.[1] https://isites.harvard.edu/fs/docs/icb.topic515975.files/OLS...
One of my favourite math lectures.
It comes up so often! Whenever someone asks what's new, the answer, of course, is "E over h." :-)
https://en.wikipedia.org/wiki/Planck%E2%80%93Einstein_relati...
Of course that isn't the real point of the Y combinator, but this thought, this "equation" completely changed the way I think about programming.
Gauss's law for magnetism is so simultaneously simple and profound at the same time. It says that there is no such thing as magnetic charge (monopole).
Just the fact that energy can be converted to mass and mass can be converted into energy is very interesting
E² = m²c⁴ + p²c²
We sure have, LHC creates matter/anti-matter particle pairs using enormous amounts of concentrated energy.
Just the fact that energy can be converted to mass and mass can be converted into energy is very interesting
Using Einstein summation notation and covariant (raising and lowering operator) derivatives the curvature of spacetime can be compactly described.
It just really took me by surprise when I learned it, and it's useful too.
This little fact makes numerically solving second-order differential equations using a sin(x) basis really cheap, just need to pay the upfront FFT cost... (O(n log n))