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

     y f = f (y f)

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.

    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

    (λ (r) (λ (h) (h h)) (λ (g) (r (λ (x) (g g) x))

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).

    0 = d/dB [ (Y - XB)'(Y - XB) ]
      = d/dB [ Y′Y − Y′XB − B′X′Y + B′X′XB ]
      = d/dB [ Y′Y − 2Y′XB + B′X′XB ]            Y′XB and B′X′Y are just scalars :)
      = − 2Y′X + 2′X′XB

[1] https://isites.harvard.edu/fs/docs/icb.topic515975.files/OLS...

One of my favourite math lectures.

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

  λ = c / f

My son, when asking why knowing the speed of light matters, was impressed when I showed him we could make our own wifi antenna by using this to figure out the length it needed to be. Of course antenna design can be complex but this was still cool to show.

    λ = u / f

Oh so many reasons to love it and hate it

[1] http://www.espenhaug.com/black_scholes.html

    dV/dt + 1/2*sigma²*S²*d²V/dS² + r*S*dV/dS - r*V = 0

    e^(a * i) = cos(a) + i * sin(a)

[1] https://en.wikipedia.org/wiki/Phasor

    d^2/dx^2 sin(x) = -sin(x)

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))

For equation, probably, Maxwell's equations, because I use them in my research.

∇ ‧B = 0

I love the simplicity of the equation, and the implications that come from it.

(I only write it in places where graffiti has become acceptable, like some hike-in cabins. I also occasionally leave it in small writing in the corner of random whiteboards.)

It just really took me by surprise when I learned it, and it's useful too.

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.

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).

Using Einstein summation notation and covariant (raising and lowering operator) derivatives the curvature of spacetime can be compactly described.

  // since typing it out like this is preferable to reteaching myself LaTeX right now
  
  let favoriteEquation = arr => {
    let total = 0;
    for(let x of arr)
      total += Math.abs(x);
    let normalizedArr = [];
    for(let x of arr)
      normalizedArr.push(Math.abs(x/total));
    return normalizedArr;
  };

Just the fact that energy can be converted to mass and mass can be converted into energy is very interesting

We sure have, LHC creates matter/anti-matter particle pairs using enormous amounts of concentrated energy.

E² = m²c⁴ + p²c²

v^2 = GM(2/r - 1/a)

It's so simple, and at the same time so useful for orbital mechanics.

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...

Probably the most important equation in my field (ecology).

  a * b = exp(log a + log b)

  Equivalently: a + b = log(exp a * exp b)

  a and b = not(not a or not b)

  Equivalently: a or b = not(not a and not b)

However their is one crucial difference, in the Boolean case the duality is based on a symmetric operation, in the numerical case they are not.

  a = not b, b = not a

  a = log b, b = exp a

1) take a set of symbols, then define various adds and subtractions of those symbols you will get some result, which will be a sum of positive/negative versions of those symbols,

2) then replace all your additions and subtractions with multiplications and divisions of the same symbols,

...you will get an answer with exactly the same form, and with the same symbols, except with products/divisions instead of additions/subtractions. In isolation, addition and product behave exactly the same way!

We could call the addition/multiplier operator the combiner operator and the subtraction/divisor operator the remover operator, and forget about whether we were doing addition or multiplication and would still get the right answer either way.

It is only when you combine BOTH additions with multiplications in equations, that they operate different RELATIVE to each other, with the LOG/EXP relation "adjusting" values between their additive to multiplicative forms and back.

INSIGHT

So add and multiply are the same operations, but on values that are either the log or exp version of themselves respectively.

INSIGHT TAKES US TO AN ODD PLACE

Looked at that way, numbers live on an "Addition Ladder" where numbers start at the bottom rung where addition happens and to move up to multiplication we apply EXP before continuing to combine them with addition and then LOG them to get back to our home rung again.

Now that we have two rungs, why not operate at more levels up (EXP EXP) or below (LOG). The math is easy to do this, but for some reason nature doesn't seem to give us much use for "super-multiplication" or "sub-addition".

MY UNANSWERED QUESTION

Why doesn't nature use super-multiplication or sub-addition? Or does it somewhere and I am just not aware of it?

* If anyone has any insight on this, please pass it on! nevermark.mail at marks.house

Shannon entropy

Quoting from Amazon:

"The Babylonians invented it, the Greeks banned it, the Hindus worshiped it, and the Church used it to fend off heretics. Now it threatens the foundations of modern physics. For centuries the power of zero savored of the demonic; once harnessed, it became the most important tool in mathematics. For zero, infinity's twin, is not like other numbers. It is both nothing and everything.

In Zero, Science Journalist Charles Seife follows this innocent-looking number from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe, its rise and transcendence in the West, and its ever-present threat to modern physics. Here are the legendary thinkers—from Pythagoras to Newton to Heisenberg, from the Kabalists to today's astrophysicists—who have tried to understand it and whose clashes shook the foundations of philosophy, science, mathematics, and religion. Zero has pitted East against West and faith against reason, and its intransigence persists in the dark core of a black hole and the brilliant flash of the Big Bang. Today, zero lies at the heart of one of the biggest scientific controversies of all time: the quest for a theory of everything."

Chapter 0 intro from the book:

"Zero hit the USS Yorktown like a torpedo.

On September 21, 1997, while cruising off the coast of Virginia, the billion-dollar missile cruiser shuddered to a halt. Yorktown was dead in the water.

Warships are designed to withstand the strike of a torpedo or the blast of a mine. Though it was armored against weapons, nobody had thought to defend the Yorktown from zero. It was a grave mistake.

The Yorktown's computers had just received new software that was controlling the engines. Unfortunately, nobody had spotted the time bomb lurking in the code, a zero the engineers were supposed to remove while installing software. But for one reason or another, the zero was overlooked, and it stayed hidden in the code. Hidden, that is, until the software called it into memory --and choked."

This is a book about the history and issues surrounding the concept of nothing and it's mathematical representation, the number zero. It isn't a book about computers or programming. It's about zero from ancient times to physics.

Great book. Read it years ago. Going to read it again.

Here's the link:

https://www.amazon.com/Zero-Biography-Dangerous-Charles-Seif...