# Math and equations are fun.

August 10, 2014 9:06 AM Subscribe

The power of math: 17 Equations That Changed the World - a one table summary of the book by Ian Stewart FRS. Business Insider gives its interpretation of the importance of each equation. Brain pickings (2012) on this book and equations, and another extract from the book.

Previously on MetaFilter: the Physics World list of the top 20 equations of all time (needs registration), via this comment.

Previously on MetaFilter: the Physics World list of the top 20 equations of all time (needs registration), via this comment.

Sammyo: yes, the last/most recent equation was somewhat hmmmmm. From the Business Insider article:

Wondering if Business Insider featured this list because it gives the Black-Scholes equation some kind of Newton/Einstein credibility in their minds?

posted by Wordshore at 9:36 AM on August 10, 2014

*Another differential equation, Black-Scholes describes how finance experts and traders find prices for derivatives. Derivatives - financial products based on some underlying asset, like a stock - are a major part of the modern financial system.*Wondering if Business Insider featured this list because it gives the Black-Scholes equation some kind of Newton/Einstein credibility in their minds?

posted by Wordshore at 9:36 AM on August 10, 2014

There's possibly an argument that the equation at the heart of Google searching is a candidate for the list? Whatever that is; for historical Information Science reasons the Tanimoto Coefficient could be part of it.

posted by Wordshore at 9:41 AM on August 10, 2014 [2 favorites]

posted by Wordshore at 9:41 AM on August 10, 2014 [2 favorites]

For good or ill, Black-Scholes

posted by Etrigan at 9:47 AM on August 10, 2014 [3 favorites]

*has*changed the world.posted by Etrigan at 9:47 AM on August 10, 2014 [3 favorites]

*If we put an ice cube in a cup of hot coffee, we always see the ice cube melt, and never see the coffee freeze.*

For a moment there, I almost thought Business Insider was going to get through them all without making any obvious mistakes.

posted by sfenders at 9:52 AM on August 10, 2014 [4 favorites]

*Wondering if Business Insider featured this list because it gives the Black-Scholes equation some kind of Newton/Einstein credibility in their minds?*

The association will perhaps resonate with readers who believe economics is more of a science than a field of engineering.

posted by Mr. Six at 9:53 AM on August 10, 2014

I was going to mention the finance equation being the only one in the last forty years, looks like I've been beaten to the punch. I seem to recall that Black-Scholes was outdated even when I learned about it a decade ago, as well.

posted by Yowser at 10:09 AM on August 10, 2014

posted by Yowser at 10:09 AM on August 10, 2014

And I see Wikipedias first paragraph confirms it. Looks like its one more Finance equation that uses fudge factors. Truly finance is the dismal science, not economics!

posted by Yowser at 10:11 AM on August 10, 2014

posted by Yowser at 10:11 AM on August 10, 2014

i^2 = -1

Hm, this is less of an equation that changed the world than a

e^(i.pi) = -1

Would be the more profound, world changing equation relevant to the complex field and geometry. Still, quite a lot to cover in a chapter, I suppose.

posted by iotic at 10:14 AM on August 10, 2014 [12 favorites]

Hm, this is less of an equation that changed the world than a

*definition*. Surelye^(i.pi) = -1

Would be the more profound, world changing equation relevant to the complex field and geometry. Still, quite a lot to cover in a chapter, I suppose.

posted by iotic at 10:14 AM on August 10, 2014 [12 favorites]

BI also mentions Shannon's theory in the same paragraph as JPEGs. I'm not an EE but that just seems wrong somehow.

posted by Yowser at 10:19 AM on August 10, 2014

posted by Yowser at 10:19 AM on August 10, 2014

*Hm, this is less of an equation that changed the world than a definition*

Yes, this is true for about half of these "expressions with equal signs in them"...

posted by grog at 10:43 AM on August 10, 2014 [2 favorites]

As for the importance of math in economics and finance, boiling it down to a specific equation seems as fuzzy as those branches of math can seem to be. I don't have a specific equation but some element of von Neumann's Game Theory would seem to be a more significant/core element.

posted by sammyo at 10:44 AM on August 10, 2014

posted by sammyo at 10:44 AM on August 10, 2014

Isn't the heat equation a wee bit more important than the Black-Scholes equation? Yes, the Black-Scholes equation is simply the heat equation evolving backwards in time, but that makes it all the more strange to mention the Black-Scholes equation but not heat heat equation.

posted by jeffburdges at 11:21 AM on August 10, 2014

posted by jeffburdges at 11:21 AM on August 10, 2014

I would think that Fermat's Last Theorem (an + bn = cn) might have made the list just for all of the time that mathematicians spent trying to prove or disprove it.

posted by TDavis at 11:23 AM on August 10, 2014

posted by TDavis at 11:23 AM on August 10, 2014

In fact, I'd argue that our deeper understanding of the heat equation should be described as what changed finance because Kiyoshi Itō was not working on finance when he proved Itō lemma and developed Itō calculus, but obviously topics around the heat equation were forefront in his mind.

posted by jeffburdges at 11:28 AM on August 10, 2014 [3 favorites]

posted by jeffburdges at 11:28 AM on August 10, 2014 [3 favorites]

Including

posted by mubba at 11:43 AM on August 10, 2014 [7 favorites]

*E*=*mc*^{2}instead of*R*–_{μν}^{1}/_{2}*R g*= –8_{μν}*π**G T*is kind of like passing over_{μν}*Paul’s Boutique*in favor of “(You Gotta) Fight for Your Right (to Party!)”posted by mubba at 11:43 AM on August 10, 2014 [7 favorites]

That was a really poor description of special versus general relativity. I feel like you should at least try to communicate that one is an elaboration of the other.

posted by vibratory manner of working at 12:07 PM on August 10, 2014 [1 favorite]

posted by vibratory manner of working at 12:07 PM on August 10, 2014 [1 favorite]

@Yowser: Claude Shannon is basically a god for Electrical and Computer Engineers. I can't think of a single thing in modern electronics that doesn't build in some way on Shannon's theories.

So in that sense, yes, JGEP is an outcome of Shannon's work. A silly example, perhaps, but it is one. I might have just said "all modern communications" instead.

posted by sbutler at 12:14 PM on August 10, 2014 [3 favorites]

So in that sense, yes, JGEP is an outcome of Shannon's work. A silly example, perhaps, but it is one. I might have just said "all modern communications" instead.

posted by sbutler at 12:14 PM on August 10, 2014 [3 favorites]

I have known so many finance people who think Black-Scholes is sufficiently advanced as to be indistinguishable from magic.

posted by benzenedream at 1:10 PM on August 10, 2014 [3 favorites]

posted by benzenedream at 1:10 PM on August 10, 2014 [3 favorites]

> I'd argue that our deeper understanding of the heat equation should be described as what changed finance because Kiyoshi Itō ...

Considering that the Heat Equation was also what lead to the Fourier Transform, I'd say that it's Heat Equation for the win.

posted by benito.strauss at 1:27 PM on August 10, 2014 [7 favorites]

Considering that the Heat Equation was also what lead to the Fourier Transform, I'd say that it's Heat Equation for the win.

posted by benito.strauss at 1:27 PM on August 10, 2014 [7 favorites]

Yay #6, my favorite: in any spherical polyhedron the sum of the # of vertices and the number of faces is always 2 more than the number of edges.

(Can't remember how many edges the icosahedron's got? Well, being the

Proofs

posted by leahwrenn at 2:16 PM on August 10, 2014 [3 favorites]

(Can't remember how many edges the icosahedron's got? Well, being the

**icosa**hedron, it's got 20 faces. And it's dual to the**dodeca**hedron, which has 12 faces, so the icosahedron must have 12 vertices. Therefore, the icosahedron must have 20+12-2=30 edges. Ain't math fun?)Proofs

posted by leahwrenn at 2:16 PM on August 10, 2014 [3 favorites]

Honestly, Black-Scholes is overrated. It's a formula for people who were using bad formulas and seeking better ones. One day it will be replaced by a better model. It's remarkable mainly in that it was purloined from missile guidance to a completely different field. If you want a formula that really changed the world, in a way people weren't expecting or hoping for. How about tracing the modern accounting equation back to the 1600s. Most businesses can operate without pricing options or futures, but few modern businesses can operate without accountants!

There's a similar book I read last year, Nine Algorithms that changed the Future, which I found to be a good read for layman.

posted by pwnguin at 2:18 PM on August 10, 2014 [3 favorites]

There's a similar book I read last year, Nine Algorithms that changed the Future, which I found to be a good read for layman.

posted by pwnguin at 2:18 PM on August 10, 2014 [3 favorites]

leahwrenn, I see you subscribe to the same school of math learning as I do: Learn just enough facts so that you can use them, along with general principles, to deduce it all.

Like if you find yourself stranded on an island and the locals threaten to kill you unless you can tell them all the angle sum trig formulas. Euler's Formula + complex multiplication ==> Saved!

posted by benito.strauss at 3:44 PM on August 10, 2014 [1 favorite]

Like if you find yourself stranded on an island and the locals threaten to kill you unless you can tell them all the angle sum trig formulas. Euler's Formula + complex multiplication ==> Saved!

posted by benito.strauss at 3:44 PM on August 10, 2014 [1 favorite]

benito.strauss - the animation of the square wave and it's fourier components in that wikipedia article is awesome. thanks for that link.

posted by joeblough at 3:58 PM on August 10, 2014 [1 favorite]

posted by joeblough at 3:58 PM on August 10, 2014 [1 favorite]

I would like to renew my objection to Bayes' Theroem not being on the list.

P(A|B)=P(B|A) * P(A) / P(B) given P(B) != 0.

posted by jeffamaphone at 4:27 PM on August 10, 2014 [3 favorites]

P(A|B)=P(B|A) * P(A) / P(B) given P(B) != 0.

posted by jeffamaphone at 4:27 PM on August 10, 2014 [3 favorites]

Just in case anyone else did a double-take on the comment regarding Fermat's Last Theorem, it's actually a little more complicated than "an + bn = cn" and that equation makes no sense without each "n" being an exponent and a little more background:

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation aposted by aydeejones at 5:04 PM on August 10, 2014 [1 favorite]^{n}+ b^{n}= c^{n}for any integer value of n greater than two.

And this kind of reflects the problem with the underlying "equations that changed the world" concept because the equations are often just expressive pieces ofpuzzle-bits underlying a greater puzzle, or definitions that lie at the precipice of a much larger chasm to explore. i

posted by aydeejones at 5:10 PM on August 10, 2014

^{2}= -1 is one good example; it's like they didn't want put Euler on the list a second time but wanted to recognize the awesomeness intrinsic to the complex number plane -- Fractals! Fundamental relationships between mathematical constructs including the intuitive (0, 1), simple (-1), and mind-bending (ln -1, i, etc), constructs wrapped in impossible-looking equations! A bunch of other shit I can't speak to!posted by aydeejones at 5:10 PM on August 10, 2014

Also the more I see "Business Insider" coming up as a source for random things...well, I realize what it is and don't want to get Meta, but at first I thought they came up with the list simply because of the emphasis on the financial equation.

To finish a trailing thought, it almost starts to give me a "Brawndo Corporation" impression; in the movie Idiocracy a beverage company inexplicably dominates the Capital-Fascist landscape rather than Wal-Mart or a Financial Institution. I could see Business Insider getting increasingly cited on Wikipedia (coincidentally I just saw WikiHow cited on a Wikipedia article that had factual errors and had a "WTF" down the rabbit hole moment) and taking over all Google Search results and eventually turning into the IGNORANCE MUTILATOR

Then everyone goes to Business Insider CBT v6.0 for K-12 and higher education, with Tiny Twerps Business Noob education starting at 18mo

posted by aydeejones at 5:18 PM on August 10, 2014

To finish a trailing thought, it almost starts to give me a "Brawndo Corporation" impression; in the movie Idiocracy a beverage company inexplicably dominates the Capital-Fascist landscape rather than Wal-Mart or a Financial Institution. I could see Business Insider getting increasingly cited on Wikipedia (coincidentally I just saw WikiHow cited on a Wikipedia article that had factual errors and had a "WTF" down the rabbit hole moment) and taking over all Google Search results and eventually turning into the IGNORANCE MUTILATOR

Then everyone goes to Business Insider CBT v6.0 for K-12 and higher education, with Tiny Twerps Business Noob education starting at 18mo

posted by aydeejones at 5:18 PM on August 10, 2014

The thing about i

posted by Elementary Penguin at 5:25 PM on August 10, 2014 [3 favorites]

^{2}= -1 is that the very idea that you could have a number with that property was controversial. Even before you figure out e^{ix}= cos x + i * sin x, the introduction of imaginary numbers makes solving polynomial equations much easier. In fact, mathematicians only really accepted the idea of imaginary numbers because it makes it possible to find a general solution to a cubic equation, even in the case where the equation has three real solutions.posted by Elementary Penguin at 5:25 PM on August 10, 2014 [3 favorites]

Someone has posted an "Algorithms that changed the world" AskMe, if there's any computery people here.

posted by Wordshore at 6:21 PM on August 10, 2014

posted by Wordshore at 6:21 PM on August 10, 2014

Wow! I never knew that Fourier Transforms owed so much to the Heat equation, thanks benito.strauss. Yeah he fucked up by omitting the Heat equation.

Also, Bayes' theorem "is to the theory of probability what Pythagoras's theorem is to geometry" according to Sir Harold Jeffreys, actually that's probably an understatement, certainly supersedes the normal distribution.

I'm happy he omitted Fermat's Last Theorem because :

(a) It's an inequation not an equation, and he allows a system of equations elsewhere, so why not go to first order formula.

(b) If you're discussing equations then discuss their actual consequences, not the consequences of possibly unrelated efforts. E=mc

(c) I'm doubtful about Fermat's Last Theorem's real influence upon mathematics myself. There were not soo many people who knew enough about it's potential to be influenced before it was proven.

Did FLT influence mid 20th century algebraic geometry? Ain't so clear : I'd assume that Serre ported sheaf theory to algebraic geometry simply because it was beautiful. And then Alexander Grothendieck generalized the living fuck out of it because it was beautiful if you were as crazy as him. And everybody else went along with them because it was fun.

posted by jeffburdges at 6:31 PM on August 10, 2014 [1 favorite]

Also, Bayes' theorem "is to the theory of probability what Pythagoras's theorem is to geometry" according to Sir Harold Jeffreys, actually that's probably an understatement, certainly supersedes the normal distribution.

I'm happy he omitted Fermat's Last Theorem because :

(a) It's an inequation not an equation, and he allows a system of equations elsewhere, so why not go to first order formula.

(b) If you're discussing equations then discuss their actual consequences, not the consequences of possibly unrelated efforts. E=mc

^{2}(c) I'm doubtful about Fermat's Last Theorem's real influence upon mathematics myself. There were not soo many people who knew enough about it's potential to be influenced before it was proven.

Did FLT influence mid 20th century algebraic geometry? Ain't so clear : I'd assume that Serre ported sheaf theory to algebraic geometry simply because it was beautiful. And then Alexander Grothendieck generalized the living fuck out of it because it was beautiful if you were as crazy as him. And everybody else went along with them because it was fun.

posted by jeffburdges at 6:31 PM on August 10, 2014 [1 favorite]

It kind of bugs me that, say, Noether's first theorem doesn't make the list, while Black–Scholes does.

I can think of other, similar examples. I've gotten really tired of looking at lists like this and feeling like women have been erased from history.

posted by kyrademon at 5:10 AM on August 11, 2014 [4 favorites]

I can think of other, similar examples. I've gotten really tired of looking at lists like this and feeling like women have been erased from history.

posted by kyrademon at 5:10 AM on August 11, 2014 [4 favorites]

*I've gotten really tired of looking at lists like this and feeling like women have been erased from history.*

Good point. There's an FPP from 2012 on Amalie Emmy Noether, and the Wikipedia page is detailed. From the NYT article in the FPP:

"Some consider Noether’s theorem, as it is now called, as important as Einstein’s theory of relativity; it undergirds much of today’s vanguard research in physics, including the hunt for the almighty Higgs boson."

posted by Wordshore at 7:33 AM on August 11, 2014

There was a time in my life when i could read most of those and 'get it'

You guys speaking about Math in an intelligent way really makes me miss those days.

posted by DigDoug at 9:48 AM on August 11, 2014

You guys speaking about Math in an intelligent way really makes me miss those days.

`*[ALT-TAB]* back to Outlook, for my next stupid meeting.`

posted by DigDoug at 9:48 AM on August 11, 2014

I think I'd actually put the normal distribution ahead of Bayes' Theorem, because it's hard to overstate its importance to statistics (and so science in general) in the era before computers. Nowadays people like physicists working on the LHC can produce confidence intervals using Markov chain Monte Carlo methods and incredibly complicated models, but back in Gauss's day and for a long time afterward, weighted linear regression with normally distributed errors was the state of the art tool for quantifying experimental uncertainty.

posted by mubba at 7:11 PM on August 11, 2014 [1 favorite]

posted by mubba at 7:11 PM on August 11, 2014 [1 favorite]

*"Mathematicians don't really care about 'the answer' to any particular question; even the most sought-after theorems, like Fermat's Last Theorem, are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them."*

posted by jeffburdges at 10:20 PM on August 12, 2014

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posted by sammyo at 9:27 AM on August 10, 2014 [1 favorite]