Earlier this year, Mathematician Ian Stewart came out with an excellent and deeply researched book titled "In Pursuit of the Unknown: 17 Equations That Changed the World
" that takes a look at the most pivotal equations of all time, and puts them in a human, rather than technical context.
We asked Professor Stewart why he decided to do this book:
"Equations definitely CAN be dull, and they CAN seem complicated, but that’s because they are often presented in a dull and complicated way. I have an advantage over school math teachers: I’m not trying to show you how to do the sums yourself. You can appreciate the beauty and importance of equations without knowing how to solve them..... The intention is to locate them in their cultural and human context, and pull back the veil on their hidden effects on history. Equations are a vital part of our culture. The stories behind them --- the people who discovered/invented them and the periods in which they lived --- are fascinating."
This should be particularly relevant to anybody affected by the financial crisis.
Click here to see the 17 equations >
Black Scholes, a derivative pricing equation and number 17 on this list helped cause it.
From an email exchange with Professor Stewart:
"It’s actually a fairly simple equation, mathematically speaking. What caused trouble was the complexity of the system the mathematics was intended to model.... You don’t really need to be a rocket scientist to understand that lending hundreds of billions of dollars to people who have no prospect of ever paying it back is not a great idea...."
People took a theoretical equation too seriously, overreached its assumptions, used it to justify poor decisions, and built a trillion dollar house of cards on it. This made the crisis inevitable:
"I think that the crisis became inevitable once the financial instruments being traded in gigantic quantities became so complex that no one could understand either their value or the risks they entailed. When markets trade real goods for real money, excesses can only grow to the limits of what is actually out there. When they trade virtual goods (derivatives) for virtual money (leverage), there’s no real-world limit, so the markets can gallop off into Cloud Cuckoo Land."
You can buy the full book here.
Most of the commenters on this site would not read this at all, and even if they did, when they get to the Navier-Stokes equation they would think it had something to do with copulation, and money shots.
if you think the world revolves around BS finance
These are important equations all right - too bad you blew the first one.
The SQUARE of the hypotenuse of a RIGHT triangle is equal to the SUM of the squares of its legs.
lol
First, you way, way miss the point about FTC. The page shows the definition of the derivate (instead of the equation representing the FTC!), but calculus isn't just about the derivative. Is it that integration and differentiation are _inverse_ operations, like multiplication and division. This insight is what lead to the mathematical tool set known as calculus and helped give birth to physics as a science.
The discovery of logarithms and exponentiation were useful, but I don't think they had nearly as much effect as the author claims.
It isn't the equation i^2 = -1 that is as important as the analysis of complex numbers. A better representation of that field would have been the Cauchy integral theorem. If you wanted to go whole hog, the author could have picked the generalized Stokes' theorem to represent calculus in multiple dimensions (which is topology).
Navier-Stokes is a wee bit of a stretch. The study of fluid mechanics allows modern aircraft, civil, and chemical engineering, but I don't know how much it changed the world.
Each of Maxwell's Equations (though not all are due to him) should be treated seperately. Next to Newton's explanation of mechanics (which isn't even represented here!), they are the most astonoishing part of classical physics.
The laws of thermodynamics are important, but they are derived from the application of statistics to mechanics, so I think the entropy equation should not be here.
The population growth equation is important, but also probably not world changing. Same goes for Black-Scholes. F=ma got comlpetely snuffed here, as did the conservation laws. Einstein's field equations, representing general relativity, could have made an appearance here, but general relativity has had less of an impact on everyday life than special relativity.
And if you're talking about special relativity, it's important to note that Maxwell's equations are just relativity effects due to different frames of reference.
Finally, it's not special relativity that's important to things like GPS, it's general relativity that describes the frame dragging there.
As to GPS and relativity, well I suppose that in the sense that special relativity is a special case of general, general relativity by itself is enough. But the calculations as normally plugged in show two distinct effects pulling in opposite directions. If only one or the other were used, the system would have its required accuracy for no more than a few minutes.
Also note that while general relativity is necessary for GPS to work correctly, many more technologies exist for which special relativity is needed but for which general relativity is not, such as TV tubes, aircraft gyroscopes, and radiography.
The equation you give is valid only for a particle in the same reference frame (i.e. motionless) or if you allow the mass to be specified by some other means relative to your reference frame (ila's "relativistic mass" in the comment above), so it's a special case and really not very general or intuitive.
And special relativity isn't by itself enough to give GPS corrections. You need the frame drag to get everything right. Special relativity is simple by comparison to general relativity, which is why it came first (it's really just a special case).
This is bringing back memories of many a night of homework: prove the geometric laws F^(ab)_(,b) = 4*pi*J^a and F_(ab,g)+F_(bg,a)+F_(ga,b)=0 reduce to Maxwell's laws. Those I remember from the old MTW textbook even after all these years.
The wave equation is incorrect - the speed of the wave (v) should replace c. C is the speed of light *in vacuum*, if not in vacuum the speed is c/n (where n is the index of refraction).
The limits of integration on the Foirier transform are wrong, one of the limits must be *plus* infinity.
The Navir stokes equation needs to be corrected to denote *vector* quantities.
Maxwells equations in differential form are all kinds of messed up...Missing all kinds of mathematical terms.
E=mc^2 is incomplete. The more correct term is E^2=m^c^4+p^2m^2 (for an object not at rest).
Schrodinger's equation is missing a psi... the first term on the left should have a psi after the d/dt...
i give up...
From an email exchange with Professor Stewart:
"It’s actually a fairly simple equation, mathematically speaking. What caused trouble was the complexity of the system the mathematics was intended to model.... You don’t really need to be a rocket scientist to understand that lending hundreds of billions of dollars to people who have no prospect of ever paying it back is not a great idea...."
The professor is wrong. This sort of stuff will always happen given enough time. The problem is that the Black-Scholes equation assumes people don't copy each other. It assumes that people act independently, which does not happen in real life. This works most of the time, until something big happens making most people act in unison. Then everything just blows up.
The Black-Scholes equation is a time-bomb waiting to blow up.
It is the Normal distribution equation that will put a knife in the heart of modern finance. It should be banned from all economics and financial equations.
How about Jean D'Alembert and Daniel Bernoulli. Out of 4 names (2 first names and 2 last names) the author manages to get author to get 3 wrong!! Come on! check your facts!
If it compels someone to go out and buy the book, that's great. The world can always use another scientist or mathematician.