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Wednesday 19 June, 2013

# 17 equations that changed the course of humanity

## Some you may remember from school, but some are lesser known ones – like one perfected by an Irishman which allowed for development of modern passenger jets.

Image: Dusit via Shutterstock

MATHEMATICIAN IAN STEWART’s recent book In Pursuit of the Unknown: 17 Equations That Changed the World

takes a close look at some of the most important equations of all time.

A great example of the human impact of math is the financial crisis. Black Scholes, number 17 on this list, is a derivative pricing equation that played a role.

“It’s actually a fairly simple equation, mathematically speaking,” Professor Stewart told us. ”What caused trouble was the complexity of the system the mathematics was intended to model.”

Numbers have power. In this case, people depended on a theoretical equation too seriously and overreached its assumptions.

Without the equations on this list, we wouldn’t have GPS, computers, passenger jets, or countless inventions in between.

You can find the book here.

1. The Pythagorean Theorem

What does it mean: The square of the hypotenuse of a right triangle is equal to the SUM of the squares of its legs.

History: Attributed to Pythagoras, it isn’t certain that he first proved it. The first clear proof came from Euclid, and it is possible the concept was known 1000 years before Pythoragas by the Babylonians.

Importance: The equation is at the core of geometry, links it with algebra, and is the foundation of trigonometry. Without it, accurate surveying, mapmaking, and navigation would be impossible.

Modern use: Triangulation is used to this day to pinpoint relative location for GPS navigation.

2. The logarithm and its identities

What does it mean: You can multiply numbers by adding related numbers.

History: The initial concept was discovered by the Scottish Laird John Napier of Merchiston in an effort to make the multiplication of large numbers, then incredibly tedious and time consuming, easier and faster. It was later refined by Henry Briggs to make reference tables easier to calculate and more useful.

Importance: Logarithms were revolutionary, making calculation faster and more accurate for engineers and astronomers. That’s less important with the advent of computers, but they’re still an essential to scientists.

Modern use: Logarithms still inform our understanding of radioactive decay.

3. The fundamental theorem of calculus

What does it mean?: Allows the calculation of an instantaneous rate of change.

History: Calculus as we currently know it was described around the same in the late 17th century by Isaac Newton and Gottfried Leibniz. There was a lengthy debate over plagiarism and priority which may never be resolved. We use the leaps of logic and parts of the notation of both men today.

Importance: According to Stewart, “More than any other mathematical technique, it has created the modern world.” Calculus is essential in our understanding of how to measure solids, curves, and areas. It is the foundation of many natural laws, and the source of differential equations.

Modern use: Any mathematical problem where an optimal solution is required. Essential to medicine, economics, and computer science.

4. Newton’s universal law of gravitation

What does it mean?: Calculates the force of gravity between two objects.

History: Isaac Newton derived his laws with help from earlier work by Johannes Kepler. He also used, and possibly plagiarized the work of Robert Hooke.

Importance: Used techniques of calculus to describe how the world works. Even though it was later supplanted by Einstein’s theory of relativity, it is still essential for practical description of how objects interact with each other. We use it to this day to design orbits for satellites and probes.

Value: When we launch space missions, the equation is used to find optimal gravitational “tubes” or pathways so they can be as energy efficient as possible. Also makes satellite TV possible.

5. The origin of complex numbers

What does it mean?: The square of an imaginary number is negative.

History: Imaginary numbers were originally posited by famed gambler/mathematician Girolamo Cardano, then expanded by Rafael Bombelli and John Wallis. They still existed as a peculiar, but essential problem in math until William Hamilton described this definition.

Importance: According to Stewart “…. most modern technology, from electric lighting to digital cameras could not have been invented without them.” Imaginary numbers allow for complex analysis, which allows engineers to solve practical problems working in the plane.

Modern use: Used broadly in electrical engineering and complex mathematic theory.

6. Euler’s formula for polyhedra

What does it mean?: Describes a space’s shape or structure regardless of alignment.

History: The relationship was first described by Descartes, then refined, proved, and published by Leonhard Euler in 1750.

Importance: Fundamental  to the development of topography, which extends geometry to any continuous surface. An essential tool for engineers and biologists.

Modern use: Topology is used to understand the behavior and function of DNA.

7. The normal distribution

What does it mean?: Defines the standard normal distribution, a bell shaped curve in which the probability of observing a point is greatest near the average, and declines rapidly as one moves away.

History: The initial work was by Blaise Pascal, but the distribution came into its own with Bernoulli. The bell curve as we currently comes from Belgian mathematician Adolphe Quetelet.

Importance: The equation is the foundation of modern statistics. Science and social science would not exist in their current form without it.

Modern use: Used to determine whether drugs are sufficiently effective relative to negative side effects in clinical trials.

8. The wave equation

What does it mean?: A differential equation that describes the behavior of waves, originally the behavior of a vibrating violin string.

History: The mathematicians Daniel Bournoulli and Jean D’Alembert were the first to describe this relationship in the 18th century, albeit in slightly different ways.

Importance: The behavior of waves generalizes to the way sound works, how earthquakes happen, and the behavior of the ocean.

Modern use: Oil companies set off explosives, then read data from the ensuing sound waves to predict geological formations.

9. The Fourier transform

What does it mean?: Describes patterns in time as a function of frequency.

History: Joseph Fourier discovered the equation, which extended from his famous heat flow equation, and the previously described wave equation.

Importance: The equation allows for complex patterns to be broken up, cleaned up, and analysed. This is essential in many types of signal analysis.

Modern use: Used to compress information for the JPEG image format and discover the structure of molecules.

10. The Navier-Stokes equations

What does it mean?: The left side is the acceleration of a small amount of fluid, the right indicates the forces that act upon it.

History: Leonhard Euler made the first attempt at modelling fluid movement, French engineer Claude-Louis Navier and Irish mathematician George Stokes made the leap to the model still used today.

Importance: Once computers became powerful enough to solve this equation, it opened up a complex and very useful field of physics. It is particularly useful in making vehicles more aerodynamic.

Modern use: Among other things, allowed for the development of modern passenger jets.

11. Maxwell’s equations

What does it mean?: Maps out the relationship between electric and magnetic fields.

History: Michael Faraday did pioneering work on the connection between electricity and magnetism, James Clerk Maxwell translated it into equations, fundamentally altering physics.

Importance: Helped predict and aid the understanding of electromagnetic waves, helping to create many technologies we use today.

Modern use: Radar, television, and modern communications.

12. Second law of thermodynamics

What does it mean?: Energy and heat dissipate over time.

History: Sadi Carnot first posited that nature does not have reversible processes. Mathematician Ludwig Boltzmann extended the law, and William Thomson formally stated it.

Importance: Essential to our understanding of energy and the universe via the concept of entropy. It helps us realise the limits on extracting work from heat, and helped lead to a better steam engine.

Modern use: Helped prove that matter is made of atoms, which has been somewhat useful.

13. Einstein’s theory of relativity

What does it mean?: Energy equals mass times the speed of light squared.

History: The less known (among non-physicists) genesis of Einstein’s equation was an experiment by Albert Michelson and Edward Morley that proved light did not move in a Newtonian manner in comparison to changing frames of reference. Einstein followed up on this insight with his famous papers on special relativity (1905) and general relativity (1915).

Importance: Probably the most famous equation in history. Completely changed our view of matter and reality.

Modern use: Helped lead to nuclear weapons, and if GPS didn’t account for it, your directions would be off thousands of yards.

14. The Schrödinger equation

What does it mean?: Models matter as a wave, rather than a particle.

History: Louis-Victor de Broglie pinpointed the dual nature of matter in 1924. The equation you see was derived by Erwin Schrodinger in 1927, building off of the work of physicists like Werner Heisenberg.

Importance: Revolutionized the view of physics at small scales. The insight that particles at that level exist at a range of probable states was revolutionary.

Modern use: Essential to the use of the semiconductor and transistor, and thus, most modern computer technology.

15. Shannon’s information theory

What does it mean?: Estimates the amount of data in a piece of code by the probabilities of its component symbols.

History: Developed by Bell Labs engineer Claude Shannon in the years after World War 2.

Importance: According to Stewart, “It is the equation that ushered in the information age.” By stopping engineers from seeking codes that were too efficient, it established the boundaries that made everything from CDs to digital communication possible.

Modern use: Pretty much anything that involves error detection in coding. Anybody use the internet lately?

16. The logistic model for population growth

What does it mean?: Estimates the change in a population of creatures across generations with limited resources.

History: Robert May was the first to point out that this model of population growth could produce chaos in 1975. Important work by mathematicians Vladimir Arnold and Stephen Smale helped with the realization that chaos is a consequence of differential equations.

Importance: Helped in the development of chaos theory, which has completely changed our understanding of the way that natural systems work.

Modern use: Used to model earthquakes and forecast the weather.

17. The Black-Scholes model

What does it mean?: Prices a derivative based on the assumption that it is riskless and that there is no arbitrage opportunity when it is priced correctly.

History: Developed by Fischer Black and Myron Scholes, then expanded by Robert Merton. The latter two won the 1997 Nobel Prize in Economics for the discovery.

Importance: Helped create the now multi trillion dollar derivatives market. It is argued that improper use of the formula (and its descendants) contributed to the financial crisis. In particular, the equation maintains several assumptions that do not hold true in real financial markets.

Modern use: Variants are still used to price most derivatives, even after the financial crisis.

Bonus equations: From an email interview with Dr Stewart.

• At one stage I planned to include the Hodgkin-Huxley equations, which gave mathematical biology a huge boost by using equations to model the way nerve cells send signals to each other. It formed the basis of theoretical neuroscience, and is still important. But it made the book too long, and in the end I felt that its impact on human history has not yet been quite great enough. However, that is likely to change by the middle of this century, as mathematical methods become a major part of mainstream biology — which I think they will.
• My current candidate for and 18th equation (number 1 in ‘Seventeen MORE Equations That Changed the World’ — I joke… I think…) is the basic equation behind Google. This describes how to rate the importance of a website in terms of the links to it, and it’s a clever application of basic undergraduate linear algebra. It deserved to be in the book, but I was running out of space — and worried that my readers’ enthusiasm for yet another equation might be drying up.

- Max Nisen

•

#### Published with permission from:

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Ordered By: Popularity
• 27/01/13 #

Great article.

• 27/01/13 #

They didn’t teach me that in pass maths

• 27/01/13 #

You are mixing up your pluses and minuses. The sum is 30 = 25 + 3 +2. What missing 1?

• 27/01/13 #

@ben They give 10€ each.. 3 by 10 = 30.. They get it for 25€ so there is 5€ change they take 1€ each back out of the change so then there’s 2€ left .. If they gave 10€ and got 1€ back that means they paid 9€ each 3 by 9 = 27+2 = 29 in any language

• 27/01/13 #

Stop messing about. The 2 has to be subtracted from 25, not added to it.

• 27/01/13 #

Number 3 is not the Fundamental Theorem of Calculus, but the definition of the derivative in terms of difference quotients. The Fundamental Theorem of Calculus relates derivatives to integrals (or, if you prefer, rates of change to areas under curves).

• 27/01/13 #

We’ll take you at your word on that one ;-)

• 27/01/13 #

I still can’t find X

• 27/01/13 #

The more complex the mathematics, the less numbers are involved. :-/

• 27/01/13 #

I’m still working on a formula to find out how long is a piece of string

• 27/01/13 #

What about Boyles law PV/t is a constant

• 27/01/13 #

@Eddie, ‘fraid not mate, the €30 ceased to exist when they got back the fiver, meaning the merchant recieved €25. So we have 3x €9 = €27 less the €2 in hand equals the €25 taken by the merchant.

Isn’t pleasantly surreal to be arguing about algebra at Sunday lunchtime?

• 27/01/13 #

@ Ben are ya hungover whahaha

• 27/01/13 #

@Eddie, hungover! never. Maybe a bit secondhand……..

• 28/01/13 #

I think the formula was pressure is proportional to the reciprocal of the volume. Charle’s Law brought in Temperature into the equation later.

• 27/01/13 #

Now I get it !!!

• 27/01/13 #

3 house mates were going to by a radio between them, it cost 30€ so they gave 10€ each, the shop keeper said I will give it to ye for 25€ so there was 5€ in change they each took 1€ then there was 2€ left right , so really they only paid 9€ each cos they give 10 and got 1 back each so 3 times 9 = 27 + the 2 that was left over from the 5 change = 29
Where’s the missing 1€????

• 27/01/13 #

It was put in the tip jar as a kind gesture as thanks for the €5 discount ;)

• 27/01/13 #

• 27/01/13 #

Should be compulsory for teachers to introduce secondary school maths with this article/book digest. Might make a few more people appreciative of the usefulness of maths.

• 27/01/13 #

That’s what ‘Project Maths’ is about – making Maths real, eg. finding the height of a cliff by walking away from it (rather than the length of 1 side of a triangle).

• 27/01/13 #

What about all of murphys laws???

• 27/01/13 #

Great article.

• 04/02/13 #

Equation 9 (The Fourier Transform) is printed incorrectly.

There should NOT be a minus sign in front of the top infinity symbol.
(That is the top sideways ” figure 8″).

As printed the equation is meaningless.

• 07/06/13 #

Not completely meaningless; the RHS of int(f(x),a,a) is just zero….

• 27/01/13 #

What about Boyle’s Law? Robert Boyle, Irishman and the father of chemistry? There should be 18 laws listed!

• 27/01/13 #

Not THAT important compared to those listed. Great experiment proving it in LC Physics though.

• 27/01/13 #

Black = White

• 27/01/13 #

Great article…

• 27/01/13 #

Quadratic equations ruined by life. x=-b plus or minus the square root of b squared minus four ac divided by twice a. I hate you all.

• 27/01/13 #

Whats sad about our education system is that I DO remember this equation 25 years on, but could not remember what it was for (or even if our teachers told us).

• 27/01/13 #

That’s the major problem I have with math lessons, they teach the formulae, but no context within which I can be used.
http://www.udacity.com have a great introduction to physics course (free), they show you how to calculate the height of a building, the distance from the earth to the sun, or how to aim a cannon at attacking pirates, using trigonometry. fun course.

• 27/01/13 #

*No context in which IT can be used.

• 27/01/13 #

It’s how you get the solutions to a quadratic equation (one where the highest power of x is 2) when it can’t be factorised..

• 27/01/13 #

It’s how you get the solutions to a quadratic equation (one where the highest power of x is 2) when it can’t be factorised..

• 27/01/13 #

Folks spend too much time schoolin’. All anybody needs to know is that Merica is undefeated in world wars and 50 stars + 13 bars = Merica.

• 27/01/13 #

School is irrelevant to the most popular trades,I have never uses French, Irish ,biology,physics,religion,they were forced as no other options.
The ones I do need and to a considerable degree maths, mechanical engineering,carpentry.
I wanted to do business but the class was full(most anoying some never turned up).
Pe was a waste of time anyone who didn’t want to go had 80min wasted.
The education system is a joke its a revolving door system.
The guidance cllr said I would be best suited to “a vet because I had a liking of animals” I since have got a business degree and run 2 small steady businesses not that I have a 50k merc or mansion.
Over half what I done in school is beyond useless

• 27/01/13 #

So we should give up the other subjects and the whole country should do the same job as you Eric? I use business, economics and accounting in my job, my mates jobs range from lab scientists (physics), doctors and vets (biology), personal trainers (PE) to a document translator (Irish). There’s more to making the world go round than just what you do.

• 27/01/13 #

@amy you missed my point all subjects are not suited to everyone..I know guys who wanted to be carpenter’s but were told to do the leaving insted of leaving at 16 for a apprenticeship and saving 4yrs

• 27/01/13 #

@ Eric

To be fair, those mates of yours who wanted to be carpenters would probably be unemployed now and find it much harder to secure non-carpentry work without the Leaving Cert. It might not be fair but it’s the reality.

• 27/01/13 #

@chris,one working for himself and 2 lads working for him.two others still work for the same guys.
Leaving is useless in trade work if I was hiring a carpenter and had a choice of 2x 24yr olds one fresh from college papers to beat the band or a guy who started his apprenticeship at 16 and qualified at 20 with 4yrs qualified to date exp,IL take the guy that’s over all has most hands on experience the guy with the degree has no exp closing all doors bar jobs that need no qualifications

• 27/01/13 #

The leaving absolutely is not useless in trade work. Maths for example is essential in nearly all trade work, and geography in some. Your friends are the lucky ones I know a carpenter and a plumber who lost their jobs and had to retrain. One had the leaving and so got straight into college doing sound engineering, the other had no leaving and went to work in a shop and has to claw his way up the ladder. Kids have no idea as young teenagers what they want to go on to do as adults, that’s why there is a variety of subjects to help them decide which they prefer to train in as they come into adulthood. It also makes for more rounded individuals who aren’t completely ignorant of all other professions and a certain amount of life knowledge. That’s what school is for. College and apprenticeships are for the intensive career training.

• 27/01/13 #

@amy.maths is part of a apprenticeship anyway..I never mentioned geography but it wasn’t on offer to me and I don’t know a carpenter,plumber,electrician,or other trade that need geography

• 27/01/13 #

Geography isn’t just about which town is where. It’s also about types of soil, lay of the land, aerial locationing, (site analysis) all of which need to be taken into account in any large scale building project. Pipes are laid underground. Electric lines can’t go through lakes. A house cant be built on marshland. And so on.

• 27/01/13 #

@amy,that is true but they are specialist jobs for architects and engineers so my point still stands to the jobs I mentioned first,by your logic I could bring meteorology into the conversation and say well every needs to know the weather

• 27/01/13 #

As is quite evident from your post.

• 28/01/13 #

Eric,

If you’re looking for a carpenter and have a choice between a carpenter and someone who isn’t a carpenter, it’s a bit if a no-brainer.

• 27/01/13 #

Dont need to buy the book now!!!

• 27/01/13 #

Yuck algebra!!!

• 27/01/13 #

Think the Black Scoles model is the one the A beautiful mind is loosely based.

• 27/01/13 #

Different mathematician, read up on John Nash.

• 27/01/13 #

Huh?

• 27/01/13 #

Ewwwwww hate maths

• 01/02/13 #

Number 6 Euler’s formula for polyhedra only applies to convex polyhedra