Equations that changed the world

Most people will have heard of some of history's most famous equations: Newton's Law of Gravity, for instance, or Einstein's theory of relativity. These equations and more represent the world all around us, and equations have long been a driving force behind nearly every aspect of our lives. This page explains seventeen of humanities most important equations, such as the Wave Equation that allowed engineers to measure a building's response to earthquakes, saving countless lives, and the Black-Scholes model, used by bankers to track the price of financial derivatives over time. Inspired by In Pursuit of the Unknown: 17 Equations That Changed the World by Ian Stewart.

\[a^2 + b^2 = c^2\]

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

History: Though attributed to Pythagoras, it is not certain that he was the first person to prove it. The first clear proof came from Euclid, and it is possible the concept was known 1000 years before Pythagoras by the Babylonians!

Importance: The Pythagorean theorem forms the basis of trigonometry and, when applied to arithmetic, it connects the fields of algebra and geometry. Without it, accurate surveying, map making, and navigation would be impossible.

Modern use: GPS navigation uses satellites to triangulate your position on the globe..

\[\log xy = \log x + \log y\]

What does it mean: Logarithms are the inverses, or opposites, of exponential functions. A logarithm for a particular base tells you what power you need to raise that base to to get a number. For example, the base 10 logarithm of 1 is log(1) = 0, since 1 = 100; log(10) = 1, since 10 = 101; and log(100) = 2, since 100 = 102. 

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.

\[{d\int \over d{t}} = \lim \limits_{h \to 0} \frac{f\left( {{t + h}} \right) -f(t)}{f\left( {{h}} \right)} \]

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

What does it mean?: Allows the calculation of an instantaneous rate of change. The formula above is the definition of the derivative in calculus. The derivative measures the rate at which a quantity is changing. For example, we can think of velocity, or speed, as being the derivative of position — if you are walking at 2 miles per hour, then every hour, you have changed your position by 2 miles.

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: 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.

"More than any other mathematical technique, calculus has created the modern world."

\[F = G \frac{m_1 m_2}{r^2}\]

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.

Modern Use: When we launch space missions, Newton's law is used to find optimal gravitational "tubes" or pathways so they can be as energy efficient as possible. Also makes satellite TV possible. 

\[i^2 = -1\]

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. 

\[V - E + F = 2\]

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. 

\[\Phi(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{{{ - \left( {x - \mu } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {x - \mu } \right)^2 } {2\sigma ^2 }}} \right. -} {2\sigma ^2 }}}\]

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. 

\[\frac{{\partial ^2 u}}{{\partial t^2 }} = {{c^2 }}\frac{{\partial ^2 u}}{{\partial x^2 }}\]

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.

\[\frac{{\partial ^2 u}}{{\partial t^2 }} = {{c^2 }}\frac{{\partial ^2 u}}{{\partial x^2 }}\]

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 analyzed. 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. 

\[ \rho \left( \frac{\partial v}{\partial t} + v \cdot \nabla v \right)= -\nabla p + \nabla \cdot T + f\]

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 modeling 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. 

\[\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}\]

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. 

\[d S \geq 0\]

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 realize 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. ⚛

\[E=mc^2\]

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. 

\[i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi\]

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.

\[H = - \sum_{i=1}^{k}{p_{x} \log(p_{x})}\]

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: 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?

\[x_{t+1}= kx_t(1-x_t)\]

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.

\[\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}+ rS\frac{\partial C}{\partial S}-rC=0\]

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.