Education and Skills

# These equations changed the course of history

The real-world applications of mathematical equations. Image: REUTERS/Tobias Schwarz

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In 2012, Mathematician Ian Stewart came out with an excellent and deeply researched book titled "In Pursuit of the Unknown: 17 Equations That Changed the World."

His book takes a look at the most pivotal equations of all time, and puts them in a human, rather than technical context.

"Equations definitely can be dull, and they can seem complicated, but that’s because they are often presented in a dull and complicated way," Stewart told Business Insider. "I have an advantage over school math teachers: I'm not trying to show you how to do the sums yourself."

He explained that anyone 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."

Stewart continued that "equations are a vital part of our culture. The stories behind them — the people who discovered or invented them and the periods in which they lived — are fascinating."

Here are 17 equations that have changed the world:

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: 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 1,000 years before Pythoragas by the Babylonians.

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

In terms of pure math, the Pythagorean Theorem defines normal, Euclidean plane geometry. For example, a right triangle drawn on the surface of a sphere like the Earth doesn't necessarily satisfy the theorem.

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

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, and the related exponential functions, are used to model everything from compound interest to biological growth to radioactive decay.

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 time 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, physics, engineering, and computer science.

Newton's universal law of gravitation

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

History: Isaac Newton derived his laws based on earlier astronomical and mathematical 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 a practical description of how objects in space, like stars, planets, and human-made spacecraft, interact with each other. We use it to this day to design orbits for satellites and probes.

Philosophically, Newton's law is important because it describes how gravity works everywhere, from a ball falling to the ground on Earth to the evolution of galaxies and the universe as a whole. While we take the idea of universal laws for granted today, in earlier eras the idea that the terrestrial and celestial worlds shared the same properties was revolutionary.

Modern use: Although, as mentioned above, for practical uses Newton's law has been augmented by Einstein's theories, the basic idea of Newtonian gravity is still a useful approximation for how things behave in space.

Complex numbers

What does it mean? Mathematicians can expand our idea of what numbers are by introducing the square roots of negative numbers.

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.

The imaginary and complex numbers are mathematically very elegant. Algebra works perfectly the way we want it to — any equation has a complex number solution, a situation that is not true for the real numbers : x2 + 4 = 0 has no real number solution, but it does have a complex solution: the square root of -4, or 2i. Calculus can be extended to the complex numbers, and by doing so, we find some amazing symmetries and properties of these numbers.

Importance: According to Stewart ".... most modern technology, from electric lighting to digital cameras could not have been invented without them." The extension of calculus to the complex numbers, a branch of math called "complex analysis," is essential to understanding electrical systems and a variety of modern data processing algorithms.

Modern use: Used broadly in electrical engineering and mathematical theory.

Euler's formula for polyhedra

What does it mean? Describes a numerical relationship that is true of all solid shapes of a particular type.

History: This was developed by the great 18th century mathematician Leonhard Euler. Polyhedra are the three-dimensional versions of polygons, like the cube to the right. The corners of a polyhedron are called its vertices, the lines connecting the vertices are its edges, and the polygons covering it are its faces.

A cube has 8 vertices, 12 edges, and 6 faces. If I add the vertices and faces together, and subtract the edges, I get 8 + 6 - 12 = 2.

Euler's formula states that, as long as your polyhedron is somewhat well behaved, if you add the vertices and faces together, and subtract the edges, you will always get 2. This will be true whether your polyhedron has 4, 8, 12, 20, or any number of faces.

Importance: Fundamental to the development of topology, which extends geometry to any continuous surface.

Modern use: Topology is used to understand the behavior and function of DNA, and it is an underlying part of the mathematical tool kit used to understand networks like social media and the internet.

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. Statistical experiment design relies on the properties of the normal curve, and how those properties relate to errors that can occur when taking a random sample.

Modern use: Used to determine whether drugs are sufficiently effective in clinical trials.

The wave equation

What does it mean? A differential equation that describes the behavior of waves, like 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.

The techniques developed to solve the wave equation have been very useful in solving similar types of equations as well.

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

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 solution to a differential equation describing how heat flows, and the previously described wave equation.

Importance: The equation allows for complex wave patterns, like music, speech, or images, 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.

The Navier-Stokes equations

What does it mean? The Navier-Stokes equations are the fundamental physical equation that describes how fluids work. 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 approximately solve this equation, it opened up a complex and very useful field of physics. It is particularly useful in making vehicles more aerodynamic.

While we can use modern computers to make practical approximate simulations of fluid dynamics that are useful in engineering, finding a mathematically exact solution (or even knowing whether or not an exact solution exists in all cases) is still an open question, one whose answer is attached to a million-dollar prize.

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

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, and James Clerk Maxwell translated it into these equations. Maxwell's equations were for classical electromagnetism what Newton's laws of motion were for classical mechanics.

Importance: Helped understand electromagnetic waves, helping to create most modern electrical and electronic technology.

Modern use: Radar, television, and modern communications.

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.Thermodynamic entropy is, roughly speaking, a measure of how disordered a system is. A system that starts out in an ordered, uneven state — say, a hot region next to a cold region — will always tend to even out, with heat flowing from the hot area to the cold area until evenly distributed.

Modern use: Thermodynamics underlies much of our understanding of chemistry and is essential in building any kind of power plant or engine.

Einstein's theory of relativity

What does it mean? Energy and matter are two sides of the same coin.

History: The 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).

Special relativity brought in ideas like the speed of light being a universal speed limit and the passage of time being different for people moving at different speeds.

General relativity describes gravity as a curving and folding of space and time themselves, and was the first major change to our understanding of gravity since Newton's law. General relativity is essential to our understanding of the origins, structure, and ultimate fate of the universe.

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.

The Schrödinger equation

What does it mean? This is the main equation in quantum physics. 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. It describes the way subatomic particles and atoms evolve over time.

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 quantum mechanics and general relativity are the two most successful scientific theories in history — all of the experimental observations we have made to date are entirely consistent with their predictions.

Modern use: Quantum mechanics is necessary for most modern technology — nuclear power, semiconductor-based computers, and lasers are all built around quantum phenomena.

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: The equation given here is for Shannon information entropy. As with the thermodynamic entropy given above, this is a measure of disorder. In this case, it measures the information content of a message — a book, a JPEG picture sent on the internet, or anything that can be represented symbolically. The Shannon entropy of a message represents a lower bound on how much that message can be compressed without losing some of its content.

Modern use: Shannon's entropy measure launched the mathematical study of information, and his results are central to how we communicate over networks today.

The logistic model for population growth

What does it mean? Estimates the change in a population of creatures across generations with limited resources. Importantly, this equation can lead to chaotic behavior.

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.

For certain values of k, the map shows chaotic behavior: if we start at some particular initial value of x, the process will evolve one way, but if we start at another initial value, even one very very close to the first value, the process will evolve a completely different way.

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

We see chaotic behavior — behavior sensitive to initial conditions — like this in many areas. Weather is a classic example — a small change in atmospheric conditions on one day can lead to completely different weather systems a few days later, most commonly captured in the idea of a butterfly flapping its wings on one continent causing a hurricane on another continent.

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

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.

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