Equation of the Week

Square root of minus one
 
Girolamo Cardano

What does it say?
    Even though it ought to be impossible, the square of the number i is minus one.
Why is that important?
    It led to the creation of complex numbers, which in turn led to complex analysis, one of the most powerful areas of mathematics.
What did it lead to?
    Improved methods to calculate trigonometric tables. Generalisations of almost all mathematics to the complex realm. More powerful methods to understand waves, heat, electricity, and magnetism. The mathematical basis of quantum mechanics.

For the rest of the story, see Chapter 5 of 17 Equations That Changed the World.

Pythagoras's Theorem
       
Pythagoras

What does it say?
    How the three sides of a right-angled triangle are related.
Why is that important?
    It provides a vital link between geometry and algebra, allowing us to calculate distances in terms of coordinates. It also inspired trigonometry.
What did it lead to?
    Surveying, navigation, and more recently special and general relativity—the best current theories of space, time, and gravity.

For the rest of the story, see Chapter 1 of 17 Equations That Changed the World.


Wave Equation
   
Jean Le Rond d'Alembert                                                                              Earthquake in Wenchuan, China 2008
                                                                                                                     Photo: Iris Consortium

What does it say?
    The acceleration of a small segment of a violin string is proportional to the average displacement of neighbouring segments.
Why is that important?
    It predicts that the string will move in waves, and it generalises naturally to other physical systems in which waves occur.
What did it lead to?
    Big advances in our understanding of water waves, sound waves, light waves, elastic vibrations... Seismologists use modified versions of it to deduce the structure of the interior of the Earth from how it vibrates. Oil companies use similar methods to find oil. In Chapter 11 we will see how it predicted the existence of electromagnetic waves, leading to radio, television, radar, and modern communications.

For the rest of the story, see Chapter 8 of 17 Equations That Changed the World.


Second Law of Thermodynamics
               
Ludwig Boltzmann

What does it say?
    The amount of disorder in a thermodynamic system always increases.
Why is that important?
    It places limits on how much useful work can be extracted from heat.
What did it lead to?
    Better steam engines, estimates of the efficiency of renewable energy, the ‘heat death of the universe’ scenario, proof that matter is made of atoms, and paradoxical connections with the arrow of time.

For the rest of the story, see Chapter 12 of 17 Equations That Changed the World.

Normal Distribution (Bell Curve)
 
Abraham De Moivre

What does it say?
    The probability of observing a particular data value is greatest near the mean value—the average—and dies away rapidly as the difference from the mean increases. How rapidly depends on a quantity called the standard deviation.
Why is that important?
    It defines a special family of bell-shaped probability distributions, which are often good models of common real-world observations.
What did it lead to?
    The concept of the ‘average man’, tests of the significance of experimental results, such as medical trials, and an unfortunate tendency to default to the bell curve as if nothing else existed.

For the rest of the story, see Chapter 7 of 17 Equations That Changed the World.


Euler's Formula for Polyhedra
  
Leonhard Euler

What does it say?
    The numbers of faces, edges, and vertices of a solid are not independent, but are related in a simple manner.
Why is that important?
    It distinguishes between solids with different topologies using the earliest example of a topological invariant. This paved the way to more general and more powerful techniques, creating a new branch of mathematics.
What did it lead to?
    One of the most important and powerful areas of pure mathematics: topology, which studies geometric properties that are unchanged by continuous deformations. Examples include surfaces, knots, and links. Most applications are indirect, but its influence behind the scenes is vital. It helps us understand how enzymes act on DNA in a cell, and why the motion of celestial bodies can be chaotic.

For the rest of the story, see Chapter 6 of 17 Equations That Changed the World.

Fourier Transform
 
Joseph Fourier

What does it say?
    Any pattern in space and time can be thought of as a superposition of sinusoidal patterns with different frequencies.
Why is that important?
    The component frequencies can be used to analyse the patterns, create them to order, extract important features, and remove random noise.
What did it lead to?
    Fourier’s technique is very widely used, for example in image processing and quantum mechanics. It is used to find the structure of large biological molecules like DNA, to compress image data in digital photography, to clean up old or damaged audio recordings, and to analyse earthquakes. Modern variants are used to store fingerprint data efficiently and to improve medical scanners.

For the rest of the story, see Chapter 9 of 17 Equations That Changed the World.

Relativity
 
Albert Einstein

What does it say?
    Matter contains energy equal to its mass multiplied by the square of the speed of light.
Why is that important?
   The speed of light is huge and its square is absolutely humongous. One kilogramme of matter would release about 40% of the energy in the largest nuclear weapon ever exploded. It’s part of a package of equations that changed our view of space, time, matter, and gravity.
What did it lead to?
    Radical new physics, definitely. Nuclear weapons... well, just maybe—though not as directly or conclusively as the urban myths claim. Black holes, the Big Bang, GPS and SatNav.

For the rest of the story, see Chapter 13 of 17 Equations That Changed the World.


Information Theory
    
Claude Shannon

What does it say?
    It defines how much information a message contains, in terms of the probabilities with which the symbols that make it up are likely to occur.
Why is that important?
    It is the equation that ushered in the Age of Information. It established limits on the efficiency of communications, allowing engineers to stop looking for codes that were too effective to exist. It is basic to today’s digital communications—phones, CDs, DVDs, the Internet.
What did it lead to?
    Efficient error-detecting and error-correcting codes, used in everything from CDs to space probes. Applications include statistics, artificial intelligence, cryptography, and extracting meaning from DNA sequences.

For the rest of the story, see Chapter 15 of 17 Equations That Changed the World.


Black-Scholes Equation
 
Fischer Black and Myron Scholes

What does it say?
    It describes how the price of a financial derivative changes over time, based on the principle that when the price is correct, the derivative carries no risk and no one can make a profit by selling it at a different price.

Why is that important?
    It makes it possible to trade a derivative before it matures by assigning an agreed ‘rational’ value to it, so that it can become a virtual commodity in its own right.

What did it lead to?
    Massive growth of the financial sector, ever more complex financial instruments, surges in economic prosperity punctuated by crashes, the turbulent stock markets of the 1990s, the 2008-9 financial crisis, and the ongoing economic slump.

For the rest of the story, see Chapter 17 of 17 Equations That Changed the World.


Maxwell's Equations for Electromagnetism
 
James Clerk Maxwell

What Do They Say?
    Electricity and magnetism can’t just leak away. A spinning region of electric field creates a magnetic field at right angles to the spin. A spinning region of magnetic field creates an electric field at right angles to the spin, but in the opposite direction.
Why is That Important?
    It was the first major unification of physical forces, showing that electricity and magnetism are intimately interrelated.
What Did They Lead To?
    The prediction that electromagnetic waves exist, travelling at the speed of light, so light itself is such a wave. This motivated the invention of radio, radar, television, wireless connections for computer equipment, and most modern communications.

For the rest of the story, see Chapter 11 of 17 Equations That Changed the World.

Newton's Law of Gravity
       
Sir Isaac Newton

What Does It Say?
    It determines the force of gravitational attraction between two bodies in terms of their masses and the distance between them.
Why Is That Important?
    It can be applied to any system of bodies interacting through the force of gravity, such as the solar system. It tells us that their motion is determined by a simple mathematical law.
What Did It Lead To?
    Accurate prediction of eclipses, planetary orbits, the return of comets, the rotation of galaxies. Artificial satellites, surveys of the Earth, the Hubble telescope, observations of solar flares. Interplanetary probes, Mars rovers, satellite communications and television, the Global Positioning System.

For the rest of the story, see Chapter 4 of 17 Equations That Changed the World.