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This article is published in collaboration with Business Insider.
Written by
Andy Kiersz
Published
Monday 4 April 2016
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
The 17 equations that changed the world | World Economic Forum
Written by
Andy Kiersz
Published
Monday 4 April 2016
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
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
Image: Business Insider
The 17 equations that changed the world | World Economic Forum
