Seventeen Equations that Changed the World

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

Equations in pure mathematics are generally of the first kind: they reveal deep and beautiful patterns and regularities. They are valid because, given our basic assumptions about the logical structure of mathematics, there is no alternative. Pythagoras’s Theorem, which is an equation expressed in the language of geometry, is an example. If you accept Euclid’s basic assumptions about geometry, then Pythagoras’s Theorem is true.In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Equations in applied mathematics and mathematical physics are usually of the second kind. They encode information about the real world; they express properties of the universe that could in principle have been very different. Newton’s Law of Gravity is a good example. It tells us how the attractive force between two bodies depends on their masses, and how far apart they are. Solving the resulting equations tells us how the planets orbit the Sun, or how to design a trajectory for a space probe. But Newton’s Law isn’t a mathematical theorem; it’s true for physical reasons, it fits observations. The Law of Gravity might have been different. Indeed, it is different: Einstein’s general theory of relativity improves on Newton by fitting some observations better, while not messing up those where we already know Newton’s Law does a good job.

One kind presents relations between various mathematical quantities: the task is to prove the equation is true. The other kind provides information about an unknown quantity, and the mathematician’s task is to solve it – to make the unknown known. The distinction is not clear-cut, because sometimes the same equation can be used in both ways, but it’s a useful guideline. Source: In Pursuit of the Unknown: 17 Equations That Changed the World The Greeks did not express Pythagoras’s theorem as an equation in the modern symbolic sense. That came later with the development of algebra. In ancient times, the theorem was expressed verbally and geometrically. It attained its most polished form, and its first recorded proof, in the writings of Euclid of Alexandria. Around 250 BC Euclid became the first modern mathematician when he wrote his famous Elements, the most influential mathematical textbook ever. Euclid turned geometry into logic by making his basic assumptions explicit and invoking them to give systematic proofs for all of his theorems. He built a conceptual tower whose foundations were points, lines, and circles, and whose pinnacle was the existence of precisely five regular solids. Surveys routinely employed triangulation by the late 18th century. And while we don’t explicitly use it today, it is still there in how we deduce locations from satellite data.In this article, I will show 10 of the most important and beautiful mathematical equations. This article is based on the book “In Pursuit of the Unknown: 17 Equations That Changed the World” by Ian Stewart. 1. Pythagoras’s Theorem (Pythagoras, 530 BC) 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.

Many triangles in real life are not right angles. But this does not limit the use of the equation a²+b² = c² because any triangle can be cut into two right-angled ones. Source: In Pursuit of the Unknown: 17 Equations That Changed the World For the purposes of higher mathematics, the Greeks worked with lines and areas instead of numbers. So Pythagoras and his Greek successors would decode the theorem as an equality of areas: ‘The area of a square constructed using the longest side of a right-angled triangle is the sum of the areas of the squares formed from the other two sides.’ Ignoring the advice, Hawking included E = mc² even ‘when cutting it out would have sold another 10 millions copies.” This captures our aversion to equations well. Yet, mathematician Ian Stewart argues in his book In Pursuit of the Unknown: 17 Equations That Changed the World, “[e]quations are too important to be hidden away.” 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. Equations are the lifeblood of mathematics, science, and technology. Without them, our world would not exist in its present form. However, equations have a reputation for being scary: Stephen Hawking’s publishers told him that every equation would halve the sales of A Brief History of Time, but then they ignored their own advice and allowed him to include E = mc 2 when cutting it out would allegedly have sold another ten million copies. I’m on Hawking’s side. Equations are too important to be hidden away. But his publishers had a point too: equations are formal and austere, they look complicated, and even those of us who love equations can be put off if we are bombarded with them.

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.