Seventeen Equations that Changed the World

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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. 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. 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.’ 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 .

A good example of the second kind of equation is Newton’s law of gravity. “It tells us how the attractive force between two bodies depends on their masses,” Stewart writes, “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.” This isn’t a mathematical theorem but rather it’s true for physical reasons in that it fits the observations. 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:Helped in the development of chaos theory, which has completely changed our understanding of the way that natural systems work.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.

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:Fundamental to the development of topology, which extends geometry to any continuous surface. 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. 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.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. Modern use:Helped lead to nuclear weapons, and if GPS didn't account for it, your directions would be off thousands of yards. What does it mean?Describes a numerical relationship that is true of all solid shapes of a particular type.