# The HashLearn Blog

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## 8 Equations That Must Be Revisited!

In 2013, mathematician and author Ian Stewart published a book, 17 Equations That Changed The World. If you’re wondering what has that got to do with you, read on!

Take a closer look – some of these look familiar, don’t they? We’ve already studied 8 out of the 17 equations that changed the history at school!

Let’s dive into how each of these 8 equations have changed the course of history as we know it.

1. Pythagoras’ theorem

Pythagoras’ theorem is one of the very few interesting theorems ever encountered. Even memorizing the statement isn’t too difficult and reciting a few Pythagorean triplets isn’t very uncommon during the exam season. This relationship, in some ways, actually distinguishes our normal, flat, Euclidean geometry from curved, non-Euclidean geometry. For example, a right triangle drawn on the surface of a sphere need not follow the Pythagorean theorem.

?2. Logarithms

Logarithms are the inverses, or opposites, of exponential functions. A logarithm for a particular base tells you what power you need to raise that base to to get a number. Until the development of the digital computer, this was the most common way to quickly multiply together large numbers, greatly speeding up calculations in physics, astronomy, and engineering.

Guess now we know the big fuss over f log tables!

3. Calculus

A nightmare to many of us. Calculus is the mathematical study of change, in the same way that geometry is the study of shape. A basic understanding of calculus is used in the branches of physical sciences, computer science, statistics, engineering, and in any other field wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.

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?4. Law of Gravity

Sir Issac Newton is back! We’re sure you would have written the statement of this law and derived it in vector form a thousand times. Newton’s law of gravitation was the most significant discovery in understanding our universe, till Einstein’s theory of relativity replaced it. However, it is essential to understand that the law of gravity explains why the planets move in the way they do. Also remarkable is its universal nature – this is not just how gravity works on Earth, or in our solar system, but anywhere in the universe.
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5. Square root of minus one

Mathematically, the complex numbers are supremely elegant. Algebra works perfectly the way we want it to. Any equation can have a complex number solution, a situation that is not true for the real numbers.
Imaginary numbers are an important mathematical concept, which extends the real number system to the complex number system. Complex number are used in applications such as image processing, control theory, quantum computing, fluid dynamics, vibrations, electromagnetism amongst many others.

?6. Euler’s polyhedral formula

Polyhedra are the three-dimensional versions of polygons, like octahedron in the image above.  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. The regular octahedron has 6 vertices and 12 edges and 8 faces.

On applying Euler’s formula, the sum of vertices and faces and subtract the edges (6+8-12) should be 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 observation, along with with Euler’s solution to the Bridges of Konigsburg problem, paved the way to the development of topology, a branch of math essential to modern physics.

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7. Second Law of Thermodynamics

One of the most fundamental laws of science, this law states that the entropy of a closed system (such as our universe) always remains steady or increases. Thermodynamic entropy is the measure of randomness or disorder in a system.

The second law of thermodynamics is applied to heat engines as it gives the maximum achievable efficiency.  The law is extensively used in industries to determine direction of a process or a reaction. Most common method to check whether a reaction is spontaneous is to find out change in Gibbs free energy. If this term is negative for a reaction, then the process is spontaneous.

8. Relativity

Unarguably the world’s most famous equation (E=mc^2) and its even more famous creator Albert Einstein has to be on this list.  It is very important to know that Einstein’s theory of general relativity is one of the most accurate theories explaining gravity, a feat which the radical quantum theory hasn’t been able to achieve (for now!) General relativity is essential to our understanding of the origins, structure, and ultimate fate of the universe.

Share if you are awed by the awesomeness of these equations & their application in the world!

Image source: phys.utk.edu, ?The Prospect, ?Etsystatic.
Content Source: Ian Stewart’s 17 Equations That Changed The World