## Mathematics has changed the history of mankind 12 equations to prove it

Math is all around us in our daily lives, from grocery shopping, for gas, use a GPS or place a rack on the wall. In 2013, the renowned mathematician and writer of popular books, Ian Stewart, published a book entitled “17 Equations That Changed the World”. In this article, we explain some of them and why they have been so important for the history of mankind.

1. The Pythagorean Theorem

This famous equation is the basis of geometry as we know. Dating from 530 BC and describes the relationship between the sides of a right triangle on a flat surface, stating that the sum of the square of the length of the short sides b (“Hicks”) is equal to the square of the length of (“hypotenuse”) long side c.

This theorem is particularly useful in the field of construction, but also in navigation to triangulate positions in police investigations to calculate the trajectory of a bullet or determine how far was the shooter or to find the location of a mobile phone triangulation. The GPS we use today could not function without this theorem.

2. Logarithms

Logarithms, presented by John Napier in 1610, they are the inverse (opposite) of the exponential functions. A logarithm of a given base is the power that we have to raise the base to get a number. The equation log (ab) = log (a) + log (b) is one of the most useful applications of logarithms, converting multiplication sum.

Until the development of digital computers and computing, this was the most common multiply rapidly together large numbers and greatly accelerated calculations in fields such as physics, astronomy and engineering method.

It is now used in many fields such as economics and banking, where it is used, for example, to measure the growth in time deposits; the advertising, to compile statistics of the advertising campaign; in biology, for statistical analysis of what a child will inherit genetic parent; in geology, to calculate the intensity of earthquakes; chemistry, for calculating the pH, etc.

3. The law of gravity

This law of classical physics, also called law of universal gravitation, was enunciated by Isaac Newton in 1687 and describes the mutual attractive force F experienced two bodies according to their respective masses m 1 and m 2, the distance separates r and gravitational constant G. Under this law, the greater the mass of bodies and the closer they are to each other, more strongly they attract.

This law explains almost perfectly the movement of the planets and is completely universal, ie, works not only on Earth but throughout the Universe. In our daily life, explains many things, including why the earth revolves around the sun, why a pendulum swinging movement can maintain its eternally, why if we throw any object always ends up falling down and why speed drop increases as it approaches the Earth, etc.

The law of gravity Newton remained in force until it was replaced by the theory of general relativity Albert Einstein in the early twentieth century.

4. The wave equation D’Alembert

The wave equation or D’Alembert equation is a differential equation developed by the French mathematician and philosopher Jean le Rond D’Alembert in 1746. This equation is used to describe the behavior or change shape over time a wide variety of waves, including water waves, sound and light. For example, a guitar string vibrates, the ripples in a pond after throwing a stone or light emitting incandescent bulb. Hence it is so important in fields such as acoustics, electromagnetism or fluid dynamics, among others. In addition, techniques developed to solve this equation helped others understand differential equations.

5. Theorem of polyhedrons of Euler

The mathematician Leonhard Euler in 1751 published his “Theorem of polyhedrons,” which includes a formula for amazing results. But what is a polyhedron? As it is a three-dimensional version of a polygon. For example, a cube or a polyhedron would dimensional version of the square polygon. The corners of a polyhedron are called “vertices”, the lines connecting the vertices together are called “edges” and including the remaining surfaces are the “faces” of the polyhedron.

Euler’s genius is to have found a valid formula for all polyhedra and lets us know if the polyhedron is well constructed. According to Euler, the sum of the vertices (V) and the faces (C) of a polyhedron least its edges (A) must always be equal to 2. For example, a cube has eight corners, 6 faces and 12 edges. If we apply Euler’s formula, V – E + F = 2, we see that 8-12 + 6 = 2.

Choose any polyhedron and try to apply the formula. The result will always be 2!

This observation of Euler was one of the first examples of what today we call “topological invariant”: a number or property sharing a class of similar shapes to each other; and paved the way for the development of topology, a branch of the essential to modern physics and mathematics 3D mapping.

6. The normal distribution

The normal distribution, also called Gaussian distribution or bell curve describes the behavior of certain properties or independent processes in large groups of people or things. The importance of this distribution is that constantly appears in nature and attitude of the people, allowing modeling many natural, social and psychological phenomena. Its use is very common in fields such as physics, biology, psychology or social sciences, among others.

For example, if we look up to all the people in a sample group we will see that most people are about average height and that as we move away from the average height above or below the number of people decreases, leading to chart typical bell-shaped developed by Carl Friedrich Gauss in 1810. The same will happen if we analyze their IQ.

This distribution is also a fundamental tool in laboratories, to see the percentage of effectiveness that has a drug in clinical trials.

7. The Maxwell equations

Four differential equations that describe the behavior of the electric (E) and magnetic (H) and how they relate to each other. These equations, published by James Clerk Maxwell in 1865 are the basis of the explanation of how electromagnetism works from day to day.

In everyday life, we can explain to them how information from television, the Internet and mobile phones is transmitted, how long it takes to reach Earth the starlight or how neurons work.

However, today we know that these equations provide only an approximation to electromagnetism works well in human scale but that is not accurate, which is why modern physics has replaced Maxwell’s equations by a quantum mechanical explanation.

8. The 2nd law of thermodynamics

Under this law, formulated by Boltzmann in 1874, in a closed system, entropy (S) is always constant or increasing. But what is the thermodynamic entropy? In general terms, entropy determines the amount of disorder in a system there. For example, a system of irregular and orderly like a hot region next to a Cold region always tend to equalize, heat flow from the hot zone to the cold zone, until a uniform distribution state.

Unlike the majority of which are usually reversible, physical processes the second law of thermodynamics is irreversible, only works in this direction and time dependent. So, if we prepare an iced coffee, ice cubes always will melt and freeze never coffee.

9. The theory of relativity

Albert Einstein changed the course of physics in stating his theories of special and general relativity in the early twentieth century. The famous equation E = mc2 implies that mass and energy are equivalent.

Special relativity, published in 1905, taught us that the speed of light is a universal speed limit and that the passage of time is different for people who move at different speeds. While the general theory of relativity or general relativity, published in 1915, is a new theory of gravitation replaced Newton’s Law and which describes gravity as a curvature and folding space and time to themselves. General relativity is essential to understanding the origins, structure and ultimate fate of the universe.

10. The Schrödinger equation

This equation, published by the Austrian physicist Erwin Schrödinger in 1927, governs the behavior of atoms and subatomic particles and is the basis of quantum mechanics. It is one of the most successful scientific theories of history, along with the general theory of relativity, since all experimental observations to date agrees well with predictions. In everyday life, quantum mechanics is present in the most modern technologies, such as nuclear power, semiconductor-based computers or Lasers.

11. Information Theory

This is the equation of the information entropy, Shannon published in 1949. As thermodynamic entropy, this is a measure of disorder, but applied in this case, the information contained in a message, either a book, a picture or anything else that can be represented symbolically. Data compression is applied in different fields of information, including. Compressing a message may be the case that there is any loss of content and higher compression, the more likely that losses. In this sense, Shannon entropy compression indicates the limit of a message, that is, to what extent can compress without losing part of their contents. This theory marked the beginning of the mathematical study of information and results are fundamental today for communication across networks.

12. Chaos theory

Surely you’ve ever heard that a butterfly flapping its wings in one continent can cause a hurricane in another continent. Well, that is basically chaos theory, the idea that a small event or change in initial conditions can cause a chain reaction that results in a completely different result.

The Australian mathematician and biologist Robert May developed his equation in 1976 while studying the evolution of animal populations. The equation describes a process that evolves over time, which would be the amount xt studied x (in the case of May, the number of children of each individual in the population) at the present time factor t, K is a constant chosen and xt + 1 would be the development of value x with the passage of time. For certain values ​​of K, the logistic map of May shows a chaotic behavior. According to this theory, if we start at a specific initial value of x, the process will evolve in a way, but if we start from a different initial value, even if it is extremely close to the first value, the process will evolve in a completely different way.

This highly sensitive to initial conditions chaotic behavior is present in many aspects of our daily lives. A very clear example is the weather, where a small change in atmospheric conditions one day may lead to completely different weather systems within days. Today is especially common use in the field of economics, medicine and meteorology.