List of Mathematical Equations That Changed the World

What evidence do we have of our origins? How do we know about the earth's history, present, and future? What is the foundation behind vehicle and GPS inventions?

It has been shown time and time again that all it takes is a simple formula to uncover some of the world's largest mysteries! This planet has produced brilliant mathematicians and physicists in the past, and their discoveries have made our lives simpler by providing us with essential components such as electricity. Ian Stewart, a mathematician, and scientific author wrote 17 Equations That Changed The World in 2013.

We recently came across this handy chart on Dr. Paul Coxon's Twitter account, which summarizes the equations and was created by maths instructor and blogger Larry Phillips. (We'll explain each one below):



Here's some additional information about these amazing equations that have affected mathematics and human history:

1) The Pythagorean Theorem:

• The Pythagorean Theorem is the cornerstone of our knowledge of geometry. It defines the relationship between the sides of a right triangle on a flat plane: square the short sides' lengths, a and b, add them together, and you obtain the square of the long side's length, c.
• In some senses, this connection separates our conventional, flat Euclidean geometry from curved, non-Euclidean geometry. A right triangle is drawn on the surface of a sphere, for example does not have to obey the Pythagorean theorem.

• The inverses, or opposites, of exponential functions are logarithms. A logarithm for a given base informs you what power that base has to be raised to produce a number. For example, since 1 = 100, the base 10 logarithm of 1 is log(1) = 0; log(10) = 1; and log(100) = 2, since 100 = 102.
• One of the most practical uses of logarithms is to change multiplication into addition, as shown in the figure by the equation log(ab) = log(a) + log(b).
• This was the most frequent approach to swiftly multiply together huge numbers before the introduction of the digital computer, considerably speeding up computations in physics, astronomy, and engineering.

• The derivative in calculus is defined by the formula shown above. The rate of change of a quantity is measured by the derivative. For example, we may conceive of velocity, or speed, as the derivative of position: if you're walking at 3 miles per hour, you've altered your location by 3 miles per hour.
• The derivative and integral - the other cornerstone of calculus - is at the heart of how mathematicians and scientists comprehend change.

• Newton's law of gravitation expresses the gravitational force, F, between two objects in terms of a universal constant, G, the masses of the two objects, m1 and m2, and the distance between them, r. Newton's law is a unique piece of scientific history because it explains why the planets move in the manner they do almost precisely. Its universality is also remarkable: this is how gravity operates everywhere in the universe, not only on Earth or in our solar system.
• Newton's theory of gravity stood the test of time for almost two centuries, and it wasn't updated until Einstein's theory of general relativity.

• Mathematicians have always pushed the boundaries of what numbers may be, from natural numbers through negative numbers, fractions, and finally real numbers. The complex numbers are formed by taking the square root of -1, which is generally denoted as i.
• Complex numbers are extraordinarily beautiful in mathematics. Algebra performs exactly as we want it to each equation has a complex number solution, which is not the case with real numbers: There is no real-number solution to x2 + 4 = 0, however, there is a complex answer: the square root of -2. Calculus may be extended to complex numbers, and we can discover some fascinating symmetries and features of these numbers as a result. Complex numbers are important in electronics and signal processing because of their features.

• Like the cube on the right, polyhedra are three-dimensional analogs of polygons. The vertices of a polyhedron are its corners, the edges are the lines connecting the vertices, and the faces are the polygons that cover it.
• There are 8 vertices, 12 edges, and 6 faces in a cube. I get 8 + 6 - 12 = 2 when I put the vertices and faces together and remove the edges.
• Euler's formula indicates that if you add the vertices and faces together and remove the edges, you will always obtain 2 if your polyhedron is fairly well behaved. Whether your polyhedron has four, eight, twelve, twenty, or any other number of faces is true.
• Euler's finding was one of the earliest examples of what is now known as a topological invariant, which is a number or feature shared by a group of comparable forms. V + F - E = 2 for the whole class of "well-behaved" polyhedra. This discovery, together with Euler's solution to the Bridges of Konigsburg issue, laid the way for the creation of topology, a subject of mathematics that is crucial to modern physics.

• In statistics, the normal probability distribution, which features the well-known bell curve graph on the left, is often used.
• The normal curve is used to model different features in physics, biology, and the social sciences. Because it depicts the behavior of huge groupings of independent processes, the normal curve appears frequently.

• This is a differential equation, or an equation that, as stated above, shows how a property changes over time in terms of its derivative. The wave equation covers the behavior of waves, such as those produced by a vibrating guitar string, ripples in a pond when a stone is thrown, or light from an incandescent bulb. The wave equation was an early differential equation, and the techniques used to solve it led to a better understanding of other differential equations.

• Understanding increasingly complicated wave patterns, such as human voice, requires the use of the Fourier transform. The Fourier transform allows us to split a complicated, messy wave function, such as a recording of a person talking, into a combination of a number of simple waves, substantially simplifying analysis.
• Modern signal processing and analysis, as well as data compression, rely on the Fourier transform.

11) Maxwell's Equations: This set of four differential equations describes the behavior of and relationship between electricity (E) and magnetism (H).

Maxwell's equations are to classical electromagnetism as Newton's laws of motion and law of universal gravitation are to classical mechanics - they are the foundation of our explanation of how electromagnetism works on a day to day scale. As we will see, however, modern physics relies on a quantum mechanical explanation of electromagnetism, and it is now clear that these elegant equations are just an approximation that works well on human scales.

10) Navier-Stokes Equations:

• This is a differential equation, much as the wave equation. The Navier-Stokes equations explain the behavior of flowing fluids such as water in a pipe, air over an aviation wing, and smoke rising from a cigarette. While we have approximate solutions to the Navier-Stokes equations that allow computers to reasonably accurately mimic fluid motion, the question of whether it is feasible to create mathematically perfect solutions to the equations remains an open subject (with a million dollar reward).

• This set of four differential equations defines the behavior of electricity (E) and magnetism (M) and their interaction (H).
• Maxwell's equations form the cornerstone of our understanding of how electromagnetic operates on a day-to-day scale, much as Newton's laws of motion and law of universal gravitation are in classical mechanics. However, as we will see, current physics relies on a quantum mechanical explanation of electromagnetism, and it is now evident that these beautiful equations are really a human-scale approximation.

• This asserts that entropy (S) is constantly constant or rising in a closed system. Thermodynamic entropy is a measure of how chaotic a system is, in general. A system that begins in an orderly, unequal condition - such as a hot zone close to a cold region - will always tend to smooth out, with heat moving from the hot to the cold until it is equally distributed.
• One of the rare situations in physics where time matters in this way is the second law of thermodynamics. Most physical processes are reversible, which means we can reverse the equations without causing problems. The second law, on the other hand, exclusively applies in this direction. When we put an ice cube in a cup of hot coffee, the ice cube always melts, but the coffee never freezes.

• Einstein's ideas of special and general relativity significantly altered the direction of science. E = mc2 is a well-known equation that claims that matter and energy are equal. The theory of special relativity included concepts such as the speed of light being a universal speed limit and the passage of time differing for persons traveling at different speeds.
• General relativity, which was the first fundamental advance in our understanding of gravity since Newton's law, portrays gravity as a curving and folding of space and time itself. The theory of general relativity is critical to our understanding of the universe's origins, structure, and ultimate fate.

• In quantum mechanics, this is the most important equation. This equation determines the behavior of atoms and subatomic particles because general relativity describes our cosmos at its biggest sizes.
• Modern quantum mechanics and general relativity are the two most successful scientific theories in history, with all experimental findings to date being completely compatible with their predictions. Most modern technology relies on quantum mechanics: nuclear power, semiconductor-based computers, and lasers are all dependent on quantum phenomena.

• The Shannon information entropy equation is provided here. This is a measure of disorder, similar to the thermodynamic entropy mentioned before. It quantifies the information content of a message in this context, which may be a book, a JPEG image conveyed over the internet, or anything else that can be represented symbolically. The Shannon entropy of a communication is a lower constraint on how much of its content may be compressed without losing quality. Shannon's entropy measure was the first to be used in the mathematical analysis of information, and his findings are still used in how we interact via networks today.

• It represents a process that evolves over time - the formula on the right gives xt+1, the level of some quantity x in the future time period, and it relies on xt, the current level of x. k is a predetermined value. The map exhibits chaotic behavior for certain values of k: if we start at a specific initial value of x, the process will evolve in one direction, but if we start at a different initial value, even one that is very close to the first value, the process will evolve in a completely different direction.
• In many domains, we find chaotic behavior - activity that is susceptible to beginning conditions. The image of a butterfly fluttering its wings on one continent creating a hurricane on another continent is a classic example of how a little shift in atmospheric conditions on one day may lead to radically distinct weather systems a few days later.

• Black-Scholes is a differential equation that outlines how financial professionals and traders find derivatives pricing. Derivatives, or financial instruments dependent on an underlying asset such as a stock, are an important aspect of today's financial system.
• Based on the parameters of the derivative and the underlying asset, financial experts can use the Black-Scholes equation to calculate the value of these financial instruments.

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