How many can you understand 9 equations that change the world?

Tribune South Africa shows : How many can you understand 9 equations that change the world?

How many can you understand 9 equations that change the world?

News Channels in South Africa said :  Beijing time, December 28th, according to foreign media reports, mathematics is a discipline that studies concepts such as quantity, structure, change, and space. Some groundbreaking mathematical concepts have not only changed human history, but also profoundly changed human history. The world we live in.

 

Add to cart Pakistan are also offer 50%  Off for All books , which is beneficial for students . Actually Mathematical equations are our unique window into the world. They make reality meaningful and help us see things we haven't noticed before. Therefore, new advances in mathematics are often accompanied by a further deepening of our understanding of the universe. Next, let us understand nine famous equations in history, from tiny particles to the vast universe, they have completely changed the way human beings view everything in the world.

Pythagorean Theorem

The first important trigonometric function people learn in school is the relationship between the side lengths of a right-angled triangle: the length of two right-angled sides (the shorter right-angled side is called hook length in ancient times, and the longer right-angled side is called strand length in ancient times). The sum of squares is equal to the square of the hypotenuse length (called chord length in ancient times). This theorem is usually written as: a^2 + b^2 = c^2. Since the Babylonian era, the theorem has existed for at least 3,700 years.

The Pythagorean theorem is one of the important mathematical theorems discovered and proved by mankind early. Researchers at the University of St Andrews in Scotland believe that the ancient Greek mathematician Pythagoras wrote the equation form of the theorem that is widely used today, and modern Western mathematics also calls it the "Pythagoras theorem." . In addition to its applications in construction, navigation, cartography, and other important processes, the Pythagorean theorem also helps expand the concept of numbers. In the 5th century BC, the Metapontum mathematician Hippassos noticed that if the length of the two waists of an isosceles right-angled triangle is 1, the length of its base is 2 (), which is An irrational number (no one has seen such a number in history before this). According to an article from the University of Cambridge, Hippassos is said to have been thrown into the sea because followers of Pythagoras (including Hippassos) were shocked and panicked by the so-called "infinite non-recurring decimals". . At that time, the Pythagorean school believed that "everything is number", and there are only integers and fractions (rational numbers) in the world. The discovery of Hippassos triggered the first mathematics crisis.

F = ma and the law of gravitation

Isaac Newton is one of the most outstanding figures in the history of science in the United Kingdom and even in humans. He has made a large number of discoveries that have changed the world, including Newton's second law of motion. The law states that force is equal to the mass of an object multiplied by acceleration, usually written as F = ma. Through the expansion of this law, combined with other experimental observations, Newton described what we call the law of universal gravitation today: F ​​= G (m1 * m2) / r^2, where F is between two objects Gravitation, m1 and m2 are the masses of two objects, r is the distance between them; G is a basic constant, called the gravitational constant, its value must be measured through experiments. According to records, Cavendish was the first to complete an experiment to measure the gravitational force between two objects in a laboratory, and accurately obtained the gravitational constant and the mass of the earth. Others used his experimental results to obtain the density of the earth.

Newton's second law is known as the soul of classical mechanics, which can dominate the movement of various objects and physical phenomena, and its uses are also very wide. Many concepts of Newton's law of motion are also used to understand various complex physical systems, including the motion of planets in the solar system, and how to use rockets to travel between them.

Wave equation

Using Newton's laws of motion, scientists in the 18th century began to analyze everything around them. According to a paper published in the journal Advances in Historical Studies in 2020, the knowledgeable French physicist, mathematician and astronomer Jean Leren d’Alembert deduced it in 1743 An equation describing string vibration or wave phenomenon. The equation can be written as:

1/v^2 * ∂^2y/∂t^2 = ∂^2y/∂x^2

In this equation, v is the velocity of the wave, and the other part describes the displacement of the wave in one direction. Using wave equations that extend to two or more dimensions, researchers can predict the motion of water, seismic waves, and sound waves. This equation is also the basis of the Schrodinger equation in quantum physics, which makes many modern computer devices possible.

Fourier equation

Whether you have heard of the French mathematician and physicist Baron Jean-Baptiste Joseph Fourier, his work has already affected your life. The mathematical equations he wrote in 1822 enabled researchers to decompose complex and chaotic data into simple wave combinations, making it easier to analyze. According to an article in the magazine "Yale Scientific", the basic idea of ​​the Fourier transform was a radical concept when it was put forward. Many scientists refuse to believe that complex systems can be simplified to such an elegant degree. However, in many modern scientific fields today, including data processing, image analysis, optics, communications, astronomy, engineering, finance, cryptography, oceanography and quantum mechanics, Fourier transform has a wide range of applications. For example, in signal processing, the typical use of Fourier transform is to decompose the signal into amplitude components and frequency components.

Maxwell's equations

Electricity and magnetism were still new concepts in the 19th century, and scholars at that time were studying how to capture and use these strange physical phenomena. In 1864, Scottish mathematician and physicist James Clark Maxwell published a system of 20 equations, describing how electric and magnetic fields work and how they are related to each other. This system of equations greatly promotes our understanding of these two phenomena. Now, Maxwell’s equations are composed of four first-order linear partial differential equations. They are Gauss’s law that describes how electric charges generate electric fields, Gauss’s magnetic law that shows that magnetic monopoles do not exist, and Faraday’s law of induction that explains how time-varying magnetic fields generate electric fields. , And Maxwell-Ampere’s law that explains how electric currents and time-varying electric fields produce magnetic fields. This set of equations is what all freshmen physics students must learn, and it also lays the foundation for all electronic equations in the modern technological world.

E = mc ^ 2

Without this famous equation, no list of transformation equations can be complete. In 1905, Albert Einstein first proposed the concept of mass-energy equivalence, namely E = mc^2, which was part of his pioneering special theory of relativity. E = mc^2 shows that matter and energy are two sides of the same thing. In the equation, E represents energy, m represents mass, and c represents the constant speed of light. The concepts contained in such a simple equation are still difficult for many people to understand, but without E = mc ^ 2, we would not be able to understand the existence of stars in the universe, nor would we know how to build something like the Large Hadron Collider. The giant particle accelerator in China is even less able to get a glimpse of the nature of the subatomic world. It can be said that this equation has become one of the most famous equations in human history and has become a part of culture.

Friedman equation

Using a set of equations to define the entire universe sounds like an arrogant idea, but this is exactly the important idea put forward by Russian physicist Alexander Friedman in the 1920s. Using Einstein's theory of relativity, Friedman pointed out that since the Big Bang, the characteristics of the expanding universe can be expressed by two independent equations.

These two equations combine all the important parameters of the universe, including the curvature of the universe, how much matter and energy the universe contains, and the speed at which the universe expands, with some important constants such as the speed of light, the constant of gravity, and the Hubble constant. This is a model of the expanding universe that is uniform and isotropic in space under the framework of general relativity. As we all know, Einstein did not like the idea of ​​expansion or contraction of the universe. His general theory of relativity believed that these situations occurred because of the influence of gravity. Einstein tried to add a variable labeled "λ" to Einstein's equation as a cosmological constant, so that the equation could have a static universe solution. After Hubble proposed the observation result of the expanding universe-the Hubble redshift-Einstein gave up the cosmological constant and considered this to be his "biggest mistake in his life." However, decades later, this concept was picked up again. Researchers believe that although the value of the cosmological constant is small, it may not be 0; and the constant may exist in the form of dark energy, which drives the accelerated expansion of the universe.

Shannon Information Equation

Most people are familiar with the 0s and 1s that make up the binary numbers of computers. However, this key concept would not have been developed without the pioneering work of American mathematician and engineer Claude Shannon. In an important paper in 1948, Shannon proposed an equation to show the maximum efficiency of information transmission, usually written as: C = B * 2log (1+S/N). In the formula, C is the highest error-free data speed achievable for a specific communication channel, B is the channel bandwidth, S is the average signal power, and N is the average noise power (S/N represents the signal-to-noise ratio of the system). The output of this equation is in bits per second (bps). In a 1948 paper, Shannon used bit as an abbreviation for "binary digit" and attributed its concept to mathematician John W. Tukey.

Robert May's Unimodal Image

Very simple things sometimes produce unimaginable complex results. This self-evident truth may not seem radical, but it was not until the mid-20th century that scientists fully understood the importance of this concept. At that time, the field of chaos theory was just emerging, and researchers discovered that systems with only a small number of feedbacks might produce random and unpredictable behavior. In 1976, Australian physicist, mathematician, and ecologist Robert May published an article entitled "Simple mathematical models with very complex dynamics" in the journal Nature. The paper of complicated dynamics proposed a logistic map, which can be written as: xn+1 = k * xn (1-xn) in mathematics. This is a classic example of chaos caused by simple nonlinear equations.

Xn represents a certain quantity in the current system, which feedbacks itself through the part described by (1-Xn). K is a constant, and xn+1 represents the system at the next moment. Although the equation looks simple, different values ​​of k can produce very different results, including some complex and confusing behavior. Robert May's unimodal map is used to explain the population dynamics in the ecosystem and can also generate random numbers for computer programming. (Ren Tian)

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