Do you know that these mathematical equations have changed the process of human civilization?

Tech 2023-05-08 11:40:37 Source: Network

This article is reposted from Turing Education and [Encounter Mathematics] has been authorized to forwardMathematical formulas are a wonderful thing that lurks around us and is not easily noticeable. It is incredibly concise, yet it can describe the vast world

This article is reposted from Turing Education and [Encounter Mathematics] has been authorized to forward

Mathematical formulas are a wonderful thing that lurks around us and is not easily noticeable. It is incredibly concise, yet it can describe the vast world.

From radio broadcasting to smartphones, from mapping to satellite navigation, from world travel to space exploration, equations play a crucial role. Pythagoras' theorem, the law of universal gravitation, Maxwell's equations, chaos theory, relativity... The great equations contain the laws of nature, the mysteries of the universe and subtle scientific thinking, which show the wisdom of our predecessors and open the door for future exploration. This year, Ian Stewart, a popular science writer, published his famous book 17 Equations that Change the World. Now let's list them and see what you have mastered?

1. Pythagorean theorem (Pythagorean theorem)

This theorem is the foundation of our understanding of geometry. It describes the relationship between several sides of a right triangle in the plane: two short sides a and b, the sum of their squares equals the square of the long side c.

To some extent, this equation distinguishes our usual Euclidean geometry from the non Euclidean geometry of surfaces. For example, a right triangle drawn on a sphere does not follow the Pythagorean theorem.

Bringing surveying and navigation, as well as the more recent emergence of special and general relativity - the best existing theories about space, time, and gravity.

2. Logarithmic equation

The logarithmic equation can be understood as the inverse formula of the exponential equation. It aims to find the power of a base number to obtain a given quantity. For example, a logarithm with a base of 10 is represented as log (1)=0, because here 1=100; Log (10)=1, because 10=101; Naturally, log (100)=2.

The formula log (ab)=log (a)+log (b) in the figure shows the most useful function of logarithmic equations: converting multiplication into addition. Before the widespread application of modern digital computers, this had always been a convenient means of quickly calculating large number multiplication, playing an important role in physics, astronomy, and engineering calculations.

Bringing efficient methods for calculating astronomical phenomena such as solar eclipses and planetary orbits. A method for quickly conducting scientific calculations. Slide rule, the faithful companion of engineers. Radioactive decay and psychophysics about human perception.

3. calculus

The formula in the figure is the definition of derivative in calculus. The derivative can be understood as the rate of change of a quantity. For example, we can consider velocity as the derivative of displacement. If our walking speed is 4 kilometers per hour, then our displacement changes by 4 kilometers per hour.

In fact, many studies focus on how things change. Derivative and integral (another important formula of calculus) are fundamental tools for mathematicians and scientists to understand changes.

Can be used for tangent and area calculations. Formula for solid volume and curve length. Newton's laws of motion, differential equations. Law of Conservation of Energy and MomentumMost of mathematical physics.

4. Law of Universal Gravitation

Newton's law of universal gravitation describes the gravitational action F between two objects. Where G is the constant of universal gravitation, m1 and m2 represent the masses of two objects, and r is the distance between objects. In the history of science, Newton's stroke has a pivotal position. It not only explains the gravitational forces on Earth, but also almost perfectly interprets the way planets operate. This has extended to the solar system and even the entire universe.

Newton's law of universal gravitation, as a classic, led physics for over 200 years until Einstein's theory of general relativity was replaced.

Can accurately predict solar eclipses, planetary orbits, comet returns, and galaxy rotations. Artificial satellites, Earth surveys, Hubble Space Telescope, solar flare observations. Interplanetary probes, Mars rovers, satellite communication and television, and global positioning systems.

5. Plural

Mathematicians have been subdividing numbers. Natural numbers, negative numbers, decimals, real numbers... Later, imaginary unit i appeared, which represents the square root of - 1. People are only beginning to know the plural.

Mathematically speaking, complex numbers are extremely elegant. This algebraic structure nicely solves our need that any equation has a complex solution. This is certainly impossible for real numbers. Calculus has also been extended to the complex number, by which we found the wonderful characteristics of these numbers, such as symmetry. These attributes are an important foundation for electronics and signal processing.

A better method for calculating trigonometric tables. Promote almost all mathematics to the complex field. Use more powerful methods to understand waves, heat, electricity, and magnetism. The mathematical foundation of quantum mechanics.

6. Euler's polyhedron theorem

A polyhedron is a three-dimensional version of a polygon, much like a cube to a square. Each corner of a polyhedron is called a vertex, the lines connecting the vertices are called edges, and the polygons formed by the edges are faces.

A cube has 8 vertices, 12 edges, and 6 faces. Let's calculate the number of vertices plus the number of vertices, and then subtract the number of edges, 8+6-12=2. Euler's polyhedron theorem tells us that as long as a regular polyhedron is given, the number of vertices plus the number of faces and then subtracting the number of edges will always result in 2. No matter how many faces it has.

This discovery is the first thing we later called topological invariant. In topological invariant, some attributes and quantities of objects of the same type are similar to each other. For all "regular" polyhedra, V+F-E=2. This principle and Euler's solution to the "seven bridge problem of Konesburg" laid the foundation of topology. This branch of mathematics has significant implications for modern physics.

One of the most important and powerful fields in pure mathematics: topology, which studies the invariant geometric properties under continuous deformation, such as surfaces, knots and links. Most applications are indirect, but their role behind the scenes is crucial. It helps us understand how enzymes act on deoxyribonucleic acid (DNA) in cells and why the motion of celestial bodies may be chaotic.

7. normal distribution

The normal probability distribution diagram is similar to a bell curve and is widely used in statistics.

Physics, biology, and sociology all widely use normal curves as models for different research objects. The main reason why it is so widely used is that it can describe the behavior of a large number of independent processes.

Inspired the concept of 'ordinary people', tested the significance of experimental results (such as medical experiments), and (unfortunately) assumed a tendency to form a bell shaped curve, as if no other distribution existed.

8. wave equation

The wave equation describes the behavior of waves, such as the vibration of a guitar string, the ripple of a stone thrown into a lake, or the light of an incandescent bulb. Wave equation is the most typical representative of hyperbolic partial differential equation. With the development of technology, solving this equation also opens a door for people to understand other differential equations.

Our understanding of water waves, sound waves, light waves, elastic vibrations, etc. has made a leap...... Seismologists use its improved version to infer its internal structure from the way the earth vibrates. Oil companies use similar methods to search for oil. It is also applied to predict the presence of electromagnetic waves, leading to radio, television, radar, and modern communication.

9. Fourier transform

Fourier transform is a method of understanding complex waveforms, such as those in human speech. A complex and chaotic acoustic function like human speech can be divided into several simple waveform combinations through Fourier transform. This greatly simplifies the analysis process. The Fourier transform can be called the core of modern signal processing, analysis, and data compression.

Fourier's techniques are widely used, such as image processing and quantum mechanics. It is used to discover the structure of large biological molecules such as DNA, compress image data in digital photos, clean up old or damaged recordings, and analyze earthquakes. Modern technology is used to efficiently store fingerprint data and improve medical scanners.

10. Navier Stokes equation

Like the wave equation, this is a differential equation. The Navi Stokes equation describes the behavior of fluids, such as water flowing through pipes, airflow passing over wings, or smoking from cigars. At present, people can obtain approximate solutions to equations and simulate fluid motion well through computers. However, whether an exact solution to the Navier Stokes equation can be obtained mathematically remains an unresolved issue.

Modern airliners, fast and quiet submarines, first-class equation racing at high speed on the track, and medical advances for venous and arterial blood flow. The computer method used to solve this equation, called Computational Fluid Dynamics (CFD), is widely used by engineers for technological improvements in these fields.

11. Maxwell's equations

This set of partial differential equations describes the behavior and relationship between electric field (E) and magnetic field (H).

Maxwell's equations are as important to classical electromagnetism as Newton's laws of motion and gravitation are to classical mechanics. They are the foundation for understanding electromagnetic phenomena in our daily lives. However, we know that modern physics has a quantum mechanical explanation of electromagnetism. Although these beautiful formulas are very applicable in the macro world, they are only approximate expressions.

It is predicted that electromagnetic waves exist and travel at the speed of light, so light itself is an electromagnetic wave. It propelled the invention of wireless connections for radio, radar, television, computer equipment, and most modern communication technologies.

12. Second Law of Thermodynamics

This law can be expressed as that in a closed system, entropy (S) is always stable or increasing. Roughly speaking, entropy in thermodynamics is a measure of the degree of chaos in a system. A system is initially orderly. If a high-temperature area is adjacent to a low-temperature area, the non-uniform state will tend to become uniform, meaning that heat will flow from the high-temperature area to the low-temperature area until it is evenly distributed.

The second law of thermodynamics is one of the few laws in physics that is related to time. Most physical processes are reversible, and we can easily invert the equation without any impact. However, the second law of thermodynamics can only proceed in one direction. If we put an ice cube into Hot Coffee, we will only see the ice cube melt, and never see the coffee freeze.

Better steam engine, estimation of renewable energy efficiency, scenario of "heat death", proof that matter is composed of atoms, and contradictory connection with arrow of time.

13. Relativity

Einstein revolutionized the process of physics with his special and general theories of relativity. This classic equation indicates that mass and energy are equal. Special relativity tells people that the speed limit in the universe is the speed of light, and objects moving at different speeds also experience different times.

General relativity regards gravity as the curled and folded spacetime itself. This is the first significant change in our understanding of gravity since Newton's law of universal gravitation. General relativity is the foundation for our understanding of the origin, structure, and ultimate destiny of the universe.

Of course, there is a brand new physics. Nuclear weapons... okay, maybe - but not as straightforward or straightforward as rumors suggest. Black holes, Big Bang, GPS, and satellite navigation.

14. Schr dinger equation

This is the main equation in quantum mechanics. General relativity explains our universe at the macro level, while this equation dominates the behavior of atoms and subatomic particles at the micro level.

Quantum mechanics and general relativity are the two most outstanding theories in history. At present, all phenomena observed in experiments are consistent with these two major theories. Quantum mechanics is also the foundation of many modern technologies, such as nuclear energy, semiconductor computers, and lasers.

The physics that describes the world is completely revised on a very small scale, where each particle has a "wave function" that describes the probability cloud of possible states. At this level, the world is inherently uncertain. It attempts to connect the microscopic quantum world with the macroscopic classical world, leading to philosophical issues that still have an impact today. But in experiments, quantum theory works very well. Without it, there wouldn't be today's computer chips and lasers.

15. Information Theory

This equation is Shannon information entropy. Similar to the thermodynamic entropy mentioned above, this is also a measure of the degree of chaos. It measures all information content that can be expressed, such as a book, a JPEG image on the internet, and so on. Shannon information entropy gives the lower limit of the extent to which we can lossless compress information.

This theory has sparked mathematical research in informatics, which is the foundation of our online communication today.

Efficient error detection code and error correction code are used for everything from CD to space probe. Applications include statistics, artificial intelligence, cryptography, and analysis of DNA sequences.

16. chaos theory

This formula is the unimodal mapping of biologist Robert May. It initially described that over time, the population size would change from X to Xt+1. Given a constant k, the foreground image will be chaotic: with X as the starting value, the evolution process is one way; But starting with another quantity, the evolution process will be completely different, even if this quantity is very close to X.

As we can see, chaotic behavior is very sensitive to initial conditions. Weather change is a classic example - small changes in atmospheric conditions today will lead to a completely different meteorological system in a few days, which can also be understood as the butterfly effect.

Recognize that simple nonlinear equations can create very complex dynamics, and there may be hidden orders under apparent randomness. This discovery is often called chaos theory and has numerous applications in the whole science, including the movement of planets in the solar system, weather forecasting, population dynamics in ecology, variable stars, earthquake modeling, and the efficient orbit of space probe.

17. Black Paul Scholes formula

As another differential equation, Black Paul Scholes formula describes how financial experts and traders price financial derivatives. Financial derivatives such as stocks are an important component of the modern financial system.

Based on the properties of underlying assets and derivatives, Black Paul Scholes formula can help people calculate the value of these financial products.

The financial industry has grown on a large scale, financial instruments have become increasingly complex, the boom during economic prosperity has ultimately collapsed, the stock market turmoil of the 1990s, the financial crisis from 2008 to 2009, and the ongoing economic recession.