Did you know that these mathematical equations changed the course of human civilization?

This article is transferred from Turing Education, [Meet Math] has been forwarded with permission.

Mathematical formulas are a wonderful thing, they lurk around us, and they are not easy to detect. It's incredibly concise, but it describes the world.

From radio broadcasts to smartphones, from mapping to satellite navigation, from world travel to space exploration, equations play a vital role. Pythagorean theorem, the law of gravitation, Maxwell's equations, chaos theory, relativity... The great equations contain the laws of nature, the mysteries of the universe and exquisite scientific thinking, showing the wisdom of predecessors and opening the door for future exploration. This year, popular science writer Ian Stewart's famous book "17 Equations That Changed the World" was published. Now let's list them and see what do you have?

1. Pythagorean theorem (Pythagorean theorem)

This theorem is the basis of our understanding of geometry. It describes the relationship between several sides of a right triangle in a plane: two short sides a and b, whose squares add up to the square of the long side c.

In a way, 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.

This led to mapping, navigation, and, more recently, special and general relativity—the best available theories of space, time, and gravity.

2. Logarithmic equations

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

The formula log(ab) = log(a) + log(b) shows one of the most useful functions of logarithmic equations: converting multiplication to addition. Prior to the widespread use of modern digital computers, this had been a convenient means of quickly calculating the multiplication of large numbers, playing an important role in physics, astronomy, and engineering calculations.

Brings efficient methods for calculating astronomical phenomena such as solar eclipses and planetary orbits. A fast way to perform scientific calculations. A faithful companion to engineers - slide rules. Radioactive decay and psychophysics about human perception.

3. Calculus

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

In fact, a lot of research looks at how things change. Derivatives and integrals (another important formula for calculus) are fundamental tools for mathematicians and scientists to understand change.

It can be used for tangent and area calculations. Solid volume and curve length formulas. Newton's laws of motion, differential equations. Law of conservation of energy and momentum. Much of mathematical physics.

4. The law of gravitation

Newton's law of universal gravitation describes the gravitational action F between two objects. where G is the gravitational constant, m1 and m2 represent the mass of the two objects, and r is the distance between the objects. In the history of science, Newton's stroke has a pivotal position. Not only does it explain gravity on Earth, but it also explains almost perfectly how the planets move. This has extended to the solar system and even the entire universe.

Newton's law of universal gravitation led physics as a classic for more than 200 years, until Einstein's general theory of relativity appeared.

Accurately predict solar eclipses, planetary orbits, comet returns, galaxy rotations. Sputnik, Earth Survey, Hubble Space Telescope, Solar Flare Observation. Interplanetary probes, rovers, satellite communications and television, global positioning systems.

5. Plural

Mathematicians have been subdividing numbers, natural numbers, negative numbers, decimal numbers, real numbers... Later, the imaginary unit i appeared, which represents the square root of -1. That's when people began to know the plural.

Mathematically speaking, complex numbers are extremely elegant. This algebraic structure nicely solves our need - any equation has a complex solution. This is of course impossible for real numbers. Calculus is also extended to complex numbers, and we discover the wonderful qualities of these numbers, such as symmetry. These properties are an important foundation for electronics and signal processing.

A better way to calculate trigonometric tables. Generalize almost all mathematics to the field of complex numbers. A more powerful way to understand waves, heat, electricity, and magnetism. Mathematical foundations of quantum mechanics.

6. Euler's polyhedral theorem

A polyhedron is a three-dimensional version of a polygon, much like a cube is to a square. Each corner of a polyhedron is called a vertex, the lines of 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 do the math, add the number of vertices to the above number, and then subtract the number of edges, 8+6-12=2. Euler's polyhedral theorem tells us that given a regular polyhedron , then the number of vertices plus the number of faces minus the number of prisms , the result must be 2. No matter how many faces it has.

This discovery is the first thing we later call topological invariants. In topological invariants, some properties and quantities of objects of the same type are similar to each other. For all "regular" polyhedra, V+F-E = 2. This theorem and Euler's solution to the " Königsberg Seven Bridges problem " laid the foundation of topology. This branch of mathematics has important implications for modern physics.

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

7. Normal distribution

The normal probability distribution plot 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 objects of study. 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 the "ordinary man", the test for significance of experimental results (such as medical experiments), and the (unfortunately) tendency to form bell curves by default, 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 ripples of a stone thrown into a lake, or the light of an incandescent light bulb. The wave equation is the most typical representative of hyperbolic partial differential equations, and as technology develops, solving this equation also opens the door to understanding other differential equations.

Our understanding of water waves, sound waves, light waves, elastic vibrations, etc. has taken a leap forward... Seismologists use an improved version of it to infer the Earth's internal structure from how it vibrates. Oil companies use a similar method to find oil. It was also used to predict the presence of electromagnetic waves, which led to radio, television, radar and modern communications.

9. Fourier transform

The Fourier transform is a method of understanding complex waveforms, such as those of human speech. A sound wave function as complex and chaotic as human speech can be split into combinations of several simple waveforms by means of a 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 used in a wide range of applications, such as image processing and quantum mechanics. It is used to discover the structure of large biomolecules such as DNA, compress image data in digital photographs, clean up ancient 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 Navier-Stokes equation describes the behavior of fluids, such as water flowing through a pipe, air flowing across a wing, or smoke coming out of a cigar. At present, people can get approximate solutions to equations and be able to simulate fluid motion well through computers. However, whether an exact solution to the Navier-Stokes equation can be obtained mathematically remains an unsolved problem.

Modern airliners, fast and quiet submarines, Formula 1 racing cars that stay on the track at high speeds, and medical advances against venous and arterial blood flow. The computer method used to solve this equation, called computational fluid dynamics (CFD), is widely used by engineers for technical refinement in these areas.

11. Maxwell's equations

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

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

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

12. The Second Law of Thermodynamics

The law can be stated that entropy (S) is always stable or increasing in a closed system. Roughly speaking, entropy in thermodynamics is a measure of how chaotic a system is. A system is initially ordered, and if a high temperature region is next to a low temperature area, the non-uniform state will tend to become a uniform state, that is, the heat will flow from the high temperature region to the low temperature region until it is evenly distributed.

The second law of thermodynamics is one of the few laws related to time in physics. Most physical processes are reversible, and we can turn the equation back without any effect. 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 engines, estimates of renewable energy efficiency, "heat death" scenarios, proof that matter is composed of atoms, and a contradictory connection to the arrow of time.

13. The theory of relativity

Einstein revolutionized the course of physics with his special and general theories of relativity. This classic equation shows that mass and energy are equivalent. Special relativity tells us that the speed limit in the universe is the speed of light, and that objects moving at different speeds travel different amounts of time.

General relativity views gravity as the curled and folded space-time itself. This is the first major change in our understanding of gravity since Newton's law of universal gravitation. General relativity is fundamental to our understanding of the origin of the universe, its structure, and ultimately its destiny.

Of course, there is a whole new physics. Nuclear weapon...... Well, maybe—but not as straightforward or blunt as anecdotal anecdotal. Black holes, the Big Bang, GPS and satellite navigation.

14. Schrödinger equation

This is the main equation in quantum mechanics. General relativity explains our universe at the macroscopic level, and this equation governs the behavior of atoms and subatomic particles at the microscopic level.

Quantum mechanics and general relativity are two of the most remarkable theories in history. All the phenomena observed in the current experiment are consistent with these two theories. Quantum mechanics is also fundamental to many modern technologies, such as nuclear energy, semiconductor computers, and lasers.

A complete correction of the physics that describes the world at very small scales, where each particle has a "wave function" that describes a cloud of probability of possible states. At this level, the world is inherently uncertain. Its attempt to connect the microscopic quantum world with the macroscopic classical world led to philosophical questions that still have influence today. But experimentally, quantum theory works so beautifully that without it, today's computer chips and lasers would not be possible.

15. Information theory

This equation is known as the Shannon information entropy. Similar to the thermodynamic entropy described above, this is also a measure of the degree of chaos. It measures everything that can be expressed, such as a book, a JPEG picture on the Internet, and so on. Shannon's information entropy gives a lower limit to how much we can compress information losslessly.

This theory gave rise to the mathematical study of informatics, which is the basis of our online communication today.

Efficient error detection and error correction codes for everything from CDs to space probes. Applications include statistics, artificial intelligence, cryptography, and analyzing DNA sequences.

16. Chaos theory

This formula is biologist Robert May's unimodal mapping. It originally described the evolution of the population from X to Xt+1 over time. Given the constant k, then the foreground graph will be chaotic: with X as the starting value, the evolution process is one way; But with another quantity as a starting value, 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 changes are a classic example – small changes in atmospheric conditions today will lead to a very different weather 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 that there may be hidden orders under apparent randomness. The discovery, often referred to as chaos theory, has countless applications throughout science, including planetary motion in the solar system, weather forecasting, population dynamics in ecology, variable stars, seismic modeling, and efficient orbits for space probes.

17. The Black-Scholes formula

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

Based on the properties of the underlying asset and derivatives, the Black-Scholes formula can help one calculate the value of these financial instruments.

The financial sector grew massively, financial instruments became more complex, booms culminated in collapse, stock market turmoil in the 90s of the 20th century, financial crises from 2008 to 2009, and persistent recessions.

Did you know that these mathematical equations changed the course of human civilization?