Euler (Leon Hard Euler, 1707- 1783)? A famous mathematician, Swiss, spent most of his time in Russia and France. Master's degree at the age of 65,438+07. He began to study mathematics in his early years with the appreciation of the mathematical genius Benuri, and studied mathematics after graduation. He is the most prolific writer in the history of mathematics. He published more than 700 papers before his death, leaving more than 100 to be published after his death. His works involve almost all branches of mathematics. He also solved the famous Seven Bridges problem. ? He is a master of creating mathematical symbols. Firstly, the function is represented by f(x), the addition is represented by sigma, and the imaginary unit is represented by I. 1727, firstly, e is quoted to represent the base of natural logarithm. ? There are two Euler formulas? One is about polyhedron? If the number of convex polyhedrons is f, the number of vertices is v and the number of edges is e, then V-E+F=2, which is called Euler characteristic. ? The other is about series expansion? E (I * x) = cos (x)+I * sin (x)。 Where I is the square of the imaginary unit I =- 1.
When x approaches positive infinity or negative infinity, the limit of [1+( 1/x)] X is equal to e, which is actually discovered through this limit. It is an infinite cyclic decimal. Its value is approximately equal to 2.7 1828 1828. ...
With e?
Logarithms based on e are usually used for ㏑?
And e is still a transcendental number?
E is widely used in science and technology, and the logarithm with the base of 10 is generally not used. Taking e as the base can simplify many formulas, and it is the most "natural", so it is called "natural logarithm"
Vortex or spiral is a very common form of natural things, such as: a wisp of smoke curling into the blue sky, a ripple swaying gently on the blue lake, several snails climbing slowly on the fence, and countless stars dancing in the quiet night sky. ...
The aesthetic significance of spiral, especially logarithmic spiral, can be expressed in exponential form:
φkρ=αe
Where α and k are constants, φ is polar angle, ρ is polar diameter, and E is the base of natural logarithm. For the convenience of discussion, we define the form of e or by e as "natural law". Therefore, the core of "natural law" is e, and its value is 2.7 1828 ..., which is an infinite cyclic number.
The Beauty of "Natural Law"
Is the "law of nature" E? And the form of e after some transformation and synthesis. E is the essence of "natural law", which is a function in mathematics;
( 1+ 1/x)^x
The limit when x approaches infinity.
People should study some practical problems, such as the cooling of objects, the reproduction of cells and the decay of radioactive elements.
( 1+ 1/x)^x
The x power of x, the limit when x approaches infinity. It is this finiteness obtained from infinite change that develops from two opposite directions (when X tends to positive infinity, the limit of the above formula is equal to E = 2.7 1828 ..., and when X tends to negative infinity, the result of the above formula is also equal to E = 2.7 1828 ...), which fully embodies the formation of the universe.
Modern cosmology shows that the universe originated from the Big Bang and is still expanding. This description is consistent with the entropy law, that is, the second law of thermodynamics, which is one of the two great discoveries in the second half of19th century. The law of entropy points out that the evolution of matter is always in the direction of destroying information and disintegrating order, and gradually degenerates from complex to simple and from high to low. The limit of degradation is disorderly balance, that is, the state of maximum entropy, a state of inaction and death. What is this process like? It is not difficult to understand just by looking at the photos of spiral galaxies in astrophotography. If we must find the motive force that Aristotle said, then we can regard the universe as a clockwork mechanism that is wound up in advance, or simply regard the whole universe as a huge clockwork. History is just the process of this clockwork constantly striving for freedom and releasing energy.
The evolution of organisms has opposite characteristics, which is very different from the entropy described by the second law of thermodynamics. It enables organisms to avoid the decline of trends and environment. Any life is a dissipative structure system, and the reason why it can avoid the death state approaching the maximum entropy is because life can continuously absorb negative entropy from the environment through metabolic processes such as eating, drinking and breathing. The essence of metabolism is to let the organism successfully eliminate all the entropy it has to produce when it is alive.
On the one hand, the "law of nature" embodies the collapse process of natural systems in the direction of chaos (such as the decay of elements), on the other hand, it shows the essence that life systems can maintain their stability and promote their development (such as cell reproduction) only through an orderly process. It is precisely because of this feature that order and disorder, vitality and death are in the same form, and "natural law" has important aesthetic value.
If the barren desert is the entropy increasing state of "natural law" in disorder and death, then the vast and vibrant grassland is the dynamic and stable structure of "natural law" in order and prosperity. Therefore, the desert makes people feel solemn, boundless and thought-provoking, reminding people of all kinds of difficulties and bumps in the course of life; Grassland, on the other hand, makes people excited, happy and feel the joy and happiness of life.
E = 2.7 1828 ... is a quantitative expression of "natural law". The image expression of "natural law" is a spiral. There are five mathematical expressions of spiral: (1) logarithmic spiral; (2) Archimedes spiral; (3) interlocking helix; (4) Hyperbolic spiral; (5) spiral. Logarithmic spiral is the most common spiral in nature, and other spirals are also related to logarithmic spiral, but at present we have not found the general formula of spiral. The logarithmic spiral was introduced by Descartes in 1638. Later, Swiss mathematician Jacob Bernoulli made a detailed study on it, and found that the involute and extension line of logarithmic spiral are still logarithmic spiral, and the tangent of each point and pole of logarithmic spiral is still logarithmic spiral, and so on. Bernoulli was surprised by these interesting properties and left a will on his tombstone to draw a logarithmic spiral.