In what aspects of people's lives are logarithms mainly used?

Natural logarithm

When x approaches positive infinity or negative infinity, the limit of [1+(1/x)]^x is equal to e. In fact, e is Discovered through this limit. It is an infinite non-repeating decimal. Its value is approximately equal to 2.718281828...

It is represented by e

The logarithm with e as the base is usually used for ㏑

And e is also a transcendental number

e is used a lot in science and technology, and logarithms with base 10 are generally not used. With e as the base, many formulas can be simplified. It is the most "natural" to use it, so it is called "natural logarithm".

Whirls or spirals are very common forms of natural things, such as: a wisp of smoke rising into the blue sky, a ripple gently spreading in a blue lake, and several birds climbing slowly. The snails on the fence and countless stars dancing in the quiet night sky...

The aesthetic significance of spirals, especially logarithmic spirals, can be expressed in the form of exponentials:

φkρ=αe

Among them, α and k are constants, φ is the polar angle, ρ is the polar diameter, and e is the base of the natural logarithm. For the convenience of discussion, we define e or the form of e that undergoes certain transformations and combinations as "natural law". Therefore, the core of "natural law" is e, whose value is 2.71828..., which is an infinite loop number.

The beauty of "natural law"

"Natural law" is e and the form of e that has undergone certain transformations and compounds. e is the essence of "natural law". In mathematics, it is a function:

(1+1/x)^x

The limit when X approaches infinity.

When people study some practical problems, such as the cooling of objects, the reproduction of cells, and the decay of radioactive elements, they must study

(1+1/x)^x

(1+1/x)^x

p>

X to the power of X, the limit when X approaches infinity. It is this kind of finiteness obtained from infinite changes that develops in two opposite directions (when X tends to positive infinity, the limit of the above formula is equal to e=2.71828..., when The unique form obtained (equal to e=2.71828...) fully embodies the most essential things about the formation, development and decline of the universe.

Modern cosmology shows that the universe originated from the "Big Bang" and is still expanding. This description is consistent with the law of entropy, one of the two great discoveries of the second half of the nineteenth century, the first thermodynamic law. The two laws are consistent. The law of entropy points out that the evolution of matter is always in the direction of eliminating information and disintegrating order, and is a process of gradual degradation from complex to simple, from high-level to low-level. The limit of degeneration is disordered equilibrium, that is, the state of maximum entropy, a dead state of inaction. What does this process look like? It is not difficult to understand as long as we look at the pictures of spiral galaxies in astrophotography. If we must find the kind of efficient cause mentioned by Aristotle, then we can think of the universe as being organized by various pre-tightened clockworks, or simply think of the entire universe as a huge clockwork. History is nothing but the process of this clockwork constantly fighting for freedom and releasing energy.

The evolution of living organisms has the opposite characteristics. It is very different from the entropy trend described by the second law of thermodynamics. It enables living materials to avoid trends and environmental decline. Any life is a dissipative structural system. The reason why it can avoid the death state that approaches the maximum entropy is because the living body can continuously absorb negative entropy from the environment through metabolic processes such as eating, drinking, and breathing. What is essential in metabolism is that the organism succeeds in eliminating all the entropy it has to produce while it is alive.

"Natural law" on the one hand embodies the collapse process of natural systems that continue to disintegrate toward chaos (such as the decay of elements), on the other hand it shows that living systems can only go through an ordering process. In order to maintain its own stability and promote its own development (such as cell reproduction). It is precisely with this characteristic of combining order and disorder, vitality and death in the same form that "natural law" has important value in aesthetics.

If the barren and vast desert is a disordered and dead state of entropy increase according to the "law of nature", then the vast and vibrant grassland is an orderly and prosperous dynamic stable structure of the "law of nature".

Therefore, the desert makes people feel solemn, boundless, thoughtful, and reminds people of the difficulties and ups and downs in life; while the grassland makes people excited and excited, and makes people feel the joy and happiness of life.

e=2.71828... is a quantitative expression of "natural law". The visual expression of "natural law" is a spiral. There are usually five mathematical expressions of spirals: (1) logarithmic spiral; (2) Archimedean spiral; (3) interlocking spiral; (4) hyperbolic spiral; (5) cycloidal spiral Wire. Logarithmic spirals are the most common in nature, and other spirals are also related to logarithmic spirals. However, we have not found the general formula of spirals yet. The logarithmic spiral was introduced by Descartes in 1638. Later, the Swiss mathematician Jacob Bernoulli studied it in detail and found that the involutes and involutes of the logarithmic spiral are still logarithmic spirals, and the poles Tangents at various points of a logarithmic spiral are still logarithmic spirals, and so on. Bernoulli was so amazed by these interesting properties that he left a will to draw a logarithmic spiral on his tombstone.

The famous British painter and art theorist Hogartz deeply felt that vortexes or spirals gradually narrowing to their centers are beautiful shapes. In fact, we can easily find spirals in the works of ancient and modern art masters. Why is it that our senses, our "spiritual" eyes, often instinctively and intuitively find satisfaction in the form of such a spiral? Doesn't this mean that there is an isomorphic correspondence between our spirit, our "inner" world and the outer world that is more primitive than history?

We know that protein, as the basic material of life phenomena, participates in the entire life process in living objects. The reason why its function is so complex, efficient and mysterious is closely related to its structure. Chemists have found that the polytitanium chains of proteins are mainly helical, and the structure of nucleic acids, the material that determines heredity, is also helical.

The ancient Greeks had an instrument called the aeolian, which produced beautiful and sweet tones when its strings vibrated in the wind. This tone is what's known as the "vortex wake effect." What is thought-provoking is that the inner ear structure of the hearing organ that humans have evolved over a long period of time also has a spiral shape. Is this for the convenience of appreciating the ancient Greeks’ wind harp? There are also our fingerprints, hair whorls, etc. This kind of isomorphic correspondence between the physiological structure of the aesthetic subject and the external world is the natural basis for the harmony between "inner" and "external".

Some people say that the beauty of mathematics is the brilliance of "one", which has the invariance under the action of as many transformation groups as possible, that is, it has the expression of the ordinary laws of nature, and is the "many" and "one" The unity of nature, then the "natural law" also shines with the brilliance of "one". Who can tell clearly how much convenience and success e=2.71828... brings to mathematicians? People praise the strength, clarity and frankness of straight lines, and appreciate the grace, change and implicitness of curves. Little do they know that any straight line or curve can be composed of enough parts from the spiral. Some people say that beauty is the identity of subject and object, the unity of the inner spiritual world and the external material world, then "natural law" also has this unity. Human knowledge develops according to the law of negation of negation, and the history of society and nature also follows this law of dialectical development. What gives this form a vivid and vivid expression? Spiral!

Some people say that beauty lies in the rhythm of things, and "natural law" also has this rhythm; some say that beauty is dynamic balance and eternity in change, then "natural law" is also dynamic balance, Eternity in change; some people say that beauty lies in the dynamic structure of things, so "natural laws" also have this structure - such as hairsprings in watches, springs in machinery, etc.

"Natural law" is the unity of formal cause and dynamic cause, the image appearance of things, and the simultaneous expression of concreteness and abstraction. Finite life is rooted in infinite nature, and the pulse of life consciously adjusts its movement and rhythm according to the melody of the universe... organic and inorganic, internal and external, social and natural, everything is in harmony. And as one. Is this all the aesthetic secrets revealed by "natural law"? No! "Natural law" always has inexhaustible aesthetic connotations because it symbolizes the vast and profound nature. Because of this, it attracts and deserves people's unremitting exploration, thus showing the essential power of human beings' continuous evolution.

Whirls or spirals are very common forms of natural things, such as: a wisp of smoke rising into the blue sky, a ripple gently spreading in a blue lake, and several birds climbing slowly. The snails on the fence and countless stars dancing in the quiet night sky...

The aesthetic significance of spirals, especially logarithmic spirals, can be expressed in the form of exponentials:

φkρ=αe

Among them, α and k are constants, φ is the polar angle, ρ is the polar diameter, and e is the base of the natural logarithm. For the convenience of discussion, we define e or the form of e that undergoes certain transformations and combinations as "natural law". Therefore, the core of "natural law" is e, whose value is 2.71828..., which is an infinite loop number.

Numbers, are they beautiful?

1. The beauty of numbers

People have had a deep understanding of the beauty of numbers for a long time. Among them, the Pythagorean school, which was popular in ancient Greece in the sixth century BC, had more profound insights. They first studied the harmony of music rhythm from the perspectives of mathematics and acoustics, and found that qualitative differences in sounds (such as length, height, weight, etc.) are determined by differences in the number of articulators. For example, if the sounding body (such as a string) is long, the sound will be long; if the vibration speed is fast, the sound will be high; if the vibration speed is slow, the sound will be low. The fundamental principle of music therefore lies in quantitative relations.

The Pythagoreans extended the principle of harmony in music to other arts such as architecture and sculpture. They explored what proportions would produce beautiful effects and came up with some empirical norms. For example, the "golden rule" which has a long-lasting influence in Europe is said to have been discovered by them (some people say that it was Cai Mi who proposed the so-called "golden rule" in 1854. The so-called golden rule "is to take a line Divide it into two parts, so that the square of the longer part is equal to the shorter part multiplied by the whole line segment. ""If the length and breadth of something are composed according to this ratio, then it is smaller than a rectangle composed of other proportions. Be beautiful').

This group of scholars also applied the principles of mathematics and harmony to the study of astronomy, thus forming the concept of the so-called "music of the heavens" or "harmony of the universe", believing that the stars in the sky are moving in a certain orbit. , also produces a harmonious music. They also believe that the functions of the human body are also harmonious, just like a "small universe". The beauty of the human body is due to the proper proportions and coordinated movements of its parts - head, hands, feet, facial features, etc.; the beauty of the universe is due to the speed, distance, turnover time, etc. of each material unit and each star. Wait for coordination. These are the harmony of numbers.

Ancient Chinese thinkers also had similar views. The Taoist Laozi and Zhouyi's "Xici Zhuan" both tried to explain the generation of the universe through mathematics, which later evolved into the Zhouyi Xiangshu School. In the Book of Changes, the Bi hexagram represents the beauty of simplicity, the Li hexagram represents the gorgeous beauty, and the so-called "extreme numbers determine the appearance of the world" are all conclusions similar to mathematical reasoning. Xun Qing, a Confucian, also said: "All things are the same as the universe but have different bodies. They are not suitable but useful for human beings, and they are numbered." Zhuangzi treats "small self" and "big self" equally, and "small year" and "big year" are treated equally. Similar to the Pythagoreans, the "microcosm" and "macrocosm" corroborate each other. As the saying goes, "It is obtained by the hand and applied to the heart. The mouth cannot express it, and there is no way for it to exist in the meantime." This idea of ??seeing beauty from the harmony of numbers deeply influenced later generations of Chinese aesthetics.

2. The beauty of the Golden Rule

The Golden Rule has always been dyed with a magnificent and mysterious color, and is known as the most beautiful formal proportion of "natural and reasonable". We know that the golden rule is not only a principle of composition, but also the best state of natural things. The medieval Italian mathematician Faber Neisser discovered that many plant leaves, petals, and pine cone shell petals are arranged in an approximate value of 0.618:1 from small to large. This is the famous "Febnesser sequence": 1 , 2, 3, 5, 8, 13, 21, 34... The color patterns on animals also generally conform to the golden ratio. Dance coaches and gymnastics experts select proportions of sizes for talents, such as the ratio of shoulder width to waist, and the ratio of above and below the waist, which are generally in line with the golden ratio.

Modern scientists have also discovered that when the ratio of high-frequency and low-frequency "Beta" brain waves presented by the brain is an approximate value of 1:0.618 (the ratio of 12.9 Hz to 8 Hz), the human body and mind are at their best. With pleasure. Even when the ratio of nature's temperature (23 degrees Celsius) to human body temperature of 37 degrees Celsius is 0.618:1, it is most suitable for people's physical and mental health and makes people feel most comfortable.

In addition, the optimal proportions of ingredients, optimal proportions of organizational structures, etc. proposed by mathematicians for optimizing industrial and agricultural production systems are generally in line with the golden rule.

However, this does not mean that the Golden Rule has more aesthetic significance than the "Law of Nature". We can prove that when the logarithmic spiral:

φkρ=αe

is proportional to the golden rule, that is, k=0.0765872, and the proportional P1/P2=0.618, then the spiral The ratio of adjacent polar radii on the same radius line in a line has a golden section relationship. In fact, when the function f (X) is equal to e raised to the power of contains. In other words, the "Laws of Nature" encompass the Golden Rule.

The golden rule expresses the relative static state of things, while the "natural law" expresses the universal state of the movement and development of things. Therefore, in a sense, the Golden Rule is the solidified "Law of Nature", and the "Law of Nature" is the moving Golden Rule.

3. The beauty of "natural law"

"Natural law" is e and the form of e that has undergone certain transformations and compounds. e is the essence of "natural law". In mathematics, it is a function:

1 (1+——)

X to the power of X, when X approaches infinity limit.

When people study some practical problems, such as the cooling of objects, the reproduction of cells, and the decay of radioactive elements, they must study

1 (1+——)

X to the power of X, the limit when X approaches infinity. It is this kind of finiteness obtained from infinite changes that develops in two opposite directions (when X tends to positive infinity, the limit of the above formula is equal to e=2.71828..., when The unique form obtained (equal to e=2.71828...) fully embodies the most essential things about the formation, development and decline of the universe.

Modern cosmology shows that the universe originated from the "Big Bang" and is still expanding. This description is consistent with the law of entropy, one of the two great discoveries of the second half of the nineteenth century, the first thermodynamic law. The two laws are consistent. The law of entropy points out that the evolution of matter is always in the direction of eliminating information and disintegrating order, and is a process of gradual degradation from complex to simple, from high-level to low-level. The limit of degeneration is disordered equilibrium, that is, the state of maximum entropy, a dead state of inaction. What does this process look like? It is not difficult to understand as long as we look at the pictures of spiral galaxies in astrophotography. If we must find the kind of efficient cause mentioned by Aristotle, then we can think of the universe as being organized by various pre-tightened clockworks, or simply think of the entire universe as a huge clockwork. History is nothing but the process of this clockwork constantly fighting for freedom and releasing energy.

The evolution of living organisms has the opposite characteristics. It is very different from the entropy trend described by the second law of thermodynamics. It enables living materials to avoid trends and environmental decline. Any life is a dissipative structural system. The reason why it can avoid the death state that approaches the maximum entropy is because the living body can continuously absorb negative entropy from the environment through metabolic processes such as eating, drinking, and breathing. What is essential in metabolism is that the organism succeeds in eliminating all the entropy it has to produce while it is alive.

"Natural law" on the one hand embodies the collapse process of natural systems that continue to disintegrate toward chaos (such as the decay of elements), on the other hand it shows that living systems can only go through an ordering process. In order to maintain its own stability and promote its own development (such as cell reproduction). It is precisely with this characteristic of combining order and disorder, vitality and death in the same form that "natural law" has important value in aesthetics.

If the barren and vast desert is a disordered and dead state of entropy increase according to the "law of nature", then the vast and vibrant grassland is an orderly and prosperous dynamic stable structure of the "law of nature". Therefore, the desert makes people feel solemn, boundless, thoughtful, and reminds people of the difficulties and ups and downs in life; while the grassland makes people excited and excited, and makes people feel the joy and happiness of life.

e=2.71828... is a quantitative expression of "natural law". The visual expression of "natural law" is a spiral.

There are usually five mathematical expressions of spirals: (1) logarithmic spiral; (2) Archimedean spiral; (3) interlocking spiral; (4) hyperbolic spiral; (5) cycloidal spiral Wire. Logarithmic spirals are the most common in nature, and other spirals are also related to logarithmic spirals. However, we have not found the general formula of spirals yet. The logarithmic spiral was introduced by Descartes in 1638. Later, the Swiss mathematician Jacob Bernoulli studied it in detail and found that the involutes and involutes of the logarithmic spiral are still logarithmic spirals, and the poles Tangents at various points of a logarithmic spiral are still logarithmic spirals, and so on. Bernoulli was so amazed by these interesting properties that he left a will to draw a logarithmic spiral on his tombstone.

The famous British painter and art theorist Hogartz deeply felt that vortexes or spirals gradually narrowing to their centers are beautiful shapes. In fact, we can easily find spirals in the works of ancient and modern art masters. Why is it that our senses, our "spiritual" eyes, often instinctively and intuitively find satisfaction in the form of such a spiral? Doesn't this mean that there is an isomorphic correspondence between our spirit, our "inner" world, and the outer world that is more primitive than history?

We know that protein, as the basic material of life phenomena, participates in the entire life process in living objects. The reason why its function is so complex, efficient and mysterious is closely related to its structure. Chemists have found that the polytitanium chains of proteins are mainly helical, and the structure of nucleic acids, the material that determines heredity, is also helical.

The ancient Greeks had an instrument called the aeolian, which produced beautiful and sweet tones when its strings vibrated in the wind. This tone is what's known as the "vortex wake effect." What is thought-provoking is that the inner ear structure of the hearing organ that humans have evolved over a long period of time also has a spiral shape. Is this for the convenience of appreciating the ancient Greeks’ wind harp? There are also our fingerprints, hair whorls, etc. This kind of isomorphic correspondence between the physiological structure of the aesthetic subject and the external world is the natural basis for the harmony between "inner" and "external".

Some people say that the beauty of mathematics is the brilliance of "one", which has the invariance under the action of as many transformation groups as possible, that is, it has the expression of the ordinary laws of nature, and is the "many" and "one" The unity of nature, then the "natural law" also shines with the brilliance of "one". Who can tell clearly how much convenience and success e=2.71828... brings to mathematicians? People praise the strength, clarity and frankness of straight lines, and appreciate the grace, change and implicitness of curves. Little do they know that any straight line or curve can be composed of enough parts from the spiral. Some people say that beauty is the identity of subject and object, the unity of the inner spiritual world and the external material world. Then "natural law" also has this unity. Human knowledge develops according to the law of negation of negation, and the history of society and nature also follows this law of dialectical development. What gives this form a vivid and vivid expression? Spiral!

Some people say that beauty lies in the rhythm of things, and "natural law" also has this rhythm; some people say that beauty is dynamic balance and eternity in change, so "natural law" is also dynamic balance, Eternity in change; some people say that beauty lies in the dynamic structure of things, so "natural laws" also have this structure - such as hairsprings in watches, springs in machinery, etc.

"Natural law" is the unity of formal cause and dynamic cause, the image appearance of things, and the simultaneous expression of concreteness and abstraction. Finite life is rooted in infinite nature, and the pulse of life consciously adjusts its movement and rhythm according to the melody of the universe... organic and inorganic, internal and external, social and natural, everything is in harmony. And as one. Is this all the aesthetic secrets revealed by "natural law"? No! "Natural law" always has inexhaustible aesthetic connotations because it symbolizes the vast and profound nature. Because of this, it attracts and deserves people's unremitting exploration, thus showing the essential power of human beings' continuous evolution. (Originally published in the 1984 issue 4 of "Spring of Science" magazine, the original title was: "Laws of Nature - A Treasure for Aestheticians and Artists")

2. Euler's base formula of natural logarithms< /p>

(Approximately equal to the base of the natural logarithm of 2.71828 - e)

Eula is known as the Shakespeare of numbers. He is the most prolific mathematician in history and an important figure in various fields. (Including all branches of theory and application in mathematics, as well as mechanics, optics, acoustics, water conservancy, astronomy, chemistry, medicine, etc.) The scholar with the most publications.

The eighteenth century is called the "Eurasian Era" in the history of mathematics.

Eula was born in Switzerland. He lost the vision in his right eye at the age of 31 and became blind in both eyes at the age of 59. However, his optimistic personality and amazing memory and concentration allowed him to survive in a noisy environment with 13 children. can still think accurately about complex problems.

Yura remained humble throughout his life and never named the things he discovered after himself. Only the base of the natural logarithm, which is approximately equal to 2.71828, was named e by him. However, due to his extensive contributions to mathematics, important constants, formulas and theorems named after him are often seen in many branches of mathematics.

Many of the mathematical symbols we are accustomed to now were invented and introduced by Euler, such as: function symbols f (x), π, e, ∑, logx, sinx, cosx, and the imaginary number i, etc. High school teachers often use a joke about the base e of natural logarithms to help students remember a very special differential formula: In a mental hospital, there was a patient who said to others all day long, "I differentiate you, I differentiate you." I don’t know why, but these patients all have a simple concept of calculus. They always think that one day they will be differentiated to zero and disappear like ordinary polynomial functions, so they avoid them. However, one day he encounters He was surprised when he fell in love with an unmoved person, and the person said to him calmly, "I am e raised to the power of x."

This differential formula is: e regardless of the differential of x Several times, the result is still e! No wonder mathematics students use e as a metaphor for unwavering love!

Relative to π, which is the first letter of the circle in Greek letters, the origin of e is less well-known. Some people even think that Yura took the first letter of his name as the natural logarithm.

There are two generally accepted reasons why Yura chose e: one is that after the four commonly used letters a, b, c, d, etc., the first one has not been used frequently yet. The letter of is e, so he naturally chose this symbol to represent the base of the natural logarithm; one is e, which is the first letter of the exponent. Although you may suspect that the native language of Swiss Yula is not English, but in fact The exponents in French and German are both.