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In the kingdom of mathematics, there are five numbers that are very important. Their content and role go far beyond the numerical value itself, so it is more mysterious than the general number. These five numbers are 0, 1, π, I and e.

Like pi, e is an irrational number. Its value is eBay 2.718281828459... Infinity without circulation.

At the beginning, it occasionally appeared in the calculation results, but with the development of science, people gradually found that e is of many uses, especially if it is used as the "base" to take the natural logarithm, many formulas can be simplified, and later, it is more widely used. It can be said that e is all-inclusive!

It was the Swiss mathematician Jacob Bernoulli who really introduced e into mathematical research.

Jacob Bernoulli was born on December 27, 1654 (this is the old calendar at birth, January 6, 1655 if calculated according to the new calendar). Jacob Bernoulli was born into a businessman's family in Basel, Switzerland.

In the history of science, the Bernoulli family can be called a gathering of scholars. In the three generations, there are eight world-class mathematicians. Among the eight are physicists, astronomers and geographers.

Their achievements include: the pioneer of infinite series calculation, calculus and differential equation operation, the pioneer of statistical probability theory, the founder of "law of large numbers", and in the choice problem of infinite uncertainty, the originator of the vexing St. Petersburg Paradox, the founder of the Bernoulli Theorem in fluid mechanics, the famous scholar of curve research, and so on.

Since he was a teenager, Jacob has developed a strong interest in mathematics and astronomy. During the six years from 1676 to 1682, in order to study the most advanced mathematics and science at that time, he traveled all over Europe and studied with many masters such as Robert Boyle and Robert Hook, Christian Huygens, Descartes and so on. I read carefully the papers and works of Frans Van Schopenhauer, Isaac Barrow and John Wallis.

In 1687, Jacob became professor of mathematics at the University of Basel and worked here for the rest of his life. In 1685, Jacob published books on logic and probability, and in 1687, he published another book on geometry, in which he proved that any triangle could be divided into four equal pieces of area by two lines perpendicular to each other. In 1682 and 1704, Jacob published five papers on infinite series, and in 1689 he published the most important research results of infinite series and the theorem of large numbers in statistics.

When studying infinite series in 1683, Jacob discussed an interesting problem of "compound interest" and unexpectedly found e!

The problem of compound interest is often encountered in people's daily life, such as depositing a sum of money in the bank, and after maturity, the principal plus interest becomes a new principal according to the original interest, which is called "compound interest", or "compound interest" for short. Most people may think that if they keep it in this way and keep it indefinitely, the profit will become higher and higher, so that it will reach infinity.

Source: pexels, but calculated by Jacob, this is not the case. He compiled the problem into an infinite series, which proved that if the amount of money originally deposited was 1, when the number of deposits was infinite, the sum of profits would tend to a finite value, and this value was e! 1690. Bernoulli published the results in his series of papers.

Many years later, on November 25, 1731, the great mathematician Leon hard Ora wrote to the mathematician Christian Goldbach about the number e, and gave it a name, "natural number". And take it as the "base" of the logarithm, and then there is a natural logarithm. E appeared publicly in a paper published by Euler in the journal Mechanics in 1736. After that, e began to have its own place in mathematics and was quoted as a standard constant.

To Jacob's surprise, the strange number e appears not only in his calculation of "compound interest", but also in the summation of other infinite series, such as the summation of ∑ (1 / n) and ∑ (power 2 of 1 / n). In addition, in the calculation of probability, Jacob also found that the sum of infinite series is the reciprocal of e. Then in the summation of an infinite series called the hat storage problem, this e value appears again.

The problem of "hat safekeeping" was once a topic of interest in the field of mathematics at that time. Due to the introduction of e-value, Jacob finally calculated it. This question is very interesting. it says that many guests are invited to a party. Before entering the house, everyone has to give the hat to the janitor, who will put the hat in his own box. Originally, the names of the guests were marked on each box, and the hat should be seated in the right seat, but the janitor did not know the guests. He put the hat at random and did not put it in the box according to the name.

Photo Source: pexels, the problem arises. How many times do all guests need to pick up their hats before they can find out their hats? Of course, the first person to take off a hat is the most difficult. This is also a problem of series summation. When the number of visitors tends to be infinite, there is a surprising e in Jacob's calculation. Then, in the calculation of the standard normal distribution, he discovered the e value again. In the later period of Jacob's mathematical study, he was very fond of studying all kinds of curves, including parabola, hyperbola and helix. When we study the hyperbolic function yqi1 / x, it meets the e value when calculating the area under the curve.

Later, Jacob studied the helix and ran into e again. There are five forms of helix, logarithmic helix, Archimedes helix, interlocking helix, hyperbolic helix and cyclotron helix, among which logarithmic helix is the most common in nature. When studying the logarithmic helix, Jacob found a very interesting phenomenon. The asymptote of the logarithmic helix is also a logarithmic helix, and the poles of the tangents at each point of the logarithmic helix also form a logarithmic helix. In a helix structure, he was amazed at the wonderful feature that contains multiple levels of spiral structure.

The logarithmic spiral is also favored by artists. Hogaz, a famous British painter and art theorist, once deeply felt that the spiral gradually shrinking to the center had its indescribable beauty!

Spirals often appear in famous paintings or murals left by their ancestors, they represent the ancestors' imagination of the whole universe, but also declare the feeling of beauty in the heart, and it is e that dominates the spiral form! It seems that there are inherent mathematical reasons for human doting on spirals.

In biology, the structure of conch shell, the sequence of sunflower seeds, human fingerprints and hair spins all show the characteristics of spiral.

The structure of conch shell is the protein of the basic material of life phenomenon, its function is so efficient in the whole process of life, and its mystery is also related to its spiral structure. The polypeptide chains of proteins are helical, and the nucleic acid structures that determine heredity are also helical, and the mysteries of these helical structures are controlled by e.

E also appears in physics, and there is e in the second law of thermodynamics, which unwittingly controls natural fate; in nature, from spiral nebulae and spiral galaxies, typhoons and hurricanes, to a wisp of smoke curling up and eagles soaring in the air, there is e; when a piece of music sounds beautiful, you can also find e in the rhythm. Music is loved by people, and the air vibration produced by "music" is also a spiral wake; even after a long period of evolution, the structure of the inner ear of human auditory organs is also spiral.

The airflow shape of typhoons and hurricanes seems to be omnipresent and omnipresent. E always plays a role in the beloved core of mankind. Although they are in different places, they are all linked to this natural number e in the same way.

When Jacob first introduced e in 1690, his estimate of e was only the first place after the decimal point; by 1748, when Euler used this value, it had been accurate to the 23rd place after the decimal point; in 1949, American physicist John von Neumann, using a computer, calculated e to the 2010 decimal place. By July 5, 2010, e showed a clearer face to the world, reaching the 1,000,000,000 place after the decimal point! One thing is certain, no matter how hard it is, it is impossible for mankind to see its "true value". It seems that one of the internal reasons why nature cannot fully show its true appearance lies in irrational numbers such as natural numbers e and π, which is the mystery of nature!

In Jacob's life, what he loved most was the logarithmic spiral, which he thought was the most magical and coveted mysterious line. He asked that this curve be engraved on his tombstone and marked his wish in Latin: "I will reappear in the same pattern."

Jacob's tombstone Jacob's "soul" is accompanied by e, hidden in his beloved hyperbolic helix!

Source: 365th days in the History of Science, slightly deleted by author: Wei Fengwen Wu Yi part of the picture source network copyright belongs to the original author Editor: Zhang Runxin this article comes from Wechat official account: Origin Reading (ID:tupydread), author: Wei Fengwen, Wu Yi, Editor: Zhang Runxin

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