Obviously they are all irrational numbers, why is π so much more famous than ta?

In the kingdom of mathematics, there are 5 very important numbers. Their content and functions far exceed the value itself, so they are more mysterious than ordinary numbers. These 5 numbers are 0, 1, π, i, and e. Like π, e is an irrational number. Its value is e=2.718281828459... infinite without looping.

In the beginning, it appeared in the calculation results by chance, but with the development of science, people gradually found that e has a lot of uses, especially when taking natural logarithm with it as the base, many calculation formulas can be simplified, Later, its application became more extensive. It can be said that e covers everything! It was the Swiss mathematician Jacob Bernoulli who really introduced e to mathematics research.

On December 27, 1654 (this is the old calendar at the time of birth, if calculated according to the new calendar, it should be January 6, 1655) Jacob Bernoulli was born in a merchant family in Basel, Switzerland. In the history of science, the Bernoulli family can really be called a gathering of scholars. Among the three generations of grandparents, eight world-class famous mathematicians have emerged. Among the eight people, there are also physicists, astronomers, and geographers.

Their achievements include the pioneers of infinite series calculation, calculus, and differential equation calculation, the pioneer of statistical probability theory, the creator of the Law of Large Numbers, in the infinite uncertainty decision problem, the headache The author of the St. Petersburg Paradox, the creator of the Bernoulli Theorem in fluid mechanics, and the famous scholar of curve research.

From an early age, Jacob developed a fascination with mathematics and astronomy. During the six years from 1676 to 1682, in order to learn the most advanced mathematics and science at that time, he traveled throughout Europe and followed Robert Boyle and Robert Hooke, Christian Huygens, Descartes, and many other masters. We have intensively read the papers and works of Frans Wan Schopenhauer, Isaac Barrow, and John Wallis.

In 1687, Jacob became a professor of mathematics at the University of Basel, where he worked for the rest of his life. In 1685, Jacob published a book on logic and probability, and in 1687, a book on geometry, in which he proved that any triangle can be divided by two lines perpendicular to each other into equal-area 4 blocks. In 1682 and 1704, Jacob published a total of 5 papers on the study of infinite series, and in 1689 he published the most important research results on infinite series and the theorem of large numbers in statistics.

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

The problem of compound interest is something that people often encounter in daily life. For example, when depositing a sum of money in the bank, when it matures, the principal plus interest will become the new principal and continue to be deposited at the original interest. This is called compound interest referred to as compound interest. Ordinary people may think that if you save in this way if you keep saving indefinitely, and your profits will get higher and higher, and even reach infinity.

But according to Jacob's reckoning, this is not the case. He wrote this problem into an infinite series, from which he proved that if the amount of money deposited at the beginning is 1, when the number of deposits is infinite, the sum of profits tends to a finite value, and this value is e! 1690, Bernoulli published this result in his series of papers. Many years later, on November 25, 1731, the great mathematician Lyonhard Euler talked about the number e in a letter to the mathematician Christian Goldbach, and gave it a name, called it "Natural number", and take it as the "base" of the logarithm to take the logarithm, and then there is the natural logarithm. The public appearance of e was 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 cited as a standard constant.

To Jacob's surprise, the strange number e not only appeared in the "compound interest" he calculated, but also repeatedly appeared in the sum of other infinite series, such as ∑(1 / n), ∑(1 / n's 2 power) in the summation of series; in addition, in the calculation of probability, Jacob also found that the value of the summation of an infinite series is the reciprocal of e; then in an infinite series called "hat keeping problem" In the summation, this value of e appears again. The problem of "keeping the hat" was a topic of interest to the mathematics community at that time. Because of the introduction of the e value, Jacob finally calculated it. This question is very interesting. It says that many guests are invited to a party, and everyone must hand over their hats to the doorman before entering the house, and he will put the hats in their boxes. Originally, the names of the guests were marked on each box, and the hats should be seated according to their number, but the janitor did not know these guests.

So, the question arises, when taking the hat, how many times do all the guests need to choose at most before they can find out their respective hats? Of course, the first person to take the hat is the most difficult. This is also a series summation problem. When the number of guests tends to be infinitely large, e appears in Jacob's calculation results. Then, in the calculation of the standard normal distribution, he discovered the value of e again. In the later period of Jacob's mathematical research, he was very fond of studying various curves, including parabolas, hyperbolas, and spirals. When studying the hyperbolic function y=1/x, when calculating the area contained under the curve, it meets the value of e again.

Later, Jacob studied the spiral and encountered e again by accident. There are 5 forms of spirals, logarithmic spiral, Archimedes spiral, chain spiral, hyperbolic spiral, and convolutional spiral, among which logarithmic spiral is the most common in nature. When studying the logarithmic spiral, Jacob discovered a very interesting phenomenon, the asymptote of the logarithmic spiral is also a logarithmic spiral, and the poles of the tangent lines at each point of the logarithmic spiral also constitute a logarithmic spiral, In a spiral structure, there are actually multiple layers of spiral structures. This wonderful feature made him amazed!

Logarithmic spirals are also favored by artists. The famous British painter and art theorist Hoggaz once deeply felt that the spiral shape that gradually shrinks to the center has indescribable beauty! Spirals often appear in famous paintings or murals left by ancestors. They represent the imagination of the ancestors on the entire universe, and also declare the feeling of beauty in the heart, and it is e that dominates the shape of the spiral! There is an inherent mathematical reason for the favor of lines. In biology, the structure of conch shells, the ordering of sunflower seeds, human fingerprints, and hair coils all exhibit helical characteristics.

As the basic substance of life phenomena, protein functions so efficiently in the whole process of participating in living organisms, and its mystery is also related to its helical structure. The polypeptide chain of protein is helical, and the substance that determines heredity nucleic acid structure is also helical, and the mystery in these helical structures is controlled by e. e also appears in physics, and e exists in the second law of thermodynamics that unknowingly governs the fate of nature; Rising, eagles soaring in the sky, there is e; when a piece of music sounds beautiful, if you study it carefully, you can find e from the rhythm; music is loved by people, and the air vibration produced by "music" is also Spiral trail; even after a long period of human evolution, the structure of the inner ear of the auditory organ is also spiral.

It seems that e is all-encompassing and omnipresent. At the heart of what humans love is always e at work. Although they are located in different places, they are all linked to this natural number e by different means. In 1690, when Jacob first introduced e, he estimated e to only the first decimal place; by 1748, when Euler used this value, it was accurate to the 23rd decimal place; in 1949, American physicist John von Neumann, using a computer, calculated e to the 2010th decimal place; on July 5, 2010, e showed a clearer look to the world, reaching the decimal place 1 000 000 000 000th places! One thing is for sure, no matter how hard we try, it is impossible for human beings to see its "true value". It seems that one of the internal reasons why nature cannot show its true appearance completely clearly is contained in irrational numbers like natural numbers e and π, which is the mystery of nature!

In Jacob's life, what he loved most was the logarithmic spiral. He thought it was the most magical and most desirable mysterious graph line. He asked that this curve be engraved on his tombstone and written in Latin Note his wish,

I will reappear in the same form