In 2004, Google conducted an IPO. Larry Page and Sergey Brin decided that the total amount of IPO financing was US$2718281828, which is the first ten digits of 's natural constant (Natural Constant) e (e is approximately equal to 2.718281828). For those who are proficient in mathematics For the two founders, this should be based on the appreciation of the essential meaning of the natural constant e, to place expectations on Google's bright future.
names e as "natural base" and "natural constant", and calls logarithm with e as the base "natural logarithm", which is the supremacy of e by mathematicians in their own way Evaluation and respect.
What kind of growth pattern does this e-exponential growth represent and is valued or even loved by mankind? The natural constant e itself represents what kind of practical meaning and applicable value, and it is used by various natural sciences. Well, this is a history of logical processes such as imagination, induction, proof, deduction, and so on. It is also a story about the essential integration of human civilization including many disciplines of natural science and social science.
The problem of growth is a problem that human beings have always cared about. It is both about survival and determining development. In the process of research, people found that the compound growth problem can be expressed as
, when this compound growth is carried out in an uninterrupted manner (continuity), when x→+∞, it is the limit When it grows, we can get
, one of the meanings represented by e is the limit value of compound growth. Of course, the process of obtaining e is based on the deduction and induction of ancient mathematics, rather than the extreme calculation of modern mathematics. Although this limit is the essential benchmark expression of e on the problem of exponential growth (ie continuous compound growth), But it is only the definition of e. At the same time, each growth rate can be considered to have a certain proportional relationship with e (unit growth rate),That is, e is a standard unit of complex growth, similar to the value of "1" in the positive number system. e also specifies the upper limit of all "natural growth rates". Anything that continuously increases or decays at a constant rate R is After the time t has elapsed, its quantity is always eRt.
The research on the limit of growth actually has a source in the financial field—the problem of compound interest (a form of human economy in which interest is included in the principal after a certain interest period, and the interest is recalculated), although this is not a natural constant e as an appearance of an academic concept.
7000 years ago, Mesopotamia's Sumerians (Sumerians) established a commercial system and city-state, Sumerian because of developed agriculture and trade The Erren also first invented interest. Interest is the remuneration paid by the borrower to the lender for the use of funds. It is essentially a price, that is, the use price of funds within a certain period of time. Interest can effectively adjust the balance of supply and demand of funds. The invention of interest expands the three-dimensional physical commodity transaction into a four-dimensional cross-temporal commodity transaction. However, based on the fact that there was no current economic analysis perspective on the issue of interest at that time, and at the same time, influenced by religious ethics and other ideas, the debate on the rationality of interest became the main theme at that time. Later, people finally discovered that even if the interest is continuously taken out and then deposited together with the principal to recalculate the interest, if the basic annual interest is set to as high as 100%, then this limit method of obtaining interest (commonly known as "returning interest") is adopted. That is to say, interest is issued every instant and is included in the principal. Recalculate the interest (it can be regarded as a limit usury), and the principal of 1 yuan is deposited. The limit value of the sum of principal and interest after one year is a natural constant e, that is, the interest is "e" -1”, not infinite, and this kind of limit interest calculation method does not exist in reality. This result is expected to have a compromising impact on both sides of the debate. The related issues studied in the course of this debate are also the precursors of compound interest in the current financial system.
The above is just a figure of e in history, and the real academic framework about e has not yet been established at that time. The academic discovery process of e is because a large number of calculations encountered this logarithmic operation with e as the base.That is to say, in the history of mathematics, the logarithmic function was first established, and then the exponential function was defined. It can be said that the mathematical system at that time established operations, formulas, theorems, etc., so scientists gradually discovered e. This makes people can’t help but sigh that although mathematics is a self-enclosed subject, starting from the most primitive postulates, self-derivation, self-deduction, self-development, and self-enclosure, its system foundation is so solid and complete, even if all members of mathematics are not discovered , But already included in this perfect mathematics building, it is only a matter of time before we know it.
Many things in this world follow the following law of change: the growth rate is proportional to the size of the variable itself. For example, when a radioactive element decays, the decay rate is proportional to the amount of existing radioactive material; in a society with infinite resources, the birth rate will (approximately) be proportional to the existing population, and so on. The solution determined by this kind of change law is described by the exponential growth with e as the base: if the rate of change of x is equal to the λ times of the variable x itself, then the function of the variable with time t is
, where C is any constant.
In other words, in fact, the cell division problem in biology, the radioactive decay of elements in physics, the investigation of the age of the earth in geology, the speculation of the historical age in archaeology, population science The race density and so on are all the same essential problem, that is, the problem of exponential function, which can be expressed as the problem of probability density function in probability theory and mathematical statistics. At the same time, e seems to be the hand of God, which defines the growth of all things in the world. The upper limit of the speed.
e not only has the actual meaning of exponential function change, probability density function and continuous compound growth limit, but the geometric representation in the rectangular and polar coordinate system itself is natural. Contrast existence and aesthetic meaning, and essentially embodies philosophical meaning.
In ancient Greek philosophy as the source and cornerstone of the entire modern civilization (this article especially refers to scientific thinking and scientific thinking methods),"Nature" in the philosophical sense means that all things change naturally because of their origins, that is, the regularity of all things change, and the concept of "nature" is introduced into the social field to rationally analyze the phenomena and laws in society, and at the same time give "nature" "Giving aesthetic meaning, they believe that regularity essentially has a harmonious beauty. Among them, the ratio of mathematics is a kind of beauty that can only be realized by the mind (the outstanding performance is the "golden section"), and an exponential function with the base e. One of its characteristics is that its derivative results are invariant in form (including high-order derivation), which is a self-similar or holographic form of aesthetic expression. In addition, when changing to polar coordinate , the exponential function constitutes an equiangular spiral (the name is given because the angle between the spiral and the ray is always a fixed angle in polar coordinates), that is, each time the blue line passes through When ray, the included angle is fixed, that is, equiangular.
And this kind of existence shape and movement form are common in nature, such as the biological phenomenon of "moths fighting fire", because the moths fly according to the parallel light of the natural world with the original reference light source far away , Because as long as you fly at a fixed angle, you can fly in a straight line, which saves energy the most. However, due to the emergence of artificial light sources, the closer light source emits linearly in the center. The moth thought that flying at a fixed angle with the light is a straight motion, and it will fly into an equiangular spiral, and finally fly into the fire. This phenomenon is also called the "positive phototaxis" of biological insects by humans. In fact, it stems from the human influence on the natural biological world. In addition, for example, the deviation of the fluid from the linear motion based on the divergence field and the rotation of the earth is because the logarithmic spiral is equiangular. Affected by the environment, many linear motions will be converted into equiangular spiral motion. This is also the reason for the typhoon. And water whirlpool operation mode.
If this is only the formal aesthetic embodiment of e, then the philosophical significance of the natural constant e is that, just like for a perfect circle, π is natural and is an attribute of the circle itself , Then for the fastest exponential growth, e is natural, which is the inherent attribute of continuous compound exponential growth. At the same time, scientists also found that when doing mathematical analysis,Using e as the base logarithm ln x for calculation, its form is the simplest, and further calculations are also the most available.
A natural constant e runs through mathematics, philosophy, economics, finance, aesthetics, astronomy, demography, music, art, ethics, physics, chemistry, statistics, biology, archaeology, Geology, architecture, modern computer science and other disciplines, if we say that there are many related and intersecting fields between the above disciplines, can we say more accurately and with certainty that the essence of each discipline is one, but Is there a special division based on the normative nature of the research? The natural constant e may show us that what is based on the essence and can be succinctly expressed, in essence, is the most natural and rational beauty.
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