One of its definitions is
Its numerical value is about (decimal place 100): "e ≈ 2.7182818284 59045 23536 02874 71352 66249 77572 47093 6995 95749 6999.
1690, Leibniz first mentioned the constant e in his letter. The constant E mentioned for the first time in this paper is a table in the appendix of John Napier's logarithmic works published in 16 18. But it didn't record this constant, only a list of natural logarithms calculated from it, which is generally believed to be made by William Oughtred.
It was Jacob Bernoulli who first thought that E was a constant. Euler also heard of this constant, so at the age of 27, E was "guaranteed" to calculus by publishing a paper.
The earliest known uses of the constant e are 1690 and 169 1 year, which is represented by B. In 1727, Euler began to use e to represent this constant. E was first used in publications, and it was Euler Mechanics in 1736. Although some researchers later used the letter C, E was widely used and finally became the standard.
The reason why E is used is really unknown, but it may be because E is the first letter of the word "index". Another view is that A, B, C and D have other common usages, and E is the first available letter. Another possibility is that the letter "e" refers to the first letter of Euler's name "Euler".
The important point of exponential function based on E is that its function is equal to its derivative. E is irrational number and transcendental number (see Lindemann-Weierstrass theorem). This is the first proved transcendental number, which is not deliberately constructed (compare Liouville number); Proved by Charles Hermite in 1873.
Actually, transcendental numbers mainly include natural constant (E) and pi. The popularity of natural constants is much lower than pi, because pi is easier to satisfy in real life, while natural constants are not commonly used in daily life.
Euler formula for fusing e and π
It is also the highest embodiment of the mathematical value of transcendental number e.
Natural constants are usually the base of power and logarithm in formulas. Why is this, mainly depends on its origin.
In French ordinal numbers, numbers consist of "e", for example, Première, and 2 1e is Vingt-première.
The method of obtaining natural constants is much simpler than pi. When is it?
time function
The limit of value.
Namely:
At the same time, it is also equal to
Pay attention to 1! It is also equal to 1.
Natural constants are usually used as the base of logarithm in formulas. For example, when taking the derivatives of exponential and logarithmic functions, natural constants should be used. function
The derivative of is
function
The derivative of is
Because e=2.7 1828 18284 ..., which is very close to the cyclic decimal 2.7 1828( 1828 cycle), then the cyclic decimal can be converted into 271801.
Extended data
1844, the French mathematician joseph liouville first speculated that e was a transcendental number. It was not until 1873 that the French mathematician Hermite proved that e was a transcendental number.
1727, Euler first used e as a mathematical symbol. Later, after a while, people decided to use E as the base of natural logarithm to commemorate him. Interestingly, e happens to be the first lowercase letter of Euler's name. Is it intentional or accidental coincidence? Now it can't be verified!
The application of e in natural science is not less than π value. For example, in atomic physics and geology, e is used to examine the decay law of radioactive materials or the age of the earth.
E will also be used when Tchaikovsky's formula is used to calculate rocket speed and when calculating the optimal benefits of savings and biological reproduction.
Like π, e will appear in unexpected places, such as: "How to divide a number into several equal parts to maximize the product of each equal part?" To solve this problem, we must deal with e, and the answer is: make the equal part as close as possible to the e value.
For example, 10 is divided into 10÷e≈3.7, but 3.7 is not easy to divide, so it is divided into 4, each of which is the largest product of104 = 2.5, 2.5 4 = 39.0625. If it is divided into 3 or 5 parts, the products are all the same. E just magically appeared.
1792, 15-year-old Gauss discovered the prime number theorem: "The percentage of prime numbers contained between 1 and any natural number n is approximately equal to the reciprocal of the natural logarithm of n; The larger n is, the more accurate this rule is. " This theorem was not proved by French mathematician Adama and Belgian mathematician Busan until 1896.
There are many advantages based on e, and it is best to compile a logarithmic table based on e; Calculus formulas have the simplest form. This is because only the derivative of e x is itself, that is, d/dx (e x) = e x.
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