Natural constant of e

The natural constant e is a wonderful number. Here, e is not only a letter, but also an unreasonable constant in mathematics, which is approximately equal to 2.500000000 1.50005438+0.

But have you ever wondered how this happened? Why is an irrational number called "natural constant"?

Speaking of e, we will naturally think of another unreasonable constant. By estimating the side length of the inscribed polygon and the circumscribed polygon in the figure below, we can understand it vividly.

Suppose the diameter of a circle is 1, and the perimeter of its circumscribed polygon and inscribed polygon can form the upper and lower bounds of the estimated value. The more sides of an inscribed polygon and an circumscribed polygon, the narrower the range. As long as there are enough sides, the upper and lower limits of the range will be closer.

If the calculation is intuitive, what about e? Therefore, a graphic method is also used here to understand E intuitively.

First of all, we should know that the symbol E representing the natural cardinal number was named after Euler's initials "E" by Swiss mathematician and physicist leonhard euler.

Leonhard euler (1707- 1783)

But in fact, the first person who discovered this constant was not Euler himself, but Jacob Bernoulli.

bernoulli family

Bernoulli family is a famous family in Switzerland in the18th century, among which there are many famous mathematical scientists. Jacoby Bernoulli is johann bernoulli's brother and johann bernoulli is Euler's math teacher. In short, bosses are inextricably linked.

One of the most intuitive ways to understand the origin of E is to introduce an economic name "compound interest".

Compound interest method (English: compound interest) is a method to calculate interest. According to this method, the interest will be calculated according to the principal, and the newly earned interest can also earn interest, so it is commonly called "rolling interest", "donkey rolling" or "overlapping interest". As long as the interest period is closer, the wealth will increase faster and the term will be longer, and the compound interest effect will be more obvious. 3354 Wikipedia

Before introducing the compound interest model, try to look at the more basic exponential growth model.

As we know, most bacteria reproduce by dichotomy, assuming that a certain bacteria will divide once a day, that is, a growth cycle is one day, as shown in the figure below, which means that the total number of bacteria per day is twice that of the previous day.

Obviously, if divided by x days (or x growth cycles), it is equivalent to x times. On day x, the total number of bacteria will be twice as large as before. If the initial bacterial count is 1, the bacterial count after x days is 2x:

If the initial number is k, the number of bacteria after x days is K 2x:

So as long as all bacteria divide once a day, no matter what the initial number is, the final number is 2x of the initial number. So it can also be written as:

The above formula means: on the X day, the total number of bacteria is Q times the initial number of bacteria.

If "splitting" or "doubling" is replaced by a more literary statement, it can also be said that "the growth rate is 100%". Then we can write the above formula as follows:

When the growth rate is not 100%, but 50%, 25% and so on, you only need to change the above formula from 100% to the growth rate you want. In this way, we can get a more general formula:

The mathematical connotation of this formula is: the growth rate in one growth cycle is r, and after x cycles of growth, the total amount will be q times the initial amount.

The above is a simple example of exponential growth. Let's take a look at Jacob Bernoulli's findings:

Suppose you have 1 yuan in the bank. At this time, there has been serious inflation, and the bank's interest rate has soared to 100% (exaggerated for the convenience of calculation). If the bank pays interest once a year, naturally after one year, you can get the principal of 1 yuan (blue circle) and the interest of 1 yuan (green circle), with a total balance of two yuan.

At present, the annual interest rate of the bank remains unchanged, but in order to attract customers, the bank has introduced a policy of benefiting the people and paying interest once every six months. Then in the sixth month, you can get 0.5 yuan's interest from the bank in advance.

If you are wise, you will immediately deposit the interest of 0.5 yuan in the bank again, and the interest of 0.5 yuan will also generate interest in the next settlement cycle (red circle). The technical term is "compound interest", so the balance of deposits at the end of the year is equal to 2.25 yuan.

At this point, we can look at this problem from another angle: that is, each

The settlement (growth) period is half a year, and the interest rate is 50% for half a year (or 100%/2). Interest is settled twice a year, and interest is deposited immediately after the first interest settlement. At this point, our calculation formula and results are as follows:

Go on, suppose the bank does not want to make money in the short term in order to compete with other banks, and pays interest every four months! But you are smart enough to save the interest as soon as you get it, which is similar to paying the interest once every six months: that is, every four months, the interest rate is 33.33% (or 100%/3), the interest is paid three times a year, and all the interest is paid immediately after the first two payments.

At this point, the calculation formula and results are as follows:

My God, although the annual interest rate has not changed, the money you can get from the bank at the end of the year is actually increasing with the increase of the annual interest payment!

So will it increase to infinity? Nice try, huh?

Now, suppose depositors and banks are crazy. On the premise of ensuring the annual interest rate 100%, banks continue to pay interest to depositors. Depositors stay in the bank every day, and when they get interest, they deposit it in the bank. The interest thus obtained is called "continuous compound interest".

However, you will find that there seems to be a "ceiling" that blocks your small goal of earning 1 billion yuan. This "ceiling" is E!

If we do a series of iterative operations, we will see the following results:

Where n refers to the number of interest settlement in one year.

As long as the annual interest rate remains unchanged at 100%, the balance will approach e =2.7 1828 1845?

Then, finally, we can sacrifice an important limitation of calculating E in calculus of advanced mathematics:

Looking back at this important limit now, I think there will be a more intuitive understanding.

That is to say, even if the annual interest rate of the bank is 100%, no matter how much you ask the bank to "compound interest", it is impossible to get a balance of more than e times the principal at the end of the year. In addition, I have never seen any bank with an annual interest rate of 100%.

Although normal banks will not introduce preferential policies of continuous compound interest, in essence, most things are in a state of "unconscious continuous growth". For a thing that keeps growing, if the growth rate per unit time is 100%, then it will become e times the original after one unit time. Biological growth and reproduction is similar to the process of "profit rolling".

For another example, in an equiangular spiral:

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If expressed in polar coordinates, its general mathematical expression is:

Where a and b are coefficients, r is the distance from the point on the spiral to the origin of coordinates, and θ is the rotation angle. This is an exponential function based on the natural constant e.

For example, the cross section of Nautilus shell presents a beautiful equiangular spiral:

Nautilus shell

The tropical depression also looks like an equiangular spiral:

tropical depression

Even the spiral arms of spiral galaxies are like equiangular spirals:

spiral galaxy

Perhaps this is why E is called "natural constant". Of course, the wonder of the natural constant e is far more than that, and you can't finish reading a book.

Reference:

An intuitive guide to exponential function & ampe, /articles/an- an intuitive guide to exponential function -e/

[2] Prehistoric calculus: discovery pi, /articles/ Prehistoric calculus-discovery pi/

[3] compound interest, https://en.wikipedia.org/wiki/Compound_interest

[4] leonhard euler of https://en.wikipedia.org/wiki/leonhard _ Euler

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