The meaning of the mathematical constant E

The meaning of the mathematical constant E

Nanyi

1.

E is an important constant, but I have never known what it really means.

It's not like π. As we all know, π represents the circumference of the circle and the diameter ratio of 3.14159, but if I ask you, E represents what. Can you answer it?

Wikipedia says:

"E is the base of the natural logarithm. "

But when you look at the "natural logarithm," you get the explanation:

"The natural logarithm is the base e logarithm function, and E is an irrational number, approximately equal to 2.718281828." "

This makes up a circular definition, without saying exactly what E is. Is it not strange that mathematicians choose such an irrational number as the base and claim that the logarithm is "natural"?

2.

Yesterday I read a good article, which explained the problem very clearly and understood it at a glance.

It says, what is E? Simply put, E is the limit of growth.

Here's how it's explained.

3.

Suppose there is a single-celled organism that splits once every 24 hours.

It is clear, then, that the number of such creatures is doubled every day. Today is 1, tomorrow is 2, and the day after that is 4. We can write a formula for increasing the number:

　　

The X in the formula represents the number of days. The total number of such organisms in the X-day is 2 x times. This formula can be changed to the following:

　　

Of these, 1 represents the original quantity, and 100% represents the growth rate per unit time.

4.

We continue to assume that every 12 hours, that is, halfway through the division, the newly created half of the cells can be split again.

As a result, 24 hours a day can be divided into two stages, each of which increases by 50% on the basis of the previous phase.

　　

At the end of the day, we got 2.25 cells altogether. Among them, one is original, one is new, and the other 0.25 is the split of the newborn cells into half.

If we continue to modify the hypothesis, this cell has the ability to split independently every 8 hours, that is, 1 days into 3 stages.

　　

So, in the end, we can get about 2.37 cells.

Naturally, if we further envisage that this division is continuous and that the newborn cells have the ability to continue splitting every second, how many cells can be obtained in a day?

　　

When n tends to infinity, the extremum of this equation equals 2.718281828 ....

　　

Therefore, when the growth rate is 100%, we can get up to 2.71828 cells per unit of time. Mathematicians refer to this number as E, which means the limit of the constant doubling of growth within a unit of time.

This value is the limit of natural growth, so the logarithm of base e is called the natural logarithm.

5.

With this value, it is very easy to calculate the compound interest of the bank.

Suppose there is a bank, the annual compound interest is 100%, please deposit 100 yuan, a year after the amount of money to take?

　　

The answer is 271.828 yuan, which equals 100 e.

However, in real life, the bank's interest is not so high, if the interest rate is only 5%, then 100 yuan to deposit a year can get how much money?

　　

For the sake of thought, we take n equals 50:

　　

We know that in the case of 100% interest rates, the value obtained by n=1000 is very close to E:

　　

Therefore, the 5% interest rate is equivalent to the 20 per cent of E:

　　

20 points is exactly equal to 5% of the interest rate, so we can rewrite the formula into:

　　

The rate on the formula represents growth. This means that e can be used for any calculation of growth rate, provided that it must be a continuous compound growth.

6.

Consider the time factor, how much money can you get if you deposit money in a bank for 2 years?

　　

In the case of time t, the general formula is:

　　

The equation is the universal formula for calculating growth, which can be applied at any time and at any rate of growth.

7.

Back to the above example, if the interest rate of the bank is 5% compounding, how long does it take to double the 100 yuan deposit?

　　

The result of the calculation is 13.86 years:

　　

The last equals sign indicates that the approximate time is doubled by 72 divided by the growth rate, which is the source of the 72 rule.

Finish

The meaning of the mathematical constant E