E: China's Mathematics Education

E: China's Mathematics Education

 

In our Chinese Mathematics Textbooks, we introduce the natural logarithm as follows: "If the base number of the logarithm is e, it is called natural logarithm, where E = 2. 71828 ..., is an irrational number." My classmates and I felt that the natural logarithm was too unnatural, but rather awkward. Although no statistics have been made, I think at least 90% of us as adults do not know why e is natural.

E comes from actual life, especially in biological split reproduction and astronomical computing. The following is a small example. It takes one day to assume that a cell will be continuously split and grow from just split to a complete cell. Calculate the number of cells in a day. Assume that the split cycle is one day, then there is no doubt that we will get two cells after one day, and then assume that the split cycle is half a day. Then we can get (1 + 1/2) * (1 + 1/2) = 2.25 in one day, and so on:

Split cycle 1/3 days: (1 + 1/3) 3 = 2.37037

Split cycle 1/4 days: (1 + 1/4) 4 = 2.44140

Split cycle 1/5 days: (1 + 1/5) 5 = 2.48832

Split cycle 1/6 days: (1 + 1/6) 6 = 2.52162

Split cycle 1/7 days: (1 + 1/7) 7 = 2.54649

Split cycle 1/8 days: (1 + 1/8) 8 = 2.56578

Split cycle 1/9 days: (1 + 1/9) 9 = 2.58117

Continue. The calculated data is as follows:

2.59374246

2.604199012

2.61303529

2.620600888

2.627151556

2.632878718

2.637928497

2.642414375

2.646425821

2.650034327

2.653297705

2.656263214

2.658969859

2.661450119

2.663731258

2.665836331

2.667784967

2.669593978

2.671277853

When the split cycle is 1/1000000 days, the number of cells obtained after one day is 2.71828. Are you familiar with this number? By the way, this is E. This split process actually exists in nature. It represents the number of cells we can get after a growth cycle when the classification cycle is infinitely small. We can find that, this number keeps increasing with the split cycle halved, but it is getting slower and slower, and will never reach a value. This value is E. In fact, this is also an important limit for advanced mathematics:

Lim (1 + 1/x) x (x ---> 0) = E

This limit is extremely important in calculus. the derivative of the exponent function AX = axlna is derived from this limit. The derivative means the slope. It can be seen that the slope of the exponent function is closely related to this E.

In basic elementary functions, an important constant in the triangle anti-triangle is π = 3. 14159265 ....., E = 2 is an important constant in the exponential logarithm function. 71828 ...., this is the law of memory, something constant in the mathematical system.

 

Another example is bank interest. If you save 1 yuan to the bank and the annual interest rate is 100%, you will get 2 yuan a year later. If the half-year interest rate is 50%, you will get 2.25 yuan a year later; if the annual interest rate of 1/3 is 33.3%, you will get 2.37037 yuan a year, but don't be so happy. Even if you make a profit on a daily basis, one year is (1 + 1/365) 365 = 2.714567482, it will never exceed E.

Do you think this E is natural now?

Our education will never tell you this. It will only remind you of the value, not to mention the development history of mathematics, whether the teacher did not know or intentionally did not let the students know. Another well-known irrational number is the circumference rate pi. This number is the ratio of circumference to diameter. It sounds natural, but it is not natural to calculate it. Most people think it is very simple, that is, the inner-cut polygon infinite approximation method. But I said, eldest brother, do you know how to calculate the polygon perimeter? Some things seem simple, but they are complicated. Please think about them.

In addition, why is Pi and e present? Why is it irrational data? Is it not so concise in nature? Actually not. The source of these problems is that we use decimal to calculate these values. They are designed by humans and invented by humans to explain the rules of nature, nature itself is a concise law, and different interpretations lead to different levels of interpretation formulas. in decimal notation, life is simple, and other things are not necessarily simple, as long as humans invent things themselves, there will be no perfection. Only the constant laws inherent in the objective world are the most concise and perfect.