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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), by Ji Yang

An irrational number is a number that cannot be expressed as the ratio of two integers, which seems to be "unreasonable". Many irrational numbers can be transformed into rational numbers or integers through basic operations, while some can not be "complete" no matter how hard they try. This paper introduces an irrational number, which itself is a combination of three irrational numbers, and researchers can only get its approximate value through a complex formula operation. But more importantly, we can verify it through mobile calculators!

Author Ji Yang (Institute of Semiconductors, Chinese Academy of Sciences)

As we all know, irrational numbers (irrational number) are unreasonable. According to integer standards, most of them are crippled, but many of them are idealistic and always want to pursue integrity: some rely on themselves, some rely on comrades.

For example, it's an irrational number, maybe it's an irrational number, but it's an integer, and it's "2".

For example, the natural constant e and pi are irrational numbers, but with the help of an imaginary number, they can become integers: ei π =-1; the result is a negative integer, which is not enough "2", otherwise there will be e2i π = 1.

Of course, not all efforts can be successful. The square roots of the natural constants e, pi π and 163 have also tried to be complete, but they are only a little short, very small.

How small is it?

The difference between it and an integer is that it is the 13th place after the decimal point, that is, it takes 12 nines after the decimal point to become 2.

Why is that? There is a very complicated mathematical theory that I can't understand, but I can quote a formula:

We can check whether it is correct or not with a calculator.

The calculator of the mobile phone is very powerful, especially its scientific calculator mode. For example, it can easily calculate the value of pi to 100 decimal places.

In order to facilitate your personal inspection, I will introduce the detailed operation process. First of all, turn on the calculator that comes with your phone (my phone is Huawei P30), which has two modes-the vertical version of the "standard calculator" and the horizontal version of the "scientific calculator", which can be switched at will.

When you enter the Scientific Calculator, you will automatically switch to the horizontal version, with two rows of display at the top and the keyboard at the bottom. Press the "π" key, and π appears in the first line of the display, while its value, "3.1415. 1971", is displayed in the second line. This line can display 42 numbers or characters. Through the touch screen operation, we can copy and paste the numbers.

Now, let's verify how many bits of pi that comes with the phone. Use pi to subtract the 42 digits just obtained, that is, "π-3.1415... 1971".

Press the minus key of the keyboard, and the first line of the display changes to "π -". Press the first line with your finger, and a shaded area will appear covering all the numbers, with two keys "copy" and "paste"; select "paste" (note, make sure the cursor is on the far right of the line) and you will get the result. If you continue the same operation, you can get more digits.

Select History from the menu to find previous calculations, such as the one I just did to calculate pi to 100 digits.

Is the calculation correct? I found 200 places after the pi decimal point on the Internet, as follows:

π = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 644629489 5493038196.

Compared with the picture above, you can see that the result of the mobile calculator has 115 decimal places, all correct. Using this method can obviously count more bits, but how many bits can be counted, I did not experiment, after all, it is too troublesome to copy and paste with mobile phone operation.

It is worth mentioning that this performance is better than the calculator that comes with my PC computer system. Using a similar operation, the calculator of the PC system can get "π = 3.14. 6939931148" (the 41-50 digits should be 6939937510), which is wrong starting at the 47th place after the decimal point.

Computer calculator calculates π-3.141. A value of 2795, which is incorrect at the 47th bit. Is the value of pi on the Internet necessarily correct? It may be correct, or it is possible that the mobile calculator specially retains many digits of pi, maybe where to hide it.

Next, let's do the math-- presumably this number will not be stored in advance. It turns out that the performance of the two calculators is still different. Careful observation shows that the first calculation results of the two calculators should be the same: the last four digits of the computer is 9871, while the corresponding position of the mobile phone is 9870, followed by nine digits "892347382". When you subtract with a computer calculator, the result is "- 1.076524." there is a minus sign in front of it. Considering the minus sign, there is a real difference between these two calculations up to the third-to-last place shown by our mobile phones, that is, 3 (mobile computing) and 4 (computer computing). In my opinion, compared with computers, we still believe in mobile phones. After all, it gives more digits, and the calculated value of pi is the same as that on the Internet.

Computer calculator calculation-12.767... The result of 781 (the approximate value of 30 decimal places).

The result of mobile calculator calculation has 39 decimal places.

Finally, let's do the math. First, review the previous formula:

I have done several different calculations with the scientific calculator of my mobile phone, which are the value calculated directly (using the previous method) and the approximate calculation with the formula. The results are as follows:

① calculates directly to get 18-digit integers and 23 decimal places: the first 12 decimal places are all 9, and the 13th is 2.

The first approximation of the ② formula, that is, the contribution of the first term.

The second order approximation of ③ formula, that is, the contribution of the first two terms.

The third-order approximation of ④ formula, that is, the contribution of the first three terms.

The second time ⑤ calculates directly, this time the 12 9s after subtracting 18 integers and decimal points.

The third direct calculation by ⑥ is the result of subtracting the first direct calculation, that is, 18 integers and 23 decimal places after the decimal point.

⑦ 's fourth direct calculation, minus the results of the first direct calculation and the third direct calculation, obtains the 99 decimal places after the decimal point.

A simple analysis can be drawn: direct calculation (①) is the simplest and roughest, but we have obtained the whole process of e, π and these three irrational numbers working together to pursue integrity; the formula does not tell the number of integers (which is easy to calculate), but gives more and more accurate decimal places (②, ③, ④). The calculator is more powerful than the first three terms of this formula (⑤ and ⑥) and can even be calculated to 99 decimal places (⑦). The results of the two calculations (formula calculation and direct calculation) are in good agreement, so we believe that the results of mobile phone calculation are correct.

I have also calculated with a computer calculator, although I can also witness the process of their pursuit of integrity, but dozens of places after the decimal point, still give different results.

Finally, let's talk about this formula. The fact that it is very close to integers is mentioned in some books on computational mathematics and is sometimes used as an example to illustrate the accuracy and stability of calculations. However, little is said about why it is so close to integers. And basically, first of all, this truth is not clear to ordinary people, because it involves elliptic curves, modular forms and other theories.

I found an introductory article [1], and I was confused, but it was not without success. In addition to finding the previous formula, I also found-- as you probably all know-- that mathematicians are good at mathematics, but they are not necessarily good at arithmetic and often make mistakes in symbols and so on. As the article shows, the symbol in front of "196884" in the last two lines of the formula before the success of the cheer is different, and one of them must be wrong-and I'm sure it's the last line.

"No matter whether you are careful or careful, you will not give way when you give a big gift." Even mathematicians can make mistakes, and it is not so surprising that the three efforts to pursue complete irrational numbers do not achieve particularly satisfactory results.

The last one has 99 decimal places

reference

[1] https://www.isibang.ac.in/~sury/episq163.pdf

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