The pursuit of complete irrational numbers: challenging the limits of mobile phone calculators

Irrational numbers are numbers that cannot be expressed as the ratio of two integers, which seems to be somewhat "unreasonable". Many irrational numbers can be turned into rational numbers or integers through basic operations, while some cannot be made "complete" no matter how hard we try. This article introduces an irrational number that is a combination of three irrational numbers. Researchers can only get its approximate value through a complex formula calculation. But more importantly, we can verify it with a mobile phone calculator!

Written by Ji Yang (Institute of Semiconductors, Chinese Academy of Sciences) We all know that irrational numbers are "irrational". According to integer standards, most of them are incomplete, but many of them are very idealistic and always want to pursue completeness: some rely on themselves, and some rely on their comrades.

The phone's calculator is quite powerful, especially its scientific calculator mode. For example, it can easily calculate the value of pi to 100 decimal places.
To make it easier for everyone to check it out for themselves, I will introduce the detailed operation process. First, open the calculator that comes with your phone (my phone is Huawei P30). It has two modes - the vertical version of "Standard Calculator" and the horizontal version of "Scientific Calculator", which can be switched at will.

Entering the "Scientific Calculator" will automatically switch to the horizontal version, with a two-line display on the top and a keyboard on the bottom. Press the "π" key, and π will appear on the first line of the display, while its value "3.1415...1971" will be displayed on the second line. This line can display 42 numbers or characters. Through touch screen operation, we can copy and paste numbers.

Now, let's verify how many digits the phone's built-in π actually calculated. Subtract the 42-digit number we just obtained from π, which is "π-3.1415...1971".

Press the minus key "-" on the keyboard, and the first line of the display will change to "π-". Press the first line with your finger, and a shaded area will appear covering all the numbers, and two keys "Copy" and "Paste" will appear at the same time; select "Paste" (note that the cursor must be on the far right of this line), and you will get the result.

Continue the same operation to get more digits.
Select "History" from the menu and you can find the calculations you did previously, such as the calculation of π to 100 digits that I just did.

Is the calculation correct? I found the value of π to 200 decimal places 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 6446229489 5493038196.

Comparing it with the picture above, you can see that the result of the mobile phone calculator has 115 decimal places, all correct. Obviously, this method can calculate more decimal places, but I didn't test how many decimal places it can calculate, after all, it is too troublesome to copy and paste with a mobile phone.

It is worth mentioning that this performance is better than the calculator that comes with my PC system. Using similar operations, the PC system calculator can get "π = 3.14......6939931148" (the 41st to 50th digits should be 6939937510), but it is wrong from the 47th digit after the decimal point.

A computer calculator calculated the value of π-3.141…2795 and found an error at the 47th digit.
Is the value of π on the Internet correct? It may be correct, or it is possible that the mobile phone calculator has specially reserved many digits of π, perhaps hidden somewhere for backup.

are not the same. A careful observation shows that the first calculation results of the two calculators should be the same: the last 4 digits of the computer's result are 9871; and the corresponding position of the mobile phone is 9870, followed by 9 digits "892347382". When the computer calculator is used for subtraction, the result is "-1.076524...", with a minus sign in front. Considering the minus sign, there is indeed a difference between the two calculation results until the third to last digit displayed on our mobile phone, that is, 3 (mobile phone calculation) and 4 (computer calculation). I think that compared to computers, we should trust mobile phones, after all, they give more digits, and the calculated π value is the same as the result on the Internet.

I also used a computer calculator to calculate it, and although I could see the three of them pursuing completeness, they still gave different results in the tens after the decimal point.

I found an introductory article [1], which was confusing, but not without benefit. In addition to finding the formula used above, I also discovered that - as you may know - mathematicians are good at math, but not necessarily good at arithmetic, and often make mistakes with symbols. As shown in the article, the symbols before "196884" in the last two lines of formulas before the author cheered for success were different. One of them must be wrong - and I am sure it was the one in the last line that was wrong.

References

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

This article is supported by the Science Popularization China Starry Sky Project

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.

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