Is this the right question? Emergency service phone ... = 1? This problem has appeared countless times in online communities at home and abroad, and it can attract heated debates from hundreds of people every time. It is the most enduring old problem. In fact, in academic circles, this issue is also a famous hot topic of debate. Let's see how mathematicians look at this problem. The simplest proof is as follows: 1/3 = 0.333 ... and both sides are multiplied by 3, 1 = 0.999... 1998, Fred Richman's article "0.999 ... equals to 1"? "This proof is so convincing because people take it for granted that the first step is right, because the equation of the first step is taught like this since childhood." Professor David Tall also found in the survey that many students will turn to doubt the correctness of the first equation after reading this proof. If you think about it carefully, you will find that "1/3 equals 0.333…" and "1 equals 0.999…" are actually the same and unacceptable. Just as many people think that "0.999… can only get closer and closer to 1 but not completely equal to 1", the controversy of "0.333… infinitely close but not equal to 1/3" still exists. The problem has not been solved.

Another controversial proof David Foster Wallace introduced another famous proof in his book "Everything is not just":

Let x = 0.999 ... so 10x = 9.999 ... two expressions are subtracted to get 9x = 9, so x = 1 William byers commented on this proof in How Mathematicians Think: "0.999 ... can not only represent the process of adding infinite fractions, but also. Many students only value 0.999 ... as a process, but 1 is a number. How can a process be equal to a number? This is ambiguity in mathematics? 6? 8? 6? They didn't find that this infinite process can actually be understood as a number. Those students who think that the equation holds after reading the above proof may not really understand the meaning of infinite decimal, let alone understand the meaning of this equation. "

It is proved that the proportional series has such a property that if | r |

Then we have another quick proof:

This proof first appeared in Algebraic Elements by the great mathematician leonhard euler 1770, but at that time he proved that 10 = 9.999. ...

Later, a more formal proof of limit gradually appeared in mathematics textbooks:

1846, the American textbook "University Arithmetic" says: at 0.999 ..., every 9 is closer to 1. Another textbook 1895, Learning Arithmetic, says that if it is more than 9, it is almost the same as 1. Surprisingly, these "figurative statements" have the opposite effect. Students often think that 0.999 ... is actually less than 1.

With the deepening of people's understanding of real numbers, 0.999...= 1 has some further proofs. At 1982, Robert. G. Bartle and D.R. Herbert gave a proof of interval sets in Introduction to Real Analysis: given a set of interval sets, all these intervals contain only one point on the number axis; 0.999 ... corresponds to interval sets [0, 1], [0.9, 1], [0.99, 1], [0.999, 1] ... and the only intersection of all these intervals is/kloc-.

Fred Richman's article is "0.999 ... equals 1? In the book, Dai Dejin's division is used to prove that all rational numbers less than 0.999 are less than 1, and it can be proved that all rational numbers less than 1 are always different from 0.999 ... (hence less than 0.999 ...) after the decimal point, it means 0.999 ... and 1.

In "Comprehensive Textbook of Classical Mathematics: Contemporary Interpretation" published by H. B. Griffiths and P. J. Hilton in 1970, Cauchy series gives another proof.

The endless discussion has proved to be more and more complete, but the students' doubts have never diminished. In an investigation report by Professor Pinto and Professor David Toto, when students use advanced methods to prove this equation, they will be surprised to say that this is not right, 0.999 ... obviously it should be less than 1.

On the Internet, the charm of this equation remains undiminished. Debate whether 0.999… is equal to 1 has been rated as "the most popular sport" by the discussion group sci.math, and there will always be heated discussions among netizens on various question-and-answer websites. Richard feynman, the Nobel Prize winner, also made a joke with this equation. Once he said, "If I have to recite pi, I will recite it to 762 decimal places, and then I will say 99999 and so on, so I won't recite it." There is a strange joke behind this sentence: starting from the 762nd decimal place of π, there are six nines in succession, but a "wait a minute" here will make people feel as if all the nines are behind, which is equivalent to turning π into a finite decimal place. From then on, 762 bits after π decimal point were dubbed Feynman point.