 # 0.999...is not equal to 1 in the end?

2021-04-06 | Return to Park |

Author | Yang Hao

01 How to rigorously prove that 0.999...=1?

Zhihu has a mathematical problem that triggered everyone’s discussion-"How to rigorously prove that 0.999...=1? The answers to this question are also varied and express their own opinions. The interesting thing about this question is that people with different levels of mathematics will haveDifferent understanding. In order to be able to discuss this question clearly later, we first compiled a few popular "shaking smart" answers on Zhihu.

Answer 1:

According to the PEP version of the fourth grade textbook: "How to compare the size of decimals? First compare the integer part, the larger the integer part, the larger the decimal; if the integer part is the same, then compare the decimal part...", obviously "0.999…<1 ", the two are not equal.

Answer 2:

Also based on the mutualization of elementary school scores and decimals,

There are 1/3=0.333... ,

So 1/3×3=0.333…×3,

i.e. 1=0.999^.

Answer 3: Use the ideas of junior high school algebra and equations,

Set x=0.999...,

Then 10x=0.999...,

So 10x-x=9x, 9x=9, x=1.

Answer 4: Use the idea of ​​summation and limit of high and middle ratio series to describe 0.999... : The above methods represent the mathematics knowledge level of elementary school, junior high school, and high school respectively. Within a certain range of knowledge and ability, these proofs seem to be correct.

So, which one is the proof of sufficient rigor? 0.999... is it really equal to 1? This goes back to the discussion of infinitesimals by mathematicians for hundreds of years.

02 Generation of infinitesimal quantities

The generation of infinitesimals originated from the creation of calculus in the 17th century. The birth of calculus was first to solve a series of natural science problems seeking the instantaneous rate of change, finding the tangent of a curve, etc., Newton Isaac Newton, 1643-1727 and Leibniz Gottfried Wilhelm Leibniz, 1646-1716 independently established the calculus theory system. In 1669, Newton first proposed his idea in the book "Analysis Using Infinite Polynomial Equations" this book was not published until 1711. Leibniz also introduced the same result. Newton and Leibniz both used the infinitesimal method. Although calculus was quickly popularized and widely used later, it could not conceal the lack of logic in this method.

Because the infinitesimal quantity whether it is Newton’s o or Leibniz’s dx has not been clearly defined, calculus soon ushered in a series of doubts-what is the meaning of infinitesimal quantity and 0?Difference? Why can the infinitesimal quantity be directly discarded in the inference process, but the sum of the infinitesimal quantity can be a finite quantity?

In response to these questions, Newton and Leibniz realized the problems of calculus and responded to them separately.

In 1671, Newton explained: Variables are produced by the continuous movement of points, lines, and surfaces. In 1676, he also said that the number of flows the rate of change of variables is the initial ratio of increments.

Leibniz wrote in a letter to Wallis in 1690: "It would be useful to consider such an infinitesimal quantity. When looking for their ratios, do not treat them as zero, but as long as they are incompatible withA lot of comparisons appear together, just discard them..."

It can be seen that they tried to clarify their theory, but the exact meaning of infinitesimal quantity is still very vague.

03 Infinitesimal disputes and resolution

At the beginning of the 18th century, the lack of rigor of calculus drew attacks from the church. For fear of the threat of mechanism and determinism to religion, the British Archbishop Berkeley published an article "Analytical Scholar" in 1734 and criticized Newton as "relying on duality".I got a wrong result that is not scientific but correct".

Mathematicians certainly cannot tolerate this kind of contempt for mathematics. They immediately joined the argument and continued to try to provide a rigorous basis for calculus. Although most of them failed, we cannot deny that they are getting the rightBefore the result, the contributions of some mathematicians cannot be ignored. Wallis put forward the concept of function limit in "Infinite Arithmetic", which gave birth to the bud of new ideas.

Euler liberated calculus from geometry, and made it based on arithmetic and algebra, opening the way for the fundamental argument of calculus based on the real number system. Furthermore, the concepts of derivative, integral, convergence, and infinite series have been strictly determined one by one, and the two-century controversy on infinitesimal quantities also known as the second mathematical crisis has finally ended.

But this is not the end of basic research. All related research work is based on the recognition of the real number system. The logical foundation of the real number system was gradually established in the second half of the 19th century. The founder of the real number system wasCantor also established set theory, on the basis of the rational number system, he introduced a new number class-real numbers.  Nowadays, real number theory has been further developed into real variable function theory, which has become an important branch of calculus. Real variable function is also one of the main courses for mathematics majors.

04 The relationship between infinitesimal, limit and high school mathematics

Now we can find that the problem of 0.999...<1 is essentially a mathematical problem, which reflects the denseness and completeness of real numbers. In other words, if these two numbers are not equal, then the real number theory and the calculus based on the real number system will collapse. In high school, we will also learn simple calculus knowledge, such asIn the calculation of derivative and definite integral, in mathematics, we can use derivative to solve the problem of the maximum value of a function; in physics, we can find the instantaneous velocity and acceleration according to the displacement function, and can also solve simple astrophysics motion problems. Therefore, the interpretation of derivatives in high school mathematics textbooks is actually a bit vague. In fact, in university mathematical analysis, we can learn the most clear and rigorous definitions of function continuity, derivative, and integral. Mathematics isThe most logical discipline, it took nearly 3 centuries for mathematicians to establish and perfect the theory of calculus. Even to this day, there are still some unresolved problems. I believe that after reading the article, you will be able to, Have a new understanding of mathematics, and intuitive feelings can sometimes lead to wrong results. In the process of thinking about problems in the future, we must strive to be like mathematicians, and strive to be rigorous and not paradoxical.

This article is reprinted from the WeChat public account "New Oriental Wisdom School" with authorization. The original title is "0.999... Isn't it equal to 1? More than 400 Zhihu answers, none of them are right."

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