# 0.999….is equal to 1 or not?

$$0.\bar{9}=1$$? This question shows up every year. For those who have been in math-related clubs or groups for a long time may have been a little tired of this question. This year, many students in the highschool math club on Facebook in Taiwan also had a fight for it. Then someone create a club called “0.9bar=1” and try to group these people. And they said, “One day the question shows up again, then we can easily put those “smart” people into this club to give everyone a quiet space.”

No matter you’re the people who enjoy trolling around or try to discuss the question seriously, there are more than 20 posts related to this topic this month in the math club on Facebook. It’s a $$0.\bar{9}$$ anniversary, right? Today MathPots also want to try to cook this stuff to everyone.( I want to put this article to the “soup” category originally, but I think it also meets the public tastes. Just need a little bit brain storm. So you can see this article in the “main course”. )

( I want to put this article to the “soup” category originally, but I think it also meets the public tastes. Just need a little bit brain storm. So you can see this article in the “main course”. )

## Some nouns and symbols

Before starting the topic, I want to introduce some nouns and symbols to you.

##### terminating decimal

A decimal number that has digits that end. Every terminating decimal can be expressed in the form of fraction. (So it’s rational number)

0.2851、4.1531235687130、0.11111111 the number like these are terminating decimal.

##### unlimited decimal

A decimal number that has digits that no end.

$$\pi = 3.1415926……$$、$$e=2.71828182846……$$ these are unlimited decimal.

##### Repeating decimal

A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero.

For example, 0.142857142857142857…… is a repeating decimal, it can be expressed as $$\displaystyle \frac{1}{7}$$.

In order to identify the characteristics of repeating decimals clearly, we add a line on the repeated portion (called bar).

$$\displaystyle \frac{1}{7}=0.142857142857142857……=0.\overline{142857}$$

##### Limit and converge

I could make a package of a meal for this topic XD. Hope you don’t mind that I take it briefly here:

We call a sequence $${x_n}$$, when the sequence extends to infinite (i.e., n is closed to $$\infty$$), if it will get more closed to a number (say $$x_0$$), we’ll write this situation as:

$$\lim_{n \to \infty}{x_n}=x_0$$

## $$0.\bar{9}=1$$?

Let’s start the topic. Is latex 0.\bar{9}=1\$? I can tell you, yes! And there are so many mathematicians have proved it in several different ways until today. The proof which is well-known among us is the versions that more acceptable by the public. Then why it shows up annually? One reason is that the public is not familiar with the proof method and the usual practices of mathematics. As a result, we can see some people try to prove it in a strange way (like some people want to use the concepts of biological chain or chemical molecule to explain why they are not equal. Umm, well….). But the biggest reason is that the concept of “infinity” is difficult for human brains to understand. You can see a lot of people still insist that “then why 0.00…001 disappeared? Where is it?”.

Well, all the proof I only believe is the cake on the knife!! (And I think I’ll be fired from MathPots in two days :P)

We don’t blame them, the concept of infinity is definitely not easy. David Hilbert, a German mathematician, once introduced a thought experiment which is also known as “Hilbert’s paradox of the grand hotel”. He want to show how hard it is wrap our mind around the concept of infinity. If you want to know more about the details, you can order a soup made by Alex – Hilbert’s paradox of the Grand Hotel-Welcome to the Hilbert’s hotel!

## Solutions in each stage

The following is solution sort by ordinary students at each stage.

##### The solution for elementary school students

Students in elementary school may find that some numbers are indivisible. Like the decimal part of 1÷3.
And students in this stage also know that we can divide a non-zero number on the both side of equal sign.

1÷3=0.33333333……
0.9999999……÷3=0.33333333……

The answer are the same, so 0.9999… = 1

Though this method is not rigorous at all, and there is a big flaw in logic. For example, how can you be sure that the decimal part of 1÷3 will not suddenly become a number other than 3 in a certain digit? However this method is more intuitive for students at this stage.

##### The solution for junior high school students

We have learned basic algebra concepts in junior high schools, so we can assign some numbers to some English letters.

$$\text{Let} \; a = 0.\bar{9}.\\ \text{Then} \;10a = 9.\bar{9}\\ (10a-a) = 9a = 9.\bar{9} – 0.\bar{9} = 9\\ \Rightarrow a=1$$
Therefore $$a = 0.\bar{9} = 1$$

##### The solution for high school students

We have the more complete concept of “limit” and “infinite” in senior high school, so let’s try it out!

$$0.999…\\ =\displaystyle{\lim_{n \to \infty}0.\underbrace{999…9}_{\text{n}}}\\ =\displaystyle{\lim_{n \to \infty}\sum_{k=1}^{n}\frac{9}{10^k}}\\ =\displaystyle \lim_{n \to \infty}\left( 1-\frac{1}{10^n} \right)\\ =1-\displaystyle{\lim_{n \to \infty}\frac{1}{10^n}}\\ =1$$

##### The solution for rogue

When you are fed up with the mathematical symbols and the complicated method, then you might like this method.

“infinite recurring decimal” is a key point, and we also know that the subtraction of two same number will be zero.

\begin{array}{r}
1.000000…… \\
-0.999999……\\
\hline
0.000000…… \\
\end{array}

After doing this equation, we know there is somewhere (i.e., the last digit) need to fill a 1. But the number 0.999999…… will become a terminating decimal when we finally filled a 1. Then, we don’t fill 1 until the end of 0.99999…… However that’s impossible! Because 0.9999… don’t have last 9, your answer will not have the last 1. Therefore $$1=0.\bar{9}$$.

##### Alex's Choice

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

Pulsing with drama and excitement, Infinitesimal celebrates the spirit of discovery, innovation, and intellectual achievement-and it will forever change the way you look at a simple line.

##### Seasoned with Chinese

◊ infinite – 無限
◊ repeating decimal or recurring decimal – 循環小數

Picture credit：
The feature picture is from Wikipedia.

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