0.999... = 1

[Agent Smith]

So I was exploring the kiddie section of math and encountered this:

3.0013849483127...
and
2.9986150516872...

We add these two and get 5.9999999999999... = [imath]5.\overline 9[/imath]

I "know" that [imath]0.\overline 9 = 1[/imath]. So [imath]5.\overline 9 = 5+ 1 = 6 = 3.0013849483127... + 2.9986150516872...[/imath]

This is the first problem I've worked on where [imath]0.\overline 9 = 1[/imath] shows up. It's a special result for me therefore. Question: How often does this happen in math? Is it trivial, commonplace or is it rare and should I dig a little deeper into the matter?

 

[fresh_42]

You cannot know what the dots stand for. They indicate some limit, and as soon as you calculate with limits instead of written numbers, the [imath] 0.\bar 9 [/imath] and alike will disappear. So, yes, it is rare since we usually speak about limits instead of dots.

 

[Agent Smith]

Have you ever encountered a number like [imath]5.999... = 5.\overline 9[/imath] in your mathematical life? If yes, where exactly?

Also, correct, [imath]\displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}= 1[/imath], but in my experience, to get a result like [imath]5.999...[/imath] is once a blue moon.

 

[fresh_42]

Have you ever encountered a number like [imath]5.999... = 5.\overline 9[/imath] in your mathematical life? If yes, where exactly?

In school. I played around with what periods mean, i.e. divisions by ##3,6,7,9,11## and so on, and in online forums where kids discuss whether [imath] 0.\bar 9 [/imath] and [imath]1 [/imath] are equal or not.

Also, correct, [imath]\displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}= 1[/imath], but in my experience, to get a result like [imath]5.999...[/imath] is once a blue moon.

This formula means [imath]\displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}= \lim_{n \to \infty}\sum_{k = 1} ^n \frac{9}{10^k}=1,[/imath] i.e. it is a limit and a limit is a unique number - so it exists. The question about the deficits of the decimal system (or any other) doesn't occur in mathematics. It's only a notational artifact.

I even like to say, that mathematicians actually only need very few numbers: [imath] \{\pm 2 ,\pm 1,0,\pi ,e , i\}. [/imath] Everything else are examples. [imath] 3 [/imath] is already physics. Or number theory: there you need [imath] 3,4 [/imath] and [imath] 5. [/imath]

 

[Agent Smith]

In school. I played around with what periods mean, i.e. divisions by ##3,6,7,9,11## and so on, and in online forums where kids discuss whether [imath] 0.\bar 9 [/imath] and [imath]1 [/imath] are equal or not.



This formula means [imath]\displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}= \lim_{n \to \infty}\sum_{k = 1} ^n \frac{9}{10^k}=1,[/imath] i.e. it is a limit and a limit is a unique number - so it exists. The question about the deficits of the decimal system (or any other) doesn't occur in mathematics. It's only a notational artifact.

I even like to say, that mathematicians actually only need very few numbers: [imath] \{\pm 2 ,\pm 1,0,\pi ,e , i\}. [/imath] Everything else are examples. [imath] 3 [/imath] is already physics. Or number theory: there you need [imath] 3,4 [/imath] and [imath] 5. [/imath]

Gracias for the correction. I should've said [imath]\displaystyle \lim_{n \to \infty} \sum_{k = 1} ^n \frac{9}{10^k} = 1[/imath].

 

[fresh_42]

Gracias for the correction. I should've said [imath]\displaystyle \lim_{n \to \infty} \sum_{k = 1} ^n \frac{9}{10^k} = 1[/imath].

No need to. The notation [imath] \displaystyle{\sum_{n=1}^\infty } [/imath] is totally ok.

I just wanted to emphasize that it is a limit. I have often heard / read / seen that kids think a limit is something that tends to something. But the limit itself is a number, nothing that tends. It is fixed. And as such, [imath] 0.999\ldots [/imath] does not tend towards [imath] 1.[/imath] It is one.

[imath] 0.999\ldots [/imath] is - as you already correctly mentioned - only another way to write [imath] \displaystyle{\sum_{n=1}^\infty } \dfrac{9}{10^k} [/imath] and that equals one. My comment about numbers was partly meant as a joke, but it has something true in it. The specific representation of numbers doesn't play a big role in mathematics. It is a short chapter on how decimal, binary, octal, hexadecimal, or whatever basis is used are written. Here is a nice list:

List of numeral systems - Wikipedia

en.wikipedia.org

You will see that most people have always used fingers and toes as a basis. However, the Babylonian [imath] 60 [/imath] is still present in our hours, minutes, and the calendar.

 

[pka]

[math]S=0.99\overline{9}[/math][math]10S=9.99\overline{9}[/math]_____subtract
[math]9S=9[/math] _____divide
[math]S=1[/math]

 

[Agent Smith]

[math]S=0.99\overline{9}[/math][math]10S=9.99\overline{9}[/math]_____subtract
[math]9S=9[/math] _____divide
[math]S=1[/math]

It should be, I think, [imath]10 \times 0.\overline 9 = 9. \overline 9[/imath]