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... = $5.\overline 9$

I "know" that $0.\overline 9 = 1$. So $5.\overline 9 = 5+ 1 = 6 = 3.0013849483127... + 2.9986150516872...$

This is the first problem I've worked on where $0.\overline 9 = 1$ 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 $0.\bar 9$ 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 $5.999... = 5.\overline 9$ in your mathematical life? If yes, where exactly?

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

 

[fresh_42]

Have you ever encountered a number like $5.999... = 5.\overline 9$ 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 $0.\bar 9$ and $1$ are equal or not.

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

This formula means $\displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}= \lim_{n \to \infty}\sum_{k = 1} ^n \frac{9}{10^k}=1,$ 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: $\{\pm 2 ,\pm 1,0,\pi ,e , i\}.$ Everything else are examples. $3$ is already physics. Or number theory: there you need $3,4$ and $5.$

 

[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 $0.\bar 9$ and $1$ are equal or not.



This formula means $\displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}= \lim_{n \to \infty}\sum_{k = 1} ^n \frac{9}{10^k}=1,$ 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: $\{\pm 2 ,\pm 1,0,\pi ,e , i\}.$ Everything else are examples. $3$ is already physics. Or number theory: there you need $3,4$ and $5.$

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

 

[fresh_42]

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

No need to. The notation $\displaystyle{\sum_{n=1}^\infty }$ 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, $0.999\ldots$ does not tend towards $1.$ It is one.

$0.999\ldots$ is - as you already correctly mentioned - only another way to write $\displaystyle{\sum_{n=1}^\infty } \dfrac{9}{10^k}$ 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 $60$ is still present in our hours, minutes, and the calendar.

 

[pka]

$$S=0.99\overline{9}$$$$10S=9.99\overline{9}$$_____subtract
$$9S=9$$ _____divide
$$S=1$$

 

[Agent Smith]

$$S=0.99\overline{9}$$$$10S=9.99\overline{9}$$_____subtract
$$9S=9$$ _____divide
$$S=1$$

It should be, I think, $10 \times 0.\overline 9 = 9. \overline 9$

 

[Steven G]

There are infinitely many examples, so to have seen one before is not very likely,
4.99999..., 5.999999..., 6.99999...., .... 97.99999, 314.99999..., and my favorite one: 273.9999999.....
How about -43.9999999.99999....?