How is 0.9... equal to 1

Question

How is $0.9999\ldots$ equal to $1$? I researched it a bit on Wikipedia, and I find a lot of different ways, but I'd just like to clear upon them.

Intuitive: I understand that this explanation is saying that $1$ is the highest number greater than $0.\bar{9}$, but that doesn't make sense why they are equal, for example, $0$ wouldn't be equal to $0.00000(1)$.

Discussion on completeness: I honestly didn't understand what it meant, but in the next paragraph it says the previous paragraph isn't proof.

Formal proof: I followed it for a while until to the $0\le 1-x\ldots$ line, then I got lost in how it was trying to prove.

Algebraic arguments: I don't follow how this one works, because, say $x = 0.99$ (To simplify things, it could be $0.999$ if wanted),

$10x = 9.9$

$10x = 9+0.9$

while in the argument it stated

$10x = 9+0.99$ (I believe $0.99$ should be $0.9$)

$10x = 9+x$

$9x = 9$

$x = 1$

Analytical Proof: This time I haven't got the faintest clue how this relates, and what it's saying.

I know I didn't cover most of the arguments, and there's a lot of math concepts I haven't scratch the surface of yet, but I was just curious how this is true. I didn't understand if possible for the questions, show a summary of what it was trying to say. Lastly, I just wanted to say I have a very elementary knowledge of mathematics. I still have an extremely long journey to go, so if don't understand a lot of the complicated, or even simple concepts that may help explain the problem, please be patient.

Does this answer your question? Is it true that $0.999999999\dots=1$? If not, please indicate why not. I don't see a coherent question here. — Ross Millikan, May 16 '21 at 02:58
The "algebraic argument" in your post only works if $x=0.99999\ldots$. You can't replace it with $x=0.999$ or $0.99$ or $0.9$. — angryavian, May 16 '21 at 02:59
Here’s a question for you, if $x<y$ then $x<\frac{x+y}2<y.$ How would you write a number strictly between $0.999\dots$ and $1?$ — Thomas Andrews, May 16 '21 at 03:00
I think one key question that you need to resolve for yourself is how a decimal with infinitely many digits is defined. I mention this because you bring up something called "$0.00000(1)$" which isn't well-defined. — angryavian, May 16 '21 at 03:01
Ask yourself this: what number is between $1$ and $.9999\cdots$? — Jacky Chong, May 16 '21 at 03:06
Sorry, I didn't realize there was already a question asked about this. — user15302406, May 16 '21 at 12:59

score 2 · Accepted Answer · edited May 16 '21 at 03:42

$1$ over $9$ is the same as $1/9$ which is $0.11111\ldots$

$2$ over $9$ is the same as $2/9$ which is $0.22222\ldots$

etc.. which in theory means $0.99999\ldots$ should equal $9$ over $9$ which is one.

But if you don't understand you can do:

$x = 0.99999\ldots$

which means:

$10x = 9.99999\ldots$

then, you can do:

$9.99999\ldots - 0.99999\ldots = 9$

and do:

$10x - x = 9x$

which means:

$9x = 9$

then, you simplify by dividing both by $9$:

$9x/9 = x$

$9/9 = 1$

then you write it as:

$x = 1$

and $x$ is $0.99999\ldots$

How is 0.9... equal to 1

1 Answers1