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I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc.

Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is about?

Here is the most usual proof I see that $0.999\ldots = 1$:

$x=0.999\ldots$

 

$10x=9.999\ldots$

 

$10x - x = 9$

 

$x=1$

Using this same argument template I can show $999\ldots=-1$:

$x= \ldots9999.0 $

 

$0.1x= \ldots9999.9$

 

$0.1x - x = 0.9$

 

$x=-1$

What might this mean?

Edit from one of the comments:

$$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc.

Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is about?

Here is the most usual proof I see that $0.999\ldots = 1$:

$x=0.999\ldots$

$10x=9.999\ldots$

$10x - x = 9$

$x=1$

Using this same argument template I can show $999\ldots=-1$:

$x= \ldots9999.0 $

$0.1x= \ldots9999.9$

$0.1x - x = 0.9$

$x=-1$

What might this mean?

Edit from one of the comments:

$$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999... = 2.5$ etc.

Can anyone point me to resources that would explain what the below argument that shows $999... = -1$ is about?

Here is the most usual proof I see that $0.999... = 1$.

$x=0.999...$

$10x=9.999... $

$x=1$

Using this same argument template I can show $999...=-1$. (Edited for clarity)

$x=...9999.0 $

$0.1x=...9999.9$

$x=-1$

What might this mean?

Edit from one of the comments:

$$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc.

Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is about?

Here is the most usual proof I see that $0.999\ldots = 1$:

$x=0.999\ldots$

$10x=9.999\ldots$

$10x - x = 9$

$x=1$

Using this same argument template I can show $999\ldots=-1$:

$x= \ldots9999.0 $

$0.1x= \ldots9999.9$

$0.1x - x = 0.9$

$x=-1$

What might this mean?

Edit from one of the comments:

$$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999... = 2.5$ etc.

Can anyone point me to resources that would explain what the below argument that shows $999... = -1$ is about?

Here is the most usual proof I see that $0.999... = 1$.

$x=0.999...$

$10x=9.999... $

$x=1$

Using this same argument template I can show $999...=-1$.

$x=999...9.0 $

$0.1x=999...9.9$

$x=-1$

What might this mean?

Edit from one of the comments:

$$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999... = 2.5$ etc.

Can anyone point me to resources that would explain what the below argument that shows $999... = -1$ is about?

Here is the most usual proof I see that $0.999... = 1$.

$x=0.999...$

$10x=9.999... $

$x=1$

Using this same argument template I can show $999...=-1$. (Edited for clarity)

$x=...9999.0 $

$0.1x=...9999.9$

$x=-1$

What might this mean?

Edit from one of the comments:

$$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$