Why doesn't “1 = 0.999…” break math? [duplicate]

Clash Royale CLAN TAG#URR8PPP

up vote
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This question already has an answer here:

• 
  Is it true that $0.999999999dots=1$?
  
  24 answers
  
  

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

8 = 9 : because when 2 numbers are equal, their decimal digits of the same place values are equal

So 0.999... = 1, but why doesn't that lead to nonsensical proofs? Obviously I'm getting stuff wrong here.

fake-proofs

share|cite|improve this question

edited yesterday

asked yesterday

sag

986

marked as duplicate by Xander Henderson, Shaun, Simply Beautiful Art, Foobaz John, Asaf Karagila yesterday

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  In which decimal place does the $8$ in the third line come?
  – Brian Tung
  yesterday
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  I would argue it's not quite a dupe.
  – Mason
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  youtube.com/watch?v=XKy_VTBq0yk
  – Count Iblis
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @CountIblis: And $Bbb N$ only took me one symbol. So? If you insist that mathematics should only describe reality, then how can you even justify the idea of $frac12$? How do you cut a cake in half? Do you count atoms? Wouldn't that violate uncertainty? Well, you'd argue, you know what is $1$ and $2$, and that's just the ideal ratio of these objects. Well, great, I say. Infinite sets are based on finite definitions which model the ideal notion of infinite.
  – Asaf Karagila
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  This is probably a duplicate of something but definitely isn't a duplicate of the question it is currently linked to...
  – Eric Wofsey
  yesterday
  

  
  
  
  

 | 
show 6 more comments

up vote
-3
down vote

favorite

1

This question already has an answer here:

• 
  Is it true that $0.999999999dots=1$?
  
  24 answers
  
  

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

8 = 9 : because when 2 numbers are equal, their decimal digits of the same place values are equal

So 0.999... = 1, but why doesn't that lead to nonsensical proofs? Obviously I'm getting stuff wrong here.

fake-proofs

share|cite|improve this question

edited yesterday

asked yesterday

sag

986

marked as duplicate by Xander Henderson, Shaun, Simply Beautiful Art, Foobaz John, Asaf Karagila yesterday

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  In which decimal place does the $8$ in the third line come?
  – Brian Tung
  yesterday
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  I would argue it's not quite a dupe.
  – Mason
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  youtube.com/watch?v=XKy_VTBq0yk
  – Count Iblis
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @CountIblis: And $Bbb N$ only took me one symbol. So? If you insist that mathematics should only describe reality, then how can you even justify the idea of $frac12$? How do you cut a cake in half? Do you count atoms? Wouldn't that violate uncertainty? Well, you'd argue, you know what is $1$ and $2$, and that's just the ideal ratio of these objects. Well, great, I say. Infinite sets are based on finite definitions which model the ideal notion of infinite.
  – Asaf Karagila
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  This is probably a duplicate of something but definitely isn't a duplicate of the question it is currently linked to...
  – Eric Wofsey
  yesterday
  

  
  
  
  

 | 
show 6 more comments

up vote
-3
down vote

favorite

1

up vote
-3
down vote

favorite

1

1

This question already has an answer here:

• 
  Is it true that $0.999999999dots=1$?
  
  24 answers
  
  

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

8 = 9 : because when 2 numbers are equal, their decimal digits of the same place values are equal

So 0.999... = 1, but why doesn't that lead to nonsensical proofs? Obviously I'm getting stuff wrong here.

fake-proofs

share|cite|improve this question

edited yesterday

asked yesterday

sag

986

This question already has an answer here:

• 
  Is it true that $0.999999999dots=1$?
  
  24 answers
  
  

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

8 = 9 : because when 2 numbers are equal, their decimal digits of the same place values are equal

So 0.999... = 1, but why doesn't that lead to nonsensical proofs? Obviously I'm getting stuff wrong here.

This question already has an answer here:

• 
  Is it true that $0.999999999dots=1$?
  
  24 answers
  
  

fake-proofs

share|cite|improve this question

edited yesterday

asked yesterday

sag

986

share|cite|improve this question

share|cite|improve this question

edited yesterday

edited yesterday

edited yesterday

asked yesterday

sag

986

asked yesterday

sag

986

asked yesterday

sag

986

986

marked as duplicate by Xander Henderson, Shaun, Simply Beautiful Art, Foobaz John, Asaf Karagila yesterday

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

marked as duplicate by Xander Henderson, Shaun, Simply Beautiful Art, Foobaz John, Asaf Karagila yesterday

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  In which decimal place does the $8$ in the third line come?
  – Brian Tung
  yesterday
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  I would argue it's not quite a dupe.
  – Mason
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  youtube.com/watch?v=XKy_VTBq0yk
  – Count Iblis
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @CountIblis: And $Bbb N$ only took me one symbol. So? If you insist that mathematics should only describe reality, then how can you even justify the idea of $frac12$? How do you cut a cake in half? Do you count atoms? Wouldn't that violate uncertainty? Well, you'd argue, you know what is $1$ and $2$, and that's just the ideal ratio of these objects. Well, great, I say. Infinite sets are based on finite definitions which model the ideal notion of infinite.
  – Asaf Karagila
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  This is probably a duplicate of something but definitely isn't a duplicate of the question it is currently linked to...
  – Eric Wofsey
  yesterday
  

  
  
  
  

 | 
show 6 more comments

• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  In which decimal place does the $8$ in the third line come?
  – Brian Tung
  yesterday
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  I would argue it's not quite a dupe.
  – Mason
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  youtube.com/watch?v=XKy_VTBq0yk
  – Count Iblis
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @CountIblis: And $Bbb N$ only took me one symbol. So? If you insist that mathematics should only describe reality, then how can you even justify the idea of $frac12$? How do you cut a cake in half? Do you count atoms? Wouldn't that violate uncertainty? Well, you'd argue, you know what is $1$ and $2$, and that's just the ideal ratio of these objects. Well, great, I say. Infinite sets are based on finite definitions which model the ideal notion of infinite.
  – Asaf Karagila
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  This is probably a duplicate of something but definitely isn't a duplicate of the question it is currently linked to...
  – Eric Wofsey
  yesterday
  

  
  
  
  

11

11

In which decimal place does the $8$ in the third line come?
– Brian Tung
yesterday

In which decimal place does the $8$ in the third line come?
– Brian Tung
yesterday

3

3

I would argue it's not quite a dupe.
– Mason
yesterday

I would argue it's not quite a dupe.
– Mason
yesterday

1

1

youtube.com/watch?v=XKy_VTBq0yk
– Count Iblis
yesterday

youtube.com/watch?v=XKy_VTBq0yk
– Count Iblis
yesterday

1

1

@CountIblis: And $Bbb N$ only took me one symbol. So? If you insist that mathematics should only describe reality, then how can you even justify the idea of $frac12$? How do you cut a cake in half? Do you count atoms? Wouldn't that violate uncertainty? Well, you'd argue, you know what is $1$ and $2$, and that's just the ideal ratio of these objects. Well, great, I say. Infinite sets are based on finite definitions which model the ideal notion of infinite.
– Asaf Karagila
yesterday

@CountIblis: And $Bbb N$ only took me one symbol. So? If you insist that mathematics should only describe reality, then how can you even justify the idea of $frac12$? How do you cut a cake in half? Do you count atoms? Wouldn't that violate uncertainty? Well, you'd argue, you know what is $1$ and $2$, and that's just the ideal ratio of these objects. Well, great, I say. Infinite sets are based on finite definitions which model the ideal notion of infinite.
– Asaf Karagila
yesterday

2

2

This is probably a duplicate of something but definitely isn't a duplicate of the question it is currently linked to...
– Eric Wofsey
yesterday

This is probably a duplicate of something but definitely isn't a duplicate of the question it is currently linked to...
– Eric Wofsey
yesterday

 | 
show 6 more comments

1 Answer
1

active

oldest

votes

up vote
0
down vote

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

How about we replace this line 8 = 9 with:

But shouldn't that mean that $frac810^n=frac910^n$? For some $n$?

Of course if it is the case that $frac810^n=frac910^n$ then we can always multiply both sides by $10^n$ and arrive at that conclusion.

The answer to $frac810^n=frac910^n$? is : Yes when $n=infty$ but then you can see that $frac810^infty=frac910^infty=0$. This is also the answer to Brian Tung's comment. The $8$ is somehow in the $infty$ place of the decimal.

The prevents us from taking the next step in your reasoning which is multiplying by $10^infty$.

RULE: You aren't allowed to multiply both sides of an equation by $infty$. It will have you thinking that $8=9$.

share|cite|improve this answer

answered yesterday

Mason

1,1101123

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I have to say that the use of $infty$ as if it was a number is pretty awful, giving very bad habits, and surprising in an answer that could otherwise be good.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ArnaudMortier Yeah. I appreciate the feedback. I would be more inspired to work to improve this answer if this wasn't mark as a dupe. I suppose we don't really consider duplicates closed? Or do we? Anyway: I am in full agreement. There is some good explanation in here that I/anyone who wanted to could work to distill but as it stands is only ok.
  – Mason
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  A duplicate is considered as closed, but in general it doesn't get deleted, precisely because there still could be relevant contents inside.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See e.g. this question which, although it is closed and with negative votes, was today referred to because a duplicate was posted. People are still going to read the answers there.
  – Arnaud Mortier
  21 hours ago
  

  
  
  
  

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

How about we replace this line 8 = 9 with:

But shouldn't that mean that $frac810^n=frac910^n$? For some $n$?

Of course if it is the case that $frac810^n=frac910^n$ then we can always multiply both sides by $10^n$ and arrive at that conclusion.

The answer to $frac810^n=frac910^n$? is : Yes when $n=infty$ but then you can see that $frac810^infty=frac910^infty=0$. This is also the answer to Brian Tung's comment. The $8$ is somehow in the $infty$ place of the decimal.

The prevents us from taking the next step in your reasoning which is multiplying by $10^infty$.

RULE: You aren't allowed to multiply both sides of an equation by $infty$. It will have you thinking that $8=9$.

share|cite|improve this answer

answered yesterday

Mason

1,1101123

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I have to say that the use of $infty$ as if it was a number is pretty awful, giving very bad habits, and surprising in an answer that could otherwise be good.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ArnaudMortier Yeah. I appreciate the feedback. I would be more inspired to work to improve this answer if this wasn't mark as a dupe. I suppose we don't really consider duplicates closed? Or do we? Anyway: I am in full agreement. There is some good explanation in here that I/anyone who wanted to could work to distill but as it stands is only ok.
  – Mason
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  A duplicate is considered as closed, but in general it doesn't get deleted, precisely because there still could be relevant contents inside.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See e.g. this question which, although it is closed and with negative votes, was today referred to because a duplicate was posted. People are still going to read the answers there.
  – Arnaud Mortier
  21 hours ago
  

  
  
  
  

add a comment | 

up vote
0
down vote

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

How about we replace this line 8 = 9 with:

But shouldn't that mean that $frac810^n=frac910^n$? For some $n$?

Of course if it is the case that $frac810^n=frac910^n$ then we can always multiply both sides by $10^n$ and arrive at that conclusion.

The answer to $frac810^n=frac910^n$? is : Yes when $n=infty$ but then you can see that $frac810^infty=frac910^infty=0$. This is also the answer to Brian Tung's comment. The $8$ is somehow in the $infty$ place of the decimal.

The prevents us from taking the next step in your reasoning which is multiplying by $10^infty$.

RULE: You aren't allowed to multiply both sides of an equation by $infty$. It will have you thinking that $8=9$.

share|cite|improve this answer

answered yesterday

Mason

1,1101123

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I have to say that the use of $infty$ as if it was a number is pretty awful, giving very bad habits, and surprising in an answer that could otherwise be good.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ArnaudMortier Yeah. I appreciate the feedback. I would be more inspired to work to improve this answer if this wasn't mark as a dupe. I suppose we don't really consider duplicates closed? Or do we? Anyway: I am in full agreement. There is some good explanation in here that I/anyone who wanted to could work to distill but as it stands is only ok.
  – Mason
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  A duplicate is considered as closed, but in general it doesn't get deleted, precisely because there still could be relevant contents inside.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See e.g. this question which, although it is closed and with negative votes, was today referred to because a duplicate was posted. People are still going to read the answers there.
  – Arnaud Mortier
  21 hours ago
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

How about we replace this line 8 = 9 with:

But shouldn't that mean that $frac810^n=frac910^n$? For some $n$?

Of course if it is the case that $frac810^n=frac910^n$ then we can always multiply both sides by $10^n$ and arrive at that conclusion.

The answer to $frac810^n=frac910^n$? is : Yes when $n=infty$ but then you can see that $frac810^infty=frac910^infty=0$. This is also the answer to Brian Tung's comment. The $8$ is somehow in the $infty$ place of the decimal.

The prevents us from taking the next step in your reasoning which is multiplying by $10^infty$.

RULE: You aren't allowed to multiply both sides of an equation by $infty$. It will have you thinking that $8=9$.

share|cite|improve this answer

answered yesterday

Mason

1,1101123

0.999... = 1

0.999... + 0.999... = 0.999... + 1

1.999...8 = 1.999...9

How about we replace this line 8 = 9 with:

But shouldn't that mean that $frac810^n=frac910^n$? For some $n$?

Of course if it is the case that $frac810^n=frac910^n$ then we can always multiply both sides by $10^n$ and arrive at that conclusion.

The answer to $frac810^n=frac910^n$? is : Yes when $n=infty$ but then you can see that $frac810^infty=frac910^infty=0$. This is also the answer to Brian Tung's comment. The $8$ is somehow in the $infty$ place of the decimal.

The prevents us from taking the next step in your reasoning which is multiplying by $10^infty$.

RULE: You aren't allowed to multiply both sides of an equation by $infty$. It will have you thinking that $8=9$.

share|cite|improve this answer

answered yesterday

Mason

1,1101123

share|cite|improve this answer

share|cite|improve this answer

answered yesterday

Mason

1,1101123

answered yesterday

Mason

1,1101123

answered yesterday

Mason

1,1101123

1,1101123

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I have to say that the use of $infty$ as if it was a number is pretty awful, giving very bad habits, and surprising in an answer that could otherwise be good.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ArnaudMortier Yeah. I appreciate the feedback. I would be more inspired to work to improve this answer if this wasn't mark as a dupe. I suppose we don't really consider duplicates closed? Or do we? Anyway: I am in full agreement. There is some good explanation in here that I/anyone who wanted to could work to distill but as it stands is only ok.
  – Mason
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  A duplicate is considered as closed, but in general it doesn't get deleted, precisely because there still could be relevant contents inside.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See e.g. this question which, although it is closed and with negative votes, was today referred to because a duplicate was posted. People are still going to read the answers there.
  – Arnaud Mortier
  21 hours ago
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I have to say that the use of $infty$ as if it was a number is pretty awful, giving very bad habits, and surprising in an answer that could otherwise be good.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ArnaudMortier Yeah. I appreciate the feedback. I would be more inspired to work to improve this answer if this wasn't mark as a dupe. I suppose we don't really consider duplicates closed? Or do we? Anyway: I am in full agreement. There is some good explanation in here that I/anyone who wanted to could work to distill but as it stands is only ok.
  – Mason
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  A duplicate is considered as closed, but in general it doesn't get deleted, precisely because there still could be relevant contents inside.
  – Arnaud Mortier
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See e.g. this question which, although it is closed and with negative votes, was today referred to because a duplicate was posted. People are still going to read the answers there.
  – Arnaud Mortier
  21 hours ago
  

  
  
  
  

2

2

I have to say that the use of $infty$ as if it was a number is pretty awful, giving very bad habits, and surprising in an answer that could otherwise be good.
– Arnaud Mortier
yesterday

I have to say that the use of $infty$ as if it was a number is pretty awful, giving very bad habits, and surprising in an answer that could otherwise be good.
– Arnaud Mortier
yesterday

@ArnaudMortier Yeah. I appreciate the feedback. I would be more inspired to work to improve this answer if this wasn't mark as a dupe. I suppose we don't really consider duplicates closed? Or do we? Anyway: I am in full agreement. There is some good explanation in here that I/anyone who wanted to could work to distill but as it stands is only ok.
– Mason
yesterday

@ArnaudMortier Yeah. I appreciate the feedback. I would be more inspired to work to improve this answer if this wasn't mark as a dupe. I suppose we don't really consider duplicates closed? Or do we? Anyway: I am in full agreement. There is some good explanation in here that I/anyone who wanted to could work to distill but as it stands is only ok.
– Mason
yesterday

A duplicate is considered as closed, but in general it doesn't get deleted, precisely because there still could be relevant contents inside.
– Arnaud Mortier
yesterday

A duplicate is considered as closed, but in general it doesn't get deleted, precisely because there still could be relevant contents inside.
– Arnaud Mortier
yesterday

See e.g. this question which, although it is closed and with negative votes, was today referred to because a duplicate was posted. People are still going to read the answers there.
– Arnaud Mortier
21 hours ago

See e.g. this question which, although it is closed and with negative votes, was today referred to because a duplicate was posted. People are still going to read the answers there.
– Arnaud Mortier
21 hours ago

add a comment |