Sythe Users: What Do They Know? Do They Know Things?? Lets Find Out! #7

Question, is that 0.999... repeating, as in never ending?

 

[Accounts Shop]

Question, is that 0.999... repeating, as in never ending?

yes, 0,9 followed by an infinite amount of 9.

that's why he is able to show that 0.9 = 1, if there isn't an infinite amount of 9s, it isn't equal to 1.

for example,

0,999 != 1

0.999 = X

10X = 9.99

10X - X = 9.99 - .999 = 8.991

9X = 8.991 => X = 0.999 != 1

 

[Accounts Shop]

Here's another more conventional way to think about it:

Not sure how you want to go about this,

in the above picture, Y and 10Y both are equal to infinity because this is a geometric series whos ratio is greater than 1

 

yes, 0,9 followed by an infinite amount of 9.

that's why he is able to show that 0.9 = 1, if there isn't an infinite amount of 9s, it isn't equal to 1.

for example,

0,999 != 1

0.999 = X

10X = 9.99

10X - X = 9.99 - .999 = 8.991

9X = 8.991 => X = 0.999 != 1

So then how exactly can you claim the following (pardon my poor notation):

Y = 0.999...

10Y = 10 * (0.999...)

10Y = ...99990

While the basic principle makes sense, isn't that effectively giving infinity an endpoint, which you cannot do? That's saying that where the 0 now is, that was the end of the infinitely repeating 9s. How can there be an end to that?

 

So then how exactly can you claim the following (pardon my poor notation):

Y = 0.999...

10Y = 10 * (0.999...)

10Y = ...99990

While the basic principle makes sense, isn't that effectively giving infinity an endpoint, which you cannot do? That's saying that where the 0 now is, that was the end of the infinitely repeating 9s. How can there be an end to that?

Y is ...999 where the 9s extend infinitely to the left

 

[Accounts Shop]

So then how exactly can you claim the following (pardon my poor notation):

Y = 0.999...

10Y = 10 * (0.999...)

10Y = ...99990

While the basic principle makes sense, isn't that effectively giving infinity an endpoint, which you cannot do? That's saying that where the 0 now is, that was the end of the infinitely repeating 9s. How can there be an end to that?

...99999 is an infinite amount of 9s to the left of the decimal.

if you'd like to know more about such things, i believe this could be useful

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

 

[rekface]

y = ∞
y + 1 = ∞
y - ∞ = ∞

there.

edit: Y is infinity. Therefore Y is all numbers simultaneously.

 

[Accounts Shop]

y = ∞
y + 1 = ∞
y - ∞ = ∞

there.

edit: Y is infinity. Therefore Y is all numbers simultaneously.

you fail so bad, infinity is not all numbers simultaneously

if so you could say

∞ = 2

∞ = 4

∞ = -74

and then you could say that 2 = -74 = ∞

see how that doesn't work ?

 

[OzOnline]

There isn't a single scenario where ...999999 infinite 9's are generated unless they're in the decimals. If thats what you mean, then .9 repeated.

it doesn't equal -1, but -1.1 repeated.

 

[Accounts Shop]

There isn't a single scenario where ...999999 infinite 9's are generated unless they're in the decimals. If thats what you mean, then .9 repeated.

it doesn't equal -1, but -1.1 repeated.

.9999999... = 1

not -1.1

 

[rekface]

you fail so bad, infinity is not all numbers simultaneously

if so you could say

∞ = 2

∞ = 4

∞ = -74

and then you could say that 2 = -74 = ∞

see how that doesn't work ?

You're right, it doesn't work. That's because infinity is a concept and doesn't exist in reality. Therefore it cannot be calculated logically.

Edit: The proof itself already uses that logic that you are professing against. It claims that ...999999 = -1

It does not work both ways because an infinite fraction is not an infinite number and is not comparable.

 

[Accounts Shop]

You're right, it doesn't work. That's because infinity is a concept and doesn't exist in reality. Therefore it cannot be calculated logically.

Edit: The proof itself already uses that logic that you are professing against. It claims that ...999999 = -1

It does not work both ways because an infinite fraction is not an infinite number and is not comparable.

huh ... an infinite fraction is an infinite number ... it's 0,9 followed by an infinite amount of 9s.

the proof doesn't assert that infinity = -1 or 1,

it just uses algebra to get rid of the concept of infinity

X = 0.9999...

10X = 9.99999...

10X - X = 9.9999... - 0.99999....

9X = 9 => X = 1

 

[rekface]

huh ... an infinite fraction is an infinite number ... it's 0,9 followed by an infinite amount of 9s.

the proof doesn't assert that infinity = -1 or 1,

it just uses algebra to get rid of the concept of infinity

X = 0.9999...

10X = 9.99999...

10X - X = 9.9999... - 0.99999....

9X = 9 => X = 1

And it arrives at the conclusion that -1 = 9999999 proving conventional mathematics is inadequate at handling infinite concepts. I'll agree that even infinite decimals are infinite since using the same mathematics, one is still forced to round the number to arrive at a solution, a technically incorrect one at that.

Edit: Even the original problem proposes 9.99999... - .99999... = 9. This is incorrect. This problem could never be solved because it would be subtracting infinitely meaning it would never arrive at a solution. Just in the sense that ∞ - ∞ = ∞. Infinity will always be infinity no matter what you give or take from it.

 

Sorry sythe, I have a personal policy not to appropriately participate in anything Deathsnova has already marred with his presence. Therefore, I cannot answer your question with impeccable reasoning and excellent vocabulary :c

savage

 

there is no leftmost position? otherwise we could also say 1-0.999... = 0.000...1

Like the idea that parallel lines cross at infinity, by adding 1 your one shoots off to the start of the set of 0's. It lives at infinity, but it doesn't disappear. If it could disappear at some point then it wouldn't be able to change all the 9's into 0's.

Y is a geometric serie of first term 9 and ratio 10.

10Y is a geometric serie of first term 90 and ratio 10.

Y = 9 * (1 - 10^n+1) / (1 - 10)
Y = 9 - 9*(10^n) / -9
Y = 9/-9 - 9(10^n)/-9
Y = -1 + 10^n

10Y = 90 * (1 - 10^n) / (1 - 10)
10Y = 90/-9 + 10(10^n)
10Y = -10 + 10(10^n)

Y - 9 = -1 + (10^n) - 9
Y - 9 = -10 + 10^n

Y - 9 = 10Y

-1 + 10^n - 9 = -10 + 10(10^n)

-9 = -10 + 10(10^n) + 1 - 10^n

-9 = -10 + (100^n) + 1 - 10^n

-9 = -9 + 10(10^n) - 10^n

-9 = -9 + 10^n * (10 - 1)

-9 = -9 + 10^n * 9

...

no more time to think this through, i'm probably close but so far away

Not going to check your working but the last line is 0 = 10^n. There's no n in the real numbers which gives 0. But this ties into what I'll say below.

Sorry sythe, I have a personal policy not to appropriately participate in anything Deathsnova has already marred with his presence. Therefore, I cannot answer your question with impeccable reasoning and excellent vocabulary :c

Opting out. That's a valid strategy

Not sure how you want to go about this,

in the above picture, Y and 10Y both are equal to infinity because this is a geometric series whos ratio is greater than 1

No, when we have an infinite series that does not converge we do not say it's equal to infinity. We say it does not converge. There is a very important difference. In one case we are treating infinity as a concept and in the other we are treating it as a number.

So then how exactly can you claim the following (pardon my poor notation):

Y = 0.999...

10Y = 10 * (0.999...)

10Y = ...99990

While the basic principle makes sense, isn't that effectively giving infinity an endpoint, which you cannot do? That's saying that where the 0 now is, that was the end of the infinitely repeating 9s. How can there be an end to that?

Your intuitive distrust of the number should also extend to recurring decimals.

If this is perfectly valid algebra:

Then nothing stops me having control over the least significant places of these numbers.

There isn't a single scenario where ...999999 infinite 9's are generated unless they're in the decimals. If thats what you mean, then .9 repeated.

I already give you the sum notation to show you can construct these numbers.

You're right, it doesn't work. That's because infinity is a concept and doesn't exist in reality. Therefore it cannot be calculated logically.

Edit: The proof itself already uses that logic that you are professing against. It claims that ...999999 = -1

It does not work both ways because an infinite fraction is not an infinite number and is not comparable.

huh ... an infinite fraction is an infinite number ... it's 0,9 followed by an infinite amount of 9s.

the proof doesn't assert that infinity = -1 or 1,

it just uses algebra to get rid of the concept of infinity

X = 0.9999...

10X = 9.99999...

10X - X = 9.9999... - 0.99999....

9X = 9 => X = 1

These posts are getting a lot closer to what I believe is the answer.

And it arrives at the conclusion that -1 = 9999999 proving conventional mathematics is inadequate at handling infinite concepts. I'll agree that even infinite decimals are infinite since using the same mathematics, one is still forced to round the number to arrive at a solution, a technically incorrect one at that.

Edit: Even the original problem proposes 9.99999... - .99999... = 9. This is incorrect. This problem could never be solved because it would be subtracting infinitely meaning it would never arrive at a solution. Just in the sense that ∞ - ∞ = ∞. Infinity will always be infinity no matter what you give or take from it.

This is the closest to a correct analysis so far.

I'll give my analysis below shortly.

 

[rekface]

Like the idea that parallel lines cross at infinity, by adding 1 your one shoots off to the start of the set of 0's. It lives at infinity, but it doesn't disappear. If it could disappear at some point then it wouldn't be able to change all the 9's into 0's.




Not going to check your working but the last line is 0 = 10^n. There's no n in the real numbers which gives 0. But this ties into what I'll say below.




Opting out. That's a valid strategy



No, when we have an infinite series that does not converge we do not say it's equal to infinity. We say it does not converge. There is a very important difference. In one case we are treating infinity as a concept and in the other we are treating it as a number.



Your intuitive distrust of the number should also extend to recurring decimals.

If this is perfectly valid algebra:

Then nothing stops me having control over the least significant places of these numbers.


I already give you the sum notation to show you can construct these numbers.






These posts are getting a lot closer to what I believe is the answer.



This is the closest to a correct analysis so far.

I'll give my analysis below shortly.

Would you say, instead, that in regards to 9.99999... - .99999... = 1 it would not equal 1, not because you subtract infinitely, but because no matter how close it gets, it never technically reaches the value of 1. So doctor butt is applying the same logic to a different equation to prove it is wrong?

 

Like the idea that parallel lines cross at infinity, by adding 1 your one shoots off to the start of the set of 0's. It lives at infinity, but it doesn't disappear. If it could disappear at some point then it wouldn't be able to change all the 9's into 0's.

Why can't we say then that 0.999... + x = 1 has the solution x = 0.000...1?

 

[rekface]

Why can't we say then that 0.999... + x = 1 has the solution x = 0.000...1?

Using the logic of my previous answer, I would say that 0.000...1 = 0 because, in having an infinite amount of zeros before the 1, it is infinitely small and would only contain a 1 if it had a finite number of zeros.

 

Using the logic of my previous answer, I would say that 0.000...1 = 0 because, in having an infinite amount of zeros before the 1, it is infinitely small and would only contain a 1 if it had a finite number of zeros.

yeah idk, don't understand how we can't say that ...999+1 = ...000 = 0 because there isn't a position that a leading 1 can occupy


final thoughts:

...999 represented as a geometric series does not converge. We can't say that ...999 is equal to -1, just that it behaves like -1. We can maybe think of it in terms of modular arithmetics where numbers cycle through equivalence to 0. Just like how for certain odometers 10^6 (or whatever analog odometers go up to) is equivalent 0 as 999999+1=0 --> 999999=-1. Seeing the same thing for ...999 is the result of having infinity equivalent to 0 somehow??

 

So my take is that the reals as a set doesn't have an operation for subtracting two actual infinities, and that's why you get a problem. (So essentially the subtraction of two infinite sums is not a well defined operation.) Some of your answers were along these lines but not quite all the way there.

The problem also occurs on the other side (i.e 0.9999.... = 1) but is not considered a problem because it does not give an absurd result. Or does it? I would argue that it is an absurd result: Logically 0.999... is not the same number as 1, they differ by an infinitely small non-zero amount: an infinitesimal amount. This proof "proves" that they differ by exactly zero which contradicts the premise that they are two different numbers. So if this argument is true then 0.9999... and 1 are the same number. Taken to exhaustion this concept implies that every number is the same number. I.e. If I subtract an infinitesimal amount from 0.99999... then that's still equal to 0.99999... and that's still equal to 1. I can subtract an infinite number of infinitesimal amounts and the answer will still be 1. Therefore every number is 1, that's an absurd conclusion if you ask me.

Another way to think about this is that actual infinity (i.e. infinity as a complete finished number not a concept) shouldn't really be in the number system at all. But if you are going to have it in the number system then you also need to balance it with infinitesimals.

Here's a short proof I've written which demonstrates how the reals can be supplemented with a single infinitesimal to fix the problem. Note that I use a mathematical form called dual numbers which is commonly used in physics proofs (such as perturbation theory) however it is not part of mainstream ZFC (the math you will learn in undergraduate math courses at university.)

If nothing else this little problem serves as a reminder that mathematics has never been anything more than a set of logical arguments and therefore your premises will always determine your conclusions.

Note that in this proof there is a logical jump at the first green squiggle which is a bit tricky and might not be strictly consistent with our formal definition at the top. Essentially the argument is this: If you put a 9 in every column of a base 10 number, for an infinite number of columns, then this is the largest number you can possibly express using base 10 arithmetic. (Even any sort of power etc will result in a different number which doesn't have 9's in every column and will therefore be smaller.) If you invert this number then you retrieve the smallest number you can possibly express that isn't zero. Logically that number will also be our infinitesimal.