Does 0.99999... = 1?

[Ryaja]

I heard that it does in a recreational math class(yes my school has one.) They used the proof that 1/3 = 0.333... so 3/3 = 1 and 0.999... but I remember hearing that 1/3 ≠ 0.333... and can be expressed as 0.3 repeating with a 4 at the end. So this got me thinking are there "super rational" numbers that can only be expressed as a fraction? But then with different notation maybe it could be a decimal like(pretend the strike through is a line on top of the number.) 0.34, or 0.01 for the answer too 1-0.9. Oh and I am the first to think of super rationals or not?

Edited December 18, 2022 by Ryaja

[kerbiloid]

1/3 = 0.(3);

Quote

Does 0.99999... = 1?

Asymptotically.

2 hours ago, Ryaja said:

So this got me thinking are there "super rational" numbers that can only be expressed as a fraction?

https://en.wikipedia.org/wiki/Repeating_decimal

https://en.wikipedia.org/wiki/Transcendental_number

[Ryaja]

14 minutes ago, kerbiloid said:

Asymptotically

That's my point, it continues to get closer but is never quit there, it is always(I will use the strike through as an over line.) 0.01 off from 1.

[TheSaint]

Oh, God. Not this again....

[razark]

If 3 == "no", then 2 cannot be answered.  2 is a required response, therefore the poll is invalid.

 

But 0.99... does equal 1.

[Leganeski]

https://en.wikipedia.org/wiki/0.999...#Skepticism_in_education covers this exact topic.

[Shpaget]

5 hours ago, Ryaja said:

That's my point, it continues to get closer but is never quit there, it is always(I will use the strike through as an over line.) 0.01 off from 1.

No. They are exactly the same.

8 hours ago, Ryaja said:

1/3 ≠ 0.333... and can be expressed as 0.3 repeating with a 4 at the end.

There is no end. That's the whole point.

 

5 hours ago, razark said:

If 3 == "no", then 2 cannot be answered.  2 is a required response, therefore the poll is invalid.

 

But 0.99... does equal 1.

The term "superrationality" exists in game theory.

https://en.m.wikipedia.org/wiki/Superrationality

[Ryaja]

I was thinking, if 0.9(over line is strike through.) = 1 then 1 - 0.9 = 0, but only as an asymptote so it could be expressed theoretically as 0.01, now if I multiply that by infinity (yes I am aware this is only a concept) it equals one, but 0.01 = 0 so therefore 0*∞ = 1 and therefore 1/0=∞ so if x*0=y*0 then we divide out the 0s (which would be any real number, not infinity) so, if y was 2 and x was 1 we would get 1 = 2 and this kind of breaks things.

 

(Also if the value of 0.9 ever did reach 1 it would go over one immediately after depending on which infinity we use(is it absolute infinity or a smaller infinity.))

Also I found there is already a term for something that is super rational, hypertational is a word.

Edited December 18, 2022 by Ryaja

[AlamoVampire]

Simply put: no. Not at all. 0.9 is never 1. No matter how many decimal places you add, no matter how close you push it, it is not ever going to be 1. To me its that simple. 
 

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Edited December 18, 2022 by AlamoVampire

[Ryaja]

Just now, AlamoVampire said:

Simply put: no. Not at all. 0.9 is never 1. No many how many decimal places you add, no matter how close you push it, it is not ever going to be 1. To me its that simple. 
 

125112182022

Yes but then what is 3/3 if 1/3 is 0.333...

Edited December 18, 2022 by Ryaja

[Nazalassa]

2 minutes ago, Ryaja said:

what is 3/3 if 1/3 is 0.333

It's difficult.

[Shpaget]

1 hour ago, Ryaja said:

if 0.9(over line is strike through.) = 1 then 1 - 0.9 = 0, but only as an asymptote

No ifs and buts. They are exactly the same. Not "almost but no quite". Exactly the same.

1 hour ago, Ryaja said:

it could be expressed theoretically as 0.01

No, this makes no sense. Nowhere in the sequence of 0 is there a 1. It's exactly 0.

 

1 hour ago, Ryaja said:

therefore 1/0=∞

No.

1 hour ago, AlamoVampire said:

No many how many decimal places you add, no matter how close you push it, it is not ever going to be 1

The trick is to never stop adding 9s.

[Leganeski]

36 minutes ago, Ryaja said:

Also if the value of 0.9 ever did reach 1 it would go over one

0.999... is not a process. It is a single number, in the same way that 3.14159... is a single number. It does not "reach" 1; it is 1, and then that's it.

This is not to say that there is no infinite process going on. In fact, the notation "0.999..." references an infinite sequence: 0, 0.9, 0.99, 0.999, and so on. This sequence indeed approaches 1 without ever reaching it. However, 0.999... is not part of the sequence; it is the number that the sequence is converging to. This number is 1.

59 minutes ago, AlamoVampire said:

0.9 is never 1. No many how many decimal places you add, no matter how close you push it, it is not ever going to be 1.

That is correct. 0.9 is not 1, 0.99 is not 1, 0.999 is not 1, and so on forever. But 0.999... is not any of these numbers: it is the limit of all of them.

 

[AlamoVampire]

3 hours ago, Ryaja said:

Yes but then what is 3/3 if 1/3 is 0.333...

.33… isnt 1/3 any more than .99.. is 1. yes people treat it as such, but factually it isnt. 1=/= 0.99 

2 hours ago, Shpaget said:

The trick is to never stop adding 9s.

 

3 hours ago, AlamoVampire said:

….no matter how close you push it, it is not ever going to be 1. To me its that simple. 

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[Zozaf Kerman]

Owwwww, my head… 

Edited December 18, 2022 by Zozaf Kerman

[razark]

10 hours ago, AlamoVampire said:

Not at all. 0.9 is never 1.

Correct.  0.9 is 90% of 1.  0.999... however, is not the same thing.  It's 1.

10 hours ago, AlamoVampire said:

To me its that simple. 

Insisting that being wrong is correct does not make you correct.

10 hours ago, Nazalassa said:

It's difficult.

No, it's 1.

[kerbiloid]

Tell the Ancient Romans about fractions.

https://en.m.wikipedia.org/wiki/Roman_numerals#Fractions

[Nazalassa]

2 hours ago, kerbiloid said:

Tell the Ancient Romans about fractions.

https://en.m.wikipedia.org/wiki/Roman_numerals#Fractions

...

[Ryaja]

Wait, if 0.999... = 1 then how much can that be changed does that mean it is impossible to have a sequence of infinite 9s, so any number with that has to be simplified? Like 1.09999... to 1.1 or 999.999... to 1000 and furthermore does that mean 99.999...% = 100%?

[Shpaget]

1 hour ago, Ryaja said:

does that mean it is impossible to have a sequence of infinite 9s,

Infinite 9s is a valid notation.

1 hour ago, Ryaja said:

so any number with that has to be simplified?

It doesn't have to be, but it can. You can also choose to go the other way and represent 1 as 0.9... Both are valid.

1 hour ago, Ryaja said:

Like 1.09999... to 1.1 or 999.999... to 1000 and furthermore does that mean 99.999...% = 100%?

Yup, and 4.9... = 5

[AlamoVampire]

10 hours ago, razark said:

Correct.  0.9 is 90% of 1.  0.999... however, is not the same thing.  It's 1.

Prove it. No insinuation. Mathematically prove it in plain english.  With every term canceled out to bring it to 1. While I am not positing a positive reply and am thusly not required to prove the negative, I will support my answer as to why 0.99 can never = 1.

0.999… only tends to 1. It is not equal to 1. We only have an approximation.
0.999... and 1 are not equal because they're not the same decimal. With the exception of trailing 0's, any two decimals that are written differently are different numbers.

It's not possible to add up infinitely many things, so any infinite sum is only an approximate value, not a real value.

Rebuttal: In proof 1, we cannot cancel the trailing 9's because there are infinitely many of them. We will always be left with one 9. 

Reply: Cancellation doesn't happen "term by term," where we compare the first 9 in 10A10A with the first 0 in AA. We are looking at the difference of these two numbers, and taking it all together. We get a trailing series of 0's, with no "9 at the end." 

Yet this rebuttal cancels itself in a rebuttal to a different rebuttal to why your suggestion that 0.99.. is 1 is wrong: 

Rebuttal: In proof 2, we can't just add the digits "term by term."

Reply: This argument is valid in that the perspective that we're adding "term by term" is how we are evaluating the limit.

Point is 0.99… is not now nor will it or can it ever FACTUALLY be 1. It can only be treated as an implied 1, but implication cannot now or ever change fact. 
 

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Edited December 19, 2022 by AlamoVampire

[Ryaja]

This boils down to the value of infinitesimals, that is why I said with different notation you can express things like 1/3 that is not quite 0.333... in decimals.

[Shpaget]

13 minutes ago, AlamoVampire said:

Prove it. No insinuation. Mathematically prove it in plain english.  

0.99 is closer to 1 than 0.9

0.999 is closer to 1 than 0.99

0.9999 is closer to one than 0.999

There can be nothing closer to 1 than 0.9..., therefore they are the same.

13 minutes ago, AlamoVampire said:

Point is 0.99… is not now nor will it or can it ever FACTUALLY be 1. It can only be treated as an implied 1, but implication cannot now or ever change fact. 

No, you are wrong. They are exactly the same. In the field of Mathematics, this is not a controversial opinion.

[AlamoVampire]

15 minutes ago, Ryaja said:

This boils down to the value of infinitesimals, that is why I said with different notation you can express things like 1/3 that is not quite 0.333... in decimals.

Which is why its an insinuation and implication and  =/=. 
 

Lets be fair would you (anyone) be willing to spend tons of money for a supercar but only be delivered a car that is 99.99..% complete? You get delivered the car but no wheel lugs.

@Shpaget prove it. Refer back to my challenge you replied to. Oh btw everything I used for my proofs youre wrong are accepted in the mathematical community by those who agree with me that 0.99 is not 1. So again, if you cannot meet: 

21 minutes ago, AlamoVampire said:

With every term canceled out to bring it to 1.

Then you are factually wrong. Implication and insinuation get used true enough, but, you are ignoring factual reality. You cannot cancel out infinite terms and thus cannot ever bring 0.99 to a whole 1, thus I am factually and functionally correct and will never accept implication or insinuation. Does this make me biased? Yes. Does your stance make you biased? Yes, just the reverse of me. 
 

Bottom line: factually I am right. But, that said, I NEVER ONCE SAID: implication and insinuation are not useful. I just said that 0.99.. =/= 1 in a PURELY FACTUAL way and implication and insinuation =/= factual reality. 

We are all on 2 sides of the same coin. We will never convince the other side. Nor do we have to. 

I have said now all I can or need to say and will now walk away. 
 

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[Poppa Wheelie]

The "dot dot dot" notation means:  "an infinite number of these digits".  Or, "these digits repeat infinitely".

If you do the long division to get a decimal equivalent of the fraction 1/3, you get 0.333... (that's an infinite number of 3 digits)

• 3 goes into 10 3 times, carry the 1
• 3 goes into 10 3 times, carry the 1
• 3 goes into 10 3 times, carry the 1
• dot dot dot

This process goes on forever, there is no end to it, it requires an infinite number of 3s

So that's what the "dot dot dot" notation means, "an infinite number of these"

1/3 equals 0.333...

1/3 does not approximate 0.333..., 1/3 equals 0.333...

0.333... does not "approach" 1/3, it "equals" 1/3

Computers can't deal with this.  Computers cannot represent "an infinite amount" of anything.

If you do not agree with this, then you will not believe any proofs.  But here are two demonstrations:

Mathematical:

• 1/3 = 0.333...
• 3 * 1/3 = 3 * 0.333...
• 1 = 0.999...

Algebraic:

• x = 0.999...
• 10x = 9.999...
• 10x = 9 + 0.999...
• 10x = 9 + x
• 9x = 9
• x = 1

This Wikipedia page has the Algebraic demonstration, plus several complete proofs:  https://en.wikipedia.org/wiki/0.999...#:~:text=The meaning of the notation,%3D 1.

Edited December 19, 2022 by Poppa Wheelie