## Is 1 = 0.999... ? Really?

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## Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
13
41%
No, 1 ≠ 0.999...
16
50%
Other
3
9%

### Re: Is 1 = 0.999... ? Really?

The gap of a trillion to the trillionth power surprise You?
Meno_
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### Re: Is 1 = 0.999... ? Really?

Gloominary wrote:However, if your hand is any closer to the table than 1 cm away from it, even just the teeniest, tiniest bit, you will touch both the 1 cm domino, and the 0.99> cm domino, as you wave your hand over the table, knocking them both over, because 0.999> is as close to being 1 as you possibly can come without being 1, so anything closer than 1 cm away, even if it's only 0.0000000001 cm closer less is going to be within its range to interact with it.

Regarding the bolded part, one must ask: is that true? Let's not be bound by our number system. There are number systems other than base-10 and these other number systems can represent numbers that our number system cannot, such as numbers that are larger than 0.999~ but smaller than 1. Consider base-16 number system a.k.a. hexadecimal number system. In hexadecimal number system, 0.999~ is represented the same way, as 0.999~. But since we're dealing with hexadecimals, which work with more than 10 digits, there are numbers higher than it but lower than 1 e.g. 0.AAA~, 0.BBB~, 0.CCC~ and so on. Of course, the basic idea of the quoted part of your post is correct, but I think it's important to note that 0.999~ is not the largest number that is lower than 1. Rather, it is the largest number lower than 1 in decimal number system. This means it's possible for your hand to be closer to the table than 1cm away from it and still not knock the shorter domino over.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

It can, but the mathematical use of a recurring number, in industry, would lose its utility at a certain point.
Last edited by MagsJ on Fri Dec 27, 2019 1:03 am, edited 1 time in total.
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ

MagsJ
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### Re: Is 1 = 0.999... ? Really?

I made a mistake above. 0.999~ is represented differently in base-16 systems. Nonetheless, I am sure the point remains.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:I made a mistake above. 0.999~ is represented differently in base-16 systems. Nonetheless, I am sure the point remains.

..and the point is?
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ

MagsJ
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### Re: Is 1 = 0.999... ? Really?

The point is that there are numbers larger than 0.999~ but smaller than 1. For example, 0.FFF~ (in hexadecimal system) is larger than 0.999~ but smaller than 1. In base-20 system, we have 0.JJJ~ which is larger than both hexadecimal 0.FFF~ and decimal 0.999~.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

..and the solution would be?
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ

MagsJ
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### Re: Is 1 = 0.999... ? Really?

Solution to what?
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

I thought you posed a dilemma of sorts?
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ

MagsJ
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### Re: Is 1 = 0.999... ? Really?

This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
Also all the relevant arguments have already been made which is why the thread stopped two years ago
A MIND IS LIKE A PARACHUTE : IT DOES NOT WORK UNLESS IT IS OPEN
surreptitious75
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### Re: Is 1 = 0.999... ? Really?

surreptitious75 wrote:This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
Also all the relevant arguments have already been made which is why the thread stopped two years ago

Yes, but bases can be consolidated, no?
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ

MagsJ
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Posts: 20892
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### Re: Is 1 = 0.999... ? Really?

surreptitious75 wrote:This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here

I don't agree.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
surreptitious75 wrote:This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here

I don't agree.

I don’t agree either. Other bases are fair game in mathematical debates. (Btw... Magnus is right)
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

When I was reading through this, I noticed the following that seems to bring it all to a salient conclusion.

James S Saint » Mon Jun 05, 2017 3:30 pm wrote:0.999... is not a quantity or fixed value. It is an endless series of diminishing decimal values. The ellipsis "..." means "infinitely" - "never ending". The decimals begin but never end - open ended. The fraction is never satisfied by the series.

1 is an obvious quantity and fixed value with a beginning and an obvious end.

James S Saint » Mon Jun 05, 2017 4:32 pm wrote:the only alteration required is the acknowledgement that infinitely diminishing decimal series are technically not numbers or quantities. Such can be merely stated in a foot note. Nothing very serious changes anywhere.

It makes sense that "0.999..." is just an expression signifying that some ratio cant be expressed by a fixed number of digits (unlike 1). It isn't actually a number. And neither are all of those other expressions that end with "...". And apparently that is why the Wikipedia proofs are misleading.

To me, that seems like a "game over".
You have been observed.
obsrvr524
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### Re: Is 1 = 0.999... ? Really?

Well, James was very firm that infinitesimals were useful, and usefully different than the “convergence”...

Hence, his InfA thing
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Well, James was very firm that infinitesimals were useful, and usefully different than the “convergence”...

Hence, his InfA thing

This seems to explain why he would -
James S Saint » Wed Sep 03, 2014 8:58 am wrote:
Calculus
Ancient
The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components.[2] From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus.[3] The method of exhaustion was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle.[4] In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere.[5]

Medieval
Alexander the Great's invasion of northern India brought Greek trigonometry, using the chord, to India where the sine, cosine, and tangent were conceived. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.[6] In the 14th century, Indian mathematician Madhava of Sangamagrama and the Kerala school of astronomy and mathematics stated components of calculus such as the Taylor series and infinite series approximations.[7] However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".[6]

Modern
"The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking." —John von Neumann[8]

In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise was lost until the early part of the twentieth century. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.

The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term.[9] The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1670.

Newton didn't publish work on it until 1687, 17 years after Wallis, Barrow, and Gregory had done the conceptual theorems in modern language.

Sometimes you just have to ignore religious fanatics, especially the kind who claim to not be religious yet can't explain why they know what they think they "know".

I'm no maths genius but isn't calculus just the sum of the infinitesimals? Where would science be without calculus?
You have been observed.
obsrvr524
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### Re: Is 1 = 0.999... ? Really?

Obsrvr,

I’ve been in many threads with James ...

So I’m certainly putting my own words in his mouth:

James would say that both infinitesimals and approximations are useful in their own right and for different reasons.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Calculus is awesome.. it spans multi-disciplines.. where would industry be without it.
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ

MagsJ
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:It isn't actually a number. And neither are all of those other expressions that end with "...".

I am not sure why you think it's not a number i.e. a symbol representing some quantity. It appears to me that it clearly is. It has many properties that numbers have e.g. it's greater than some numbers and less than others.

What it isn't is a finite quantity, that's for sure, and that's why it can't be 1.
Last edited by Magnus Anderson on Sat Dec 28, 2019 6:57 pm, edited 2 times in total.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

MagsJ wrote:I thought you posed a dilemma of sorts?

Not really. I just expanded upon Gloominary's post.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
obsrvr524 wrote:It isn't actually a number. And neither are all of those other expressions that end with "...".

I am not sure why you think it's not a number i.e. a symbol representing some quantity. .
'
'
What it isn't is a finite quantity, that's for sure, and that's why it can be 1.

If it isn't a "finite quantity" I imagine that it isn't a number.
You have been observed.
obsrvr524
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### Re: Is 1 = 0.999... ? Really?

The whole question boils down to a confusion between the qualitative and the quantitative.

"1" is clearly a precise quantity, but as soon as you profess 0.(9) you add in the quality of "endlessness" to describe the repetition of the quantity of "9" for each decreasing power of 10 (or whatever base you're using).

0.(9) is an attempt to restate the quantity "1" in a way that involves endlessness. As is 0.(3) to restate 1/3 when one divides 1 by 3. It's an admission that one cannot denote 1/3 etc. entirely quantitatively without the use of the quality of endlessness. Multiplying 1/3 again by 3 is obviously 1 (3/3), yet multiplying 0.(3) by 3 is not so obviously 1 (0.(9)) precisely because of the injection of the qualitative into the otherwise entirely quantitative.

Subtracting 0.(9) from 9.(9) to get the exact quantity of 9 requires the same confusion.
As soon as you allow the notion of the qualitative into the quantitative you invite possibilities such as ε as an epsilon number and so on.

This is the same kind of mistake that every extended or "new" number set allows - much to the advancement of mathematics and other utilities... but not truths. Experientialism highlights the distinction.
So we see how useful it is to make particular types of mistakes that are not true, but are useful: such as the notion that 1 =/= 0.(9)
Is it really? No.
But that's the wrong question.
The more useful question is whether any new knowledge can be gleaned from the possibility that 1 =/= 0.( 9)

Silhouette
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:0.(9) is an attempt to restate the quantity "1" in a way that involves endlessness.

That would be "1.000...", not "0.999..."
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obsrvr524
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:
Silhouette wrote:0.(9) is an attempt to restate the quantity "1" in a way that involves endlessness.

That would be "1.000...", not "0.999..."

No, I meant 0.(9)

The recurring 9 necessarily must recur endlessly without defined quantity (infinitely) for 1 to divide 3 times into 0.(3) and multiply back to 0.(9)
For 0.(9) the 9s must recur with the quality of endlessness for it to be subtracted from 9.(9) to equal 9 exactly. If they didn't, you couldn't get 9 exactly.

By contrast the quantity of 0s that you put after "1" doesn't matter, whether it's 1.0, 1.00 and so on - it doesn't affect the quantity of 1 even if you try to impose the quality of endlessness with 1.(0)
The quantity of 9s that you put after "0" most definitely matters because 0.9 is different to 0.99 and so on with any non-endless (finite) quantity of 9s after the 0 different to 0.(9) with its quality of endlessness.

So 0.(9) is 1 only with the quality of endlessness, where 1.(0) is 1 with or without the quality of endlessness.
Endlessness is irrelevant to 1 when denoted as 1 so endlessness isn't necessarily involved.
Endlessness is essential to 1 when denoted as 0.(9) so endlessness is necessarily involved.

There's no issue dividing 9 by 3 to get 3, and then multiplying it back by 3 to get 9. The 9 is the same before and after the operations.
So why would dividing 10 by 3 to get 3.(3) and multiplying it back by 3 to get 9.(9) be any different? The 10 is the also same before and after the operations.
There's no issue subtracting 1 from ten times that i.e. 10 to get 9.
So why would subtracting 0.(9) from ten times that i.e. 9.(9) not get 9?

0.(9) not being 1 requires a double standard for (mod 0) versus some other modulus, which removes a fundamental necessity that's essential to mathematics: that it's consistent.

But that's my whole point: introducing any notion of the quality of endlessness to quantities confuses everything. That's why infinities are such a minefield.
Hence why "1 = 0.(9)?" is the wrong question - the more useful question is what happens if you go against the truth that they are equal and say they aren't. It's what we did for complex numbers - there is no square root of minus one in truth, but what if there was? What usefulness can "i" provide? Turns out it provides a lot of usefulness even though it's not true that "i" exists any more or less than it's true that epsilon numbers and hyperreals exist.

The whole debate behind this thread isn't looking deep enough - and as always, Experientialism puts it all into perspective.

Silhouette
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
MagsJ wrote:I thought you posed a dilemma of sorts?
Not really. I just expanded upon Gloominary's post.

Ah..

I see that Magnus has come back to ILP much wiser than when you had departed, to KTS.
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ

MagsJ
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Posts: 20892
Joined: Wed Nov 01, 2006 2:59 pm
Location: Suryaloka / LDN Town

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