The Instigator
Con (against)
3 Points
The Contender
Pro (for)
3 Points

This House Believes 0.99* Equals 1!

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Voting Style: Open Point System: 7 Point
Started: 9/22/2014 Category: Philosophy
Updated: 2 years ago Status: Post Voting Period
Viewed: 1,295 times Debate No: 62102
Debate Rounds (1)
Comments (32)
Votes (2)




Following my opponent's declaration of intent (see ), I hereby challenge him to deliver at least ONE logically, mathematically or philosophically acceptable argument for his outlandish claim that 0,99*=1.

I say the two are not the same, and will prove so below. In order to win, my opponent has to disprove my argument and deliver one sound argument of his own, as disproving my proof is not enough to back up his claim. In this case, as it is apparently put in this site's jargon, burden of proof is shared.

My case:

The very idea of REAL numbers is that they are indeed continuous. That is why the rational numbers are superseded by the idea of real numbers by adding the irrational numbers (like pi or the square root of 2) which can not be described by a fraction.
Between any two rational numbers there's an actual infinite count of irrational numbers continuously filling up the void between fractions.

How do we achieve this continuity then?

Obviously, there must be a concept of real neighboring numbers. This poses a dilemma.
If two neighboring real numbers have actually no distance between them, they must be considered identical. If any two neighboring numbers are identical and we extend this to the infinity of all neighboring numbers, all numbers must be identical, which means there are no numbers at all. Only one number would remain.
If there is a distance, how large is it?
The answer is obvious: the difference is infinitesimally small/infinitely small.

This is very hard to understand for many, and hence there's a misconception.

Actually, there's no better example to illustrate this than the misconception of 0.99*=1.
0.99* means that we have an infinite number of decimal places with a value of 9.
Now, let's focus on the distance between 0.99* and 1.
Obviously, the number we're looking for has an infinite amount of decimals with the value of 0, except for the last decimal, which must be 1.
Now, people keep telling me that this number does not exist, because we never reach the 1, as there are INFINITE 0's in the way.
But look more closely: the 1 is in the same decimal place as the last 9, always. This means there's INFINITE MINUS ONE 0's in the way only!
Which means as long as you count the 9's on the one side, you have to accept the 1 at the end of the chain of 0's on the other side of the equation, or the whole problem does not make any sense to begin with. This turns the whole "pro" side of this debate into one giant straw man argument.
1-0.9=0.01 is just plain wrong. Same goes in extension for any added decimal places. But that is not what people who see a difference between 0.99* and 1 have ever claimed in the first place! It's only what the pro side makes it out to be - wrongfully, one must add.

What people who claim that 0.99*=1 do is to fool people with a double standard for infinity. They claim that "infinite" does away with differences and decimal places. Which is simply not how infinity works. Either we adhere to the same standards on both sides of the equation or we don't. If there is 0.99*, then so is the number that would have to be called 0.0*1 or 0.00...1.

All claims that 0.99*=1 are based on the misconception that what cannot be told apart must be identical. But it is meanwhile common knowledge that indiscriminable is not identical.
As renowned physicist Dr. John Denker so aptly puts it:
"I am going to make an important distinction between the concepts of
- identical
, which means intrinsically identical, versus
- indistinguishable
As a macroscopic illustration, consider two cars. I choose to call them intrinsically identical whenever they have the same make and model, same color, et cetera. Nevertheless you might be able to distinguish them on other grounds. For example you might be able to distinguish them on the basis of position: a car ’over here’ is distinguishable from a car ’over there’."

The incomprehensibility of the concept of "infinity" adds to the confusion.
"Suppose that you are presented with two black sticks lying half a meter in front of you on a white background, suppose moreover that their appearance is
indiscriminable to you. But if you then go nearer to them and
measure them with a ruler, you may discover that one of them is 20
cm long, while the other is 20.1 cm long. This would not be a
surprise: we commonly accept that our perceptual abilities are not so
fineR08;grained as to detect any slight difference in the physical bases
which elicit our perceptions."

Clearly, we have an extreme case of the same matter here. We have an infinite distance (to the last digit of an irrational number) and the smallest conceivable difference (between to neighbouring real numbers). The misconception does, as the above quote puts it, not come as a surprise. It's still plain wrong:

Saying that a difference actually ceases to exist only because it happens beyond an infinite distance just doesn't make any sense.
It is only natural that we don't CARE whether a hair on the head of a bank clerk in a remote town on the other side of the planet is white. That doesn't mean that the man does not have a white hair.
Infinity does not mean that things don't happen, it only means it's so far away we can't MEASURE it. Two grains of sand on a Mars don't become IDENTICAL only because we can't reach them or tell them apart. They remain two distinct grains of sand.
So, why would two numbers that differ in a single decimal place REALLY far away have to be considered identical? What far-fetched reason other than neglect could lead to such inane conclusions? Reality does not care a lot about our limited perception.

So the answer is: 0.99* is the direct real neighbour to 1, both are indistinguishable in the set of rational numbers, because 0.99* is not actually a rational number, it is just taught at school as being identical in order to help undergraduate pupils fill the gap between rational and real numbers. Anyone above that level of maths should be able to understand that those two numbers cannot be identical due to the simple matter of decimal places, as shown above.

What does this mean in application?
We can still calculate day-to-day math problems with 0.99*=1. The difference is so small we can't measure it, after all.

But it would be a critical failure to assume that this is "true" for all branches of mathematics.
A probability of 1 means something is absolutely certain and in no way a matter of chance. Not a single exception to an event with a probability of 1 can be found, it MUST happen without a fault. Something with a probability of 0.99* however does know a SINGLE exception, ever, in all of eternity.
In a way we may owe our existence to this fact with a certain probability, as the singularity some claim to be responsible for the creation of the big bang would have been this one-off chance, the one moment that made it all count.

This is why this debate is considered a philosophical debate by me, not a scientific one.

All in all, it just boils down to the question whether you accept that two things that are very much alike become the same thing. This may be a philosophy for some. It is, however, not logically sound.

And in no way can a difference in decimal places be explained away by infinity.

I leave this stage to my opponent, should he choose to accept.
Thanks to all who read this.


I am sorry for the delay. It was one thing after another, and I could not get to this sooner. I feel that in this debate, I need not disprove my opponent's entire case one by one. As I am the Proposition, if I can beyond a doubt show in my round []T which T standing for the resolution then I win. This is as simple as it gets. I will present a series of proofs. The first will be simple, an algebraic proof, the next two will be fairly complex. As for my opponent's algebraic formula I should tel him that anything into infinity is infinity, and infinity minus/plus/anything one/any other number is infinity.[1] Often novices in Mathematics make a mistake of assuming a 9 at the end. I use * to show recurring. It is not that 1-0.99*=0.00...01. This is wrong[2]. The one comes after infinity, so it both exists, and does not exist.

Let us suppose the value of 0.99* as 'x'.
So x=0.99*
Thereby 10x=9.99*
This may be written as 10x=9.00+0.99*
Then let us remove 0.99* by doing so that 9x=9.00+0.99*-0.99* (as x=0.99*)
We therefore have 9x=9
Using algebra x=9/9
Which yields in x=1.

We have therefore shown algebraically that 0.99* is in fact the same as one. It is here that we have already won this debate. Let us show you another algebraic proof.

Let us take the fraction 1/9
In decimal form it is 0.11*
Let us multiply it with 9
We get 0.99*

This can also be proved from Decedent Numbers, p-Adic numbers, Calculus works on the assumption of this. It is how limits work with k-->0. Or any such thing, to disprove the statement is impossible, and if one does it then all Mathematics shall fail.

[1]Infinity: An Essay in Metaphysics
[2]Mathematics: A Very Short Introduction by Timothy Gowers
Debate Round No. 1
32 comments have been posted on this debate. Showing 1 through 10 records.
Posted by 9spaceking 2 years ago
incredible. A purple circle ties the powerful Ajabi.
It's even a one round debate! Ajabi should have been able to insta-snipe-insta-win this thing.
Posted by BoggyDag 2 years ago
The problem with all of your alleged proofs is that theyare basically circular.

Ajabi claims that 0.99*=1 due to 1/9=0.11*.
The latter of which is not proven within this context, as it is the same problem. You use a misconception of infinity to prove itself.
In your particular case, you use Aristotle's law of non-contradiction, but we alreay know, thanks to Heisenberg, that not all matters are true or false, some are just "undetermined".
Infinity, by its definition, can not be determined in the way you do. Contradiction is not enough, as in infinity, more than the limited two alternatives you list may be possible.
First, you would have to prove that the law of non-contradiction applies to infinity.
It doesn't, as I showed in my round. Either the distance between any two numbers is larger than zero, or all numbers become the same. Yet the distance obviously cannot be measurably large, as then there would be no continuity.
Both are true. I repeat: name a number between 0.0*1 and 0!
Both are not identical. The difference is nigh zero, infinitesimally small, but NOT zero.

All you can do is quote mathematicians who were actually wrong, in making circular conclusions. Like Euler corroborating a statement about a limit using a formula that contains a limit!
What you can't for the life of you is answer a single of my questions. Which means you are just a copyist, not a thinker. Which is why you keep repeating yourself.
As long as you evade the challenge of naming the number between zero and 0.0*1, you are just pathetic.
If no other than zero infinitesimals exist and numbers are continuous, then any conceivable number must exist. Since I can imagine 0.0*1, the number must then exist, and you must back up your claim that there's room between this number and 0.
All you prove is the same neglect I have covered in my round already, which means you are biased.
Case closed, "judge".
Posted by Enji 2 years ago
Sorry if you thought my previous post was an explanation for why the two are contradictory, but the proof is roughly as follows.

First you define the real numbers using the ordered field axioms and the completeness axiom; Richard Dedekind's continuity axiom works (and tends to be used in more rigorous contexts), but the least upper bound / greatest lower bound property is similar and easier to deal with.

Next you assume the real numbers contain non-zero infinitesimals, and you attempt to define a cut above all the positive infinitesimals, or identify the supremum to the set of positive infinitesimals (positive infinitesimal numbers being numbers less than 1/m for all natural numbers m). The result is that no such Dedekind cut exists, or that there is no supremum to the set of positive infinitesimals, which contradicts how the real numbers is defined. The conclusion of this proof by contradiction, therefore, is that the real numbers do not include non-zero infinitesimals. This proof is covered in Keith Stroyan's textbook "Foundations of Infinitesimal Calculus", which, if you're interested in learning more about non-standard analysis, is an excellent resource on the topic.

This is why the real numbers can be proven to satisfy the Archimedean property, whereas number systems which contain non-zero infinitesimals (such as the hyperreals used in nonstandard analysis and your alleged 'better' real numbers which you brought up in the comments), by definition, do not.

Given this background, it's easy to see why in the real number system there is no need for neighboring numbers or for a minimum non-zero distance between two real numbers; for any distance 1/m>0, there exists a smaller distance 1/(k*m)>0 where k>1.

For future reference, the debate consists of the content presented in the debate rounds, and not any content presented in the comments. You didn't make any mention of conduct during the debate -- you first mentioned conduct in the comments section.
Posted by BoggyDag 2 years ago
@Enji: Since I specifically ASKED to award Ajabi points for conduct DURING the debate, please by all means do.
Pretentious pr*ck.
Posted by BoggyDag 2 years ago
That's a recursive false conclusion.
If - as I propose - a non-zero infinitesimal is considered infinitely small, but still adjacent, there's no PRACTICAL, only a philosophical difference. Which can be easily accepted intuitively. There needs to be a difference larger than zero between any two non-identical numbers. It just needs to be infinitely small, hence explicitly NOT Zero.
I say the exact opposite. If you accept only zero infinitesimals, you reach identity of all numbers, not a continuous set.
However, if we define what you call zero-infinitesimal AS the number 0.0*1, what's the big deal? I say it is only a philosophical one, not one that can not be mended by change of perspective.
Posted by Enji 2 years ago
The real numbers cannot be both continuous and include non-zero infinitesimals; if a number system contains non-zero infinitesimals then it is not continuous, and if it is continuous then it does not contain non-zero infinitesimals. Your attempt to merge both creates a self-contradicting number system, which is a much bigger dilemma than the problems you have with the real number system.

The facts of the debate remain as follows: (1) you specifically refer to the real number system in the debate, and you specifically refer to the three axioms which define the standard real number systems within your argument; (2) your argument relies on the existence of non-zero infinitesimals which do not exist in the real number system -- further you deal with infinite decimal places improperly, and hence you fail to satisfy your burden of proof; (3) Ajabi's second (and first) proof are correct in the context of the real numbers, thus satisfying his burden of proof.

Therefore my vote is well corroborated by the content of the debate.

The comments section is not to be considered for voting, or I would award conduct to Pro as well.
Posted by BoggyDag 2 years ago
So, what you are saying is that every old master is true, and progress should be forbidden. Arrogance, your name is Ajabi.
This is a philosophical debate. If it is not allowed to question Euler, how will we ever advance? I worship Euler, but that does not man he was not eventually bound to have a border. In this particular case, Euler needed to corroborate his idea of real numbers. It must have been difficult enough to establish the irrtional numbers. Expanding knowledge is an iterative process. One step at a time. Even Euler was only human. One of the most brilliant.
Who says I didn't already write that book? Hm?
You don't know who I am and what my merits are, and I'd prefer it to stay that way.
Posted by Ajabi 2 years ago
Boggy you should obviously write a book so you can go and collect your award at Stockholm, seeing how you basically went against everything Euler said in his Algebra. Not to mention not even going to argue.
Posted by BoggyDag 2 years ago
I quote from my comment further down:
"It's long overdue to merge non-standard analysis and standard analysis to form a proper set of axioms for the actual real numbers."
I neither adere to standard analysis nor strictly to non-standard analysis. Obviously, both have strengths that I believe can be reconciled.
And while I do not claim to have it all worked out, both can be merged to solve the dilemma at hand.
Whereas the standard theories do not suffice to do just that.
I really wish you would at least learn to read properly.
Posted by Enji 2 years ago
I know I said I wouldn't return, but I'm going to say this anyway: you reference three axioms of the real numbers in the debate: the field axioms (as you do arithmetic such as 1-0.9), the order axioms (e.g. neighbouring numbers), and the continuity axiom (as you state the real numbers are the completion of the rational numbers by combining them with the irrational numbers). It can be rigorously proven that any field (with a range from negative infinity to infinity but not including either) which satisfies these axioms must be isomorphic -- this means you must be talking about the same real numbers as 'mainstream' mathematicians in the debate. In non-standard analysis, which you mention solely in the comments, the continuity axiom is not satisfied; your alleged "actual" real numbers which include non-zero infinitesimals, therefore, are not the real numbers discussed in the content of the debate because this would contradict the continuity axiom.
2 votes have been placed for this debate. Showing 1 through 2 records.
Vote Placed by iamanatheistandthisiswhy 2 years ago
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Total points awarded:30 
Reasons for voting decision: Simply put Pros proof is flawed, even though it seems to be not flawed. Pro says we should put 0.99* as 'x'. Then we have 9x = 9. And now we divide by 9 to get x = 1 = 0.99*. However, we already placed 0.99 = x. Which means that we have x equal to two separate values and a paradox. Additionally, its evident that 1 cannot be equal to 0.99* as Con showed that 1-0.99* = 0.00*1. While this may seem illogical it in fact logically consistent with mathematical operations and the weirdness of infinity.
Vote Placed by Enji 2 years ago
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Reasons for voting decision: Pro points out the following error in Con's argument: Con argues that in the difference 1-0.(9)r there are not infinite zeroes before the 1, there are only infinite minus one zeroes therefore the difference must be greater than 0; however infinity-1=infinity so his point is flawed. Pro's simple case [1/9 = 0.1r ; 9/9 = 0.9r = 1] is sufficient to prove the resolution true (I think this proof is simpler and more straightforward than the initial proof he puts forth, but either works).