Does 0.9999... = 1? NO : Why All Proofs for 0.999...=1 are Wrong. Please Discuss On Social Media

This video explains in detail why all proofs for 0.999...=1 are wrong. It presents both sides of the argument but with a bias towards the position that 0.999... does not equal 1. It points out the problems with the foundations of mathematics that are highlighted by this dispute. Please go ahead and discuss the issues raised in this video on your chosen forums and social media platforms.

Video Guide:

00:00:00 Preamble: Very brief section questioning what it means for mathematics to be logically correct.

00:00:41 Synopsis: Says that this video will question the value of mathematical definitions and rules of logic where these things are just made up instead of having a firm basis in physical reality. As such it will question the validity of limits and abstract axiomatic systems.

00:02:43 Foreword: My experience is that mathematicians are intolerant of opposing viewpoints and I have even been told that I have been banned because my arguments might corrupt young minds (this is similar to creationists banning the teaching of evolution).

00:06:55 Intro: The symbol 0.999... has different meanings and so it is not exclusively reserved to mean 'the limit of the corresponding sequence'. And not even all mathematicians accept the concept of 'real numbers'.

00:10:00 Explains why a non-mathematician might claim it makes no sense to say that unending non-zero terms can be said to 'converge to' a constant value, using 0.999... and the so-called square root of 2 as examples.

00:16:42 This section describes the history of unending sequences in Ancient Greece and includes Zeno's paradox of Achilles and the tortoise.

00:22:37 About geometric series: It seems like the word 'limit' is also called 'sum' to make it sound like an infinite amount of non-zero terms can add up to a constant. It is far from convincing because it relies on definitions. The geometric series is effectively said to equate to a constant 'by definition'. And the basis of equivalence between different geometric series does not seem to be fair.

00:28:07 Problem 1: The first problem with the mathematician’s intuitive explanation which shows no points can exist on the number line between 0.999... and 1 is that we can use the same logic just switched around to argue that points MUST exist between 0.999... and 1.

00:32:04 Problem 2: To say that we can imagine infinitely small points on an infinitely thin number line is far from 'intuitive' and is arguably impossible.

00:33:19 Problem 3: How can all the partial sums 0.9, 0.99, 0.999 and so on exist as static points in static unchanging positions on the number line without there being a last one of these points before 1?

00:35:25 Problem 4: The formal proof version of the intuitive explanation starts with the (invalid) assumption that 0.999... must be a constant.

00:38:11 Problem 5: How can the proof that 0.999... equals a constant be valid regardless of whether or not the starting assumption that it equals a constant is valid or not?

00:38:40 Problem 6: Why it is highly dubious to claim that the division process for 1 divided by 3 yields an infinite result with no remainder part.

00:41:57 Problem 7: Why 1 minus 0.999... yields an unending series, not the constant zero.

00:43:44 Problem 8: Why the shift-and-subtract operation performed in the algebraic proof completely invalidates the so-called proof. This section includes discussion about the Riemann rearrangement theorem.

00:50:04 Problem 9: Why the argument that 1/3 has a finite representation in some bases does not mean that a corresponding representation must exist in base 10.

00:52:02 Problem 10: Why it is highly dubious to argue that the representation of a number is not the number itself, and why set theory doesn't clarify matters.

00:58:42 Problem 11: Why arguments based on methods for supposedly constructing so-called real numbers are just re-packaged versions of previous arguments that we have already found to be invalid.

01:00:10 Problem 12: Why the nested intervals theorem argument is just a re-packaging of the so-called intuitive argument. It also introduces more issues by suggesting that an arbitrarily small interval can exist, and that it can contain exactly one number.

01:01:46 Problem 13: Why the mathematician’s acceptance of arguments that are 'logically valid' make a complete mockery of mathematical logic.

01:04:57 Summary comparison showing the key areas of disagreement

01:06:36 Overview of disagreements concerning some foundational principles of mathematics. This includes why mathematicians appear to like mystery rather than clarity, they prefer usefulness over correctness, they belief mathematical proof is invincible and that it is fine to ban people they label as 'cranks'.

01:16:52 Conclusion: Unless we accept a load of absurd arguments, then 0.999... cannot equal 1. Also mathematics is not a science because it is not based on empirical evidence. It is merely a popularity contest for make-belief theories.

[KarmaPeny]

Video Guide:

00:00:00 Preamble: Very brief section questioning what it means for mathematics to be logically correct.

00:00:41 Synopsis: Says that this video will question the value of mathematical definitions and rules of logic where these things are just made up instead of having a firm basis in physical reality. As such it will question the validity of limits and abstract axiomatic systems.

00:02:43 Foreword: My experience is that mathematicians are intolerant of opposing viewpoints and I have even been told that I have been banned because my arguments might corrupt young minds (this is similar to creationists banning the teaching of evolution).

00:06:55 Intro: The symbol 0.999... has different meanings and so it is not exclusively reserved to mean 'the limit of the corresponding sequence'. And not even all mathematicians accept the concept of 'real numbers'.

00:10:00 Explains why a non-mathematician might claim it makes no sense to say that unending non-zero terms can be said to 'converge to' a constant value, using 0.999... and the so-called square root of 2 as examples.

00:16:42 This section describes the history of unending sequences in Ancient Greece and includes Zeno's paradox of Achilles and the tortoise.

00:22:37 About geometric series: It seems like the word 'limit' is also called 'sum' to make it sound like an infinite amount of non-zero terms can add up to a constant. It is far from convincing because it relies on definitions. The geometric series is effectively said to equate to a constant 'by definition'. And the basis of equivalence between different geometric series does not seem to be fair.

00:28:07 Problem 1: The first problem with the mathematician’s intuitive explanation which shows no points can exist on the number line between 0.999... and 1 is that we can use the same logic just switched around to argue that points MUST exist between 0.999... and 1.

00:32:04 Problem 2: To say that we can imagine infinitely small points on an infinitely thin number line is far from 'intuitive' and is arguably impossible.

00:33:19 Problem 3: How can all the partial sums 0.9, 0.99, 0.999 and so on exist as static points in static unchanging positions on the number line without there being a last one of these points before 1?

00:35:25 Problem 4: The formal proof version of the intuitive explanation starts with the (invalid) assumption that 0.999... must be a constant.

00:38:11 Problem 5: How can the proof that 0.999... equals a constant be valid regardless of whether or not the starting assumption that it equals a constant is valid or not?

00:38:40 Problem 6: Why it is highly dubious to claim that the division process for 1 divided by 3 yields an infinite result with no remainder part.

00:41:57 Problem 7: Why 1 minus 0.999... yields an unending series, not the constant zero.

00:43:44 Problem 8: Why the shift-and-subtract operation performed in the algebraic proof completely invalidates the so-called proof. This section includes discussion about the Riemann rearrangement theorem.

00:50:04 Problem 9: Why the argument that 1/3 has a finite representation in some bases does not mean that a corresponding representation must exist in base 10.

00:52:02 Problem 10: Why it is highly dubious to argue that the representation of a number is not the number itself, and why set theory doesn't clarify matters.

00:58:42 Problem 11: Why arguments based on methods for supposedly constructing so-called real numbers are just re-packaged versions of previous arguments that we have already found to be invalid.

01:00:10 Problem 12: Why the nested intervals theorem argument is just a re-packaging of the so-called intuitive argument. It also introduces more issues by suggesting that an arbitrarily small interval can exist, and that it can contain exactly one number.

01:01:46 Problem 13: Why the mathematician’s acceptance of arguments that are 'logically valid' make a complete mockery of mathematical logic.

01:04:57 Summary comparison showing the key areas of disagreement

01:06:36 Overview of disagreements concerning some foundational principles of mathematics. This includes why mathematicians appear to like mystery rather than clarity, they prefer usefulness over correctness, they belief mathematical proof is invincible and that it is fine to ban people they label as 'cranks'.

01:16:52 Conclusion: There are a load of arguments that we might think are absurd, and if we don't accept all of them, then we can only conclude that 0.999... cannot equal 1. Also mathematics is not a science because it is not based on empirical evidence. It is merely a popularity contest for make-belief theories.

[lordspongebobofhousesquare]

I'm a math student, so I guess I'm part of "mainstream math". Since you don't believe in the real numbers, I won't argue about 0.999... = 1 as it will be rather counterproductive. I also won't berate you.

I would rather ask about other things. At 34:50, I don't get your argument here. What do you mean by transitioning from passing a finite amount of points to an infinite amount of points? Even if you drop the irrationals, there are still an infinite amount of rationals on any given interval. If you "pass your finger" on any small amount of distance on the ruler, you'd still pass an infinite amount of rational numbers.

Your next point was that "space is granular". What do you mean by this? You can't find a direct successor of predecessor to any rational number, which you could easily proof via contradiction.

[satiremuch]

If I remove an infinitely small point from a square, have I then removed anything?

[antoneogzewalla]

In a comment elsewhere, you said, "It's absurd to talk about an infinite amount of anything. I'm trying to present arguments to point this out, but I should not need to because it is self-evident in my opinion."
In my opinion, the only thing "Infinite" means is that the infinite thing is [endless]. Like a heap, this is a concept that does not involve a magnitude. And this is the sole reason that (like a heap) we can add to or subtract from an infinite set and it is still infinite. To understand how infinity works, we can re-envision Hilbert's hotel as a hotel that is in progress. Imagine a genius who has created a team of nanobots who are capable of repairing themselves and are fully powered by the dimmest light of distant stars... so theoretically they can last forever. These bots are tasked with gathering space dust and building a hotel. It doesn't matter how many rooms you start with (1 is enough) or how fast the bots build new rooms. If all of the rooms are full and a new guest shows up to the hotel, that quest will eventually receive a room if they wait long enough, because the rooms are forever in the process of being built. The hotel is NOT infinite because of how large it is... it is infinite because the rooms will (theoretically at least) continue to be built forever.

[antoneogzewalla]

I found your video interesting. And I definitely agree with all that you said in your introduction. I've experienced all the same sort of criticisms.
I, however, think the statement [.999... = 1] is both true and false, depending on how you define [.999...].

I think the confusion is because mathematics generally fails to distinguish between the actual and the conceptual aspects of reality--which are both quite necessary to understand reality accurately. The problem is that like all reciprocal aspects, the actual and the conceptual are alike in some ways and they are the opposite in others. 1/2 is the reciprocal of 2/1, but when you take the inverse of either reciprocal it is functionally equivalent to the other. This is true with all aspects of reality in my opinion, and I believe it explains how the statement can be both true and false at the same time.

0.999... is what I like to call an infinite number. To me, this simply means there are an infinite number of 9's -- which means that the nines never stop coming. Since the nines never stop, 0.999... obviously can never be actualized. Which is to say that we can never reach the last 9 and say, "Okay, it's finished now." To me, this means that (what I like to call) the actual aspect of 0.999 is NOT equal to 1. Because that aspect is still continuing... and must always continue. Therefore, it isn't 1... yet.

This part of my answer agrees with you. However, I don't think this is the whole story, because we can think of concepts that aren't actual. For instance, we can imagine creatures called Pegasus or unicorns, for example. They aren't actually real, but we can imagine story lines that might happen if they were real. In much the same way, we can imagine what would be implied if 0.999... were actualized. Meaning that the last nine had been finished. If that were the case, then 0.999... would suddenly become equal to 1, because if point A is infinitely close to point B, then point A is point B. If it weren't, then we could put another point between point A and point B... And Point A would not be infinitely close to B. Using the same logic, because 0.999... has been actualized, it is infinitely close to 1. Which means that it is 1.

Now, since it is impossible to actualize [.999...] you might say that only the first part of my answer is correct. But, the way I see it, [...] literally means, "imaginer that all the infinite number of nines are there... thus, the way I see it, [.999...] literally means the [concept of an actualized .999...] Thus, [.999...] necessarily is equal to 1.

I'd love to hear your thoughts on this...
And, if you're interested, I also have several other untraditional views on other math topics that I'd love to share with someone who might be interested in an untraditional POV and who has the mathematical background to think about it in formal mathematics terms, instead of just from a philosophical/logical POV.

[andrewfeng]

In a sense mathematics is a mix of science and art: pure mathematics simply does not follow the scientific method, and I'm okay with that as a mathematician. I don't feel the need to base mathematics on physical entities, and I don't find it offensive that you compare orthodox mathematics to religions (I'm not religious btw, I just find math beautiful; religions? not so much). I think you did a great job pointing out exactly what in the orthodox argument that you disagree with, but I don't find them "absurd". For example, I think the definition of real numbers as limits is brilliant, creative, and neat.

Sorry to disagree.

[johnscovill]

One third plus two thirds is dual to one. One third is 0.333……, and two thirds is 😮0.666……, so 0.333… + 0.666… is equal to one.

[PeterVogopoulos]

Thank goodness for you. I was exposed to this notion of 0.9999 equaling 1 and was incredulous that this was a thing. Your video is the best one and properly examines and pokes the correct holes.

[peterosudar]

You must make a video on the Rubik's cube, and "cubes" or mathematical manipulatives(like a pyraminx) and consider the fundamental concepts of trigonometry(as Wildberger looks at with ratTrig->finite field trig) and move it towards chromogeometry.

You can take your argument to productive ends when you begin to engage with cubing.... new terminology will come in... it will make you feel like a child again... which might seem "bad" but if you think of being a "child" as good then you can... "build math back better"...

[richardlbowles]

Thank goodness for people like you, KP, who are prepared to defy accepted opinion and speak out. Even if you were wrong (and I have no reason to believe that you are), then the establishment, for want of a better word, is still wrong to try to shut you down.
At least they don't have the power to force you to drink hemlock, like some other "thorns in the flesh".

[moiqtheplayer]

Thanks to you, I, as a disbeliever or even dissident for 0.9999 recurring=1, have some proofs to show to people who believe it and other disbelievers or even dissidents

[nickd]

This video deserves more views. Thank you very much : )

[aerosoapbreeze]

Why does this channel not have way more subscribers. Total BS - Google sucks!

[m.c.]

i can't believe that people are actually arguing about this, hahaha, this make for great entertainment .
you will get infinitely close, but it is not one .

[jeffware]

One of their proofs is false: by the rule of multiplying by ten.
x= 0.999...9
10x= 9.999...0
10x= 9 + 0.999...0
9x = 9 + 0.000...1
X in the first line isn't the same as the "x" added to the 9 in line 3, due to ANY number times 10 has a virtual zero added at the end. (0.77 x 10 = 7.70)
Thus 9x will not = 9, and X will not 1. But they ignored the rule above, and claim 0.999...9 times 10 just equals 9.999...9 so it would give their x = 1 proof.