The 0.999...= 1 Controversy & Is Mathematics Fundamentally Flawed?

It's the biggest controversy in mathematics... the 0.999...=1 debate just won't go away. We examine and analyse all the common arguments and proofs for 0.999...=1. The controversial aspects are highlighted and the foundations of mathematics itself are questioned. Feel free to discuss the points raised in this video on your chosen forums and social media platforms.

Nobody can explain how the infinitely many points 0.9, 0.99, 0.999, ... can exist all at the same time, at different positions on the number line, without there being a last one. But mathematicians seem content to accept that they just can. They claim we can simply restrict ourselves to talking about ‘the limit’, and that by ignoring the problem of the last point we can claim that “there is no problem”.

This uncovers one (of the many) fundamental problems of mathematics, which is that there is no rigour about what forms an inconsistency/absurdity/contraction.

For example, say one group of people believe that it forms a trivial contradiction to claim that a decimal with endless non-zero trailing digits (such as the decimal for the square root of two, 1.41421...) can be said to equal a constant value. However, another group of people might disagree, and so which side is right?

Since maths is supposedly 'abstract' we can’t examine this scenario in the real world in order to determine which side is correct. All we have are make-believe arguments about a make-believe scenario. And so even though the second group have devised a large amount of subjective make-believe clever-sounding arguments to support their position, the first group will remain defiant.

Perhaps this is the main reason why the 0.999...=1 debate just won't go away?

[KarmaPeny]

Nobody can explain how the infinitely many points 0.9, 0.99, 0.999, ... can exist all at the same time, at different positions on the number line, without there being a last one. But mathematicians seem content to accept that they just can. They claim we can simply restrict ourselves to talking about ‘the limit’, and that by ignoring the problem of the last point we can claim that “there is no problem”.

This uncovers one (of the many) fundamental problems of mathematics, which is that there is no rigour about what forms an

For example, say one group of people believe that it forms a trivial contradiction to claim that a decimal with endless non-zero trailing digits (such as the decimal for the square root of two, 1.41421...) can be said to equal a constant value. However, another group of people might disagree, and so which side is right?

Since maths is supposedly 'abstract' we can’t examine this scenario in the real world in order to determine which side is correct. All we have are make-believe arguments about a make-believe scenario. And so even though the second group have devised a large amount of subjective make-believe clever-sounding arguments to support their position, the first group will remain defiant.

Perhaps this is the main reason why the 0.999...=1 debate just won't go away?

[johnschmidt]

Very interesting, I don't have a strong opinion about series but this seems to be related to Xeno's paradox as I understand it. I took it as a question similar to, if when going from point A to point B you pass through every point on the number line between them, what is the last point you pass-through before reaching reaching B.

He effectively goes on to point out that no such point exists, so how did you get to B?

[ThePallidor]

I think it's easiest to simply reject all ill-defined terms. "0.999..." is not clearly defined, because (for starters) is it meant to signify a process or a static object? Pinning them to this may be fairly boring but it does stop any further shenanigans from getting off the ground.

The only real objection to that is, "Maybe you're holding us to a standard of rigor that you yourself couldn't possibly hold to." So then it is necessary to show how all their "useful math" can be done without any infinities, limits, axioms, sets, or other ill-defined terms. John Gabriel did this for calculus, and there is hardly any other real feather in the mathematicians' cap.

[springinfialta]

I always thought that this was a good way to show that a number that approaches a particular integer via a series is never that number: Imagine a square with sides of 1, 1, 1, and 1/X where X starts at 1. Then let X go towards infinity. What is the sum of the angles of the polygon as X approaches infinity? Does it ever switch from 360 to 180?

[CS-moxp]

the word 'recurring' banishes any notion of '0.999...' being a fixed value, surely?

[loosebet]

OK so you question that equation. Is there any practical, real life reason in any sort of actual computation where this might make any difference?

[springinfialta]

Have you heard of Norman WIldberger? You seem to be kindred spirits with regard to the improper use that mathematicians make of the concept of infinity.

[mackeyfryguinness]

We can further raise awareness if we band together. Are you open to a future interview/discussion on the topics you raise? Delivering the content in other formats may assist the efforts. You find me in total agreement concerning your material. I just started my channel with a few criticisms but mostly building solutions. Wildberger is kind of in the right direction but fumbles hard when critisizing Euclid and his propositions are found wanting. I will make a whole playlist on him. I have solutions to the foundations of nathematics and later to calculus. Hope we talk further

[richardlbowles]

These objections seem obvious when you make them, KP. I suppose the main problem is that not enough people have thought of them. This might be why most mathematicians would oppose you: They haven't considered the arguments against their position.

[MarcusAndersonsBlog]

Hi Karma, good to see you are still getting the truth out there! I also have a new pitch.
Namely, "0.999..." is a convention, being given the status of proof, without an actual proof.
The disproof is easy :
0.999... < 1.000... by inspection.
Beyond that you need to point out that discrete maths requires there to be a unit interval that is not zero, and actually taking recurrence to infinity implies a unit interval of zero (divide by zero error). The decimal positional number system is a countable set only as long as there is a unit interval whose value is not zero. THAT is why "0.999..." is a convention only. A countable set cannot have infinite precision, by definition. This is basic.

[jadelynnmasker]

Wow, this is amazing. I have never accepted any logic that 0.999 = 1, and I did in fact come up with a similar argument that 0.999... could be an "irrational" number. I would agree that calling irrational numbers "real" is presumtious. I always hated that we came up with a symbol, √, which nicely packages up an infinite series like π. I would argue with a grown man with a master's in math, and he has gotten to the point where he refuses to talk about math because I always poke holes in the ideas.