The Truth About the Most Controversial "Number"

For centuries, humans have argued about whether 0.999 (repeating) = 1, and there are still many misconceptions on BOTH sides of that debate. In this extra-long episode, I’ll take you on a journey through infinite decimals to help clear some things up
Most of this episode is mathematical demonstrations, but there is also a philosophical edge to this topic, so leave a comment letting me know your personal opinions/beliefs about this "number" (hopefully after watching this whole episode to see all of the misconceptions I cover). And/or leave a comment if you can count how many squirrels appear in this episode haha.
A few clarifications:
-- For the definition of the Archimedean property, I accidentally spelled it Archimedian, and also I should have clarified that the numbers should be positive and that n should be an integer.
-- I mentioned the commutative and associative properties just as recognizable examples of properties that humans take for granted about the real numbers, which can sometimes be lost when switching systems. They are not specifically the properties that are lost by the hyperreal or surreal systems I mentioned (if/when I make an episode about those systems in the future, I’ll clarify more about which properties become more difficult in those).
-- When I said that every terminating decimal has two representations, I should have added "non-zero" since zero is technically terminating but doesn't have the same type of alternate form.
-- Some comments thought that I shouldn't have used "controversial" as a description and claimed that nobody disagrees about this topic nowadays. Let's appreciate the irony of people arguing that nobody is arguing haha. If you look at the comments of this video, you can see that it's clearly still a topic many people disagree/debate about.
Make sure you're also tuned in to my ‪@Domotro‬ channel for my livestreams, shorts, and other bonus content!
This episode was filmed by Carlo Trappenberg, and was directed / edited / soundtracked by me (Domotro).
Special thanks to Evan Clark and to all of my Patreon supporters:
Max, George Carozzi, Peter Offutt, Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Izeck, Beugul, OmegaRogue, Florian, William Hawkes, Michael Friemann, Claudio Fanelli, The Green Way, Julian Zassenhaus, Bailey Douglass, Jan Bosenberg, Brooks Boutwell, David Irvine, qe, George Sharabidze, Jack Dwyer, Fredrik, and Dave Brondsema!
Check out the Combo Class Patreon at / comboclass
If you want to mail me anything (such as any clocks/dice/etc. that you'd like to see in the background of Grade -2), here's my private mailbox address (not my home address). If you're going to send anything, please watch this short video first: • You Can Now Mail Me Th...
Domotro
1442 A Walnut Street, Box # 401
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and there is a subreddit here: / comboclass
If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com
In case anybody searches any of these terms, some topics mentioned in this episode include: 0.9999999 (0.9 repeating / zero point infinite nines) and whether that = 1, repeating decimals, patterns in different numeral bases, what consists of a proof vs. a demonstration, properties of the real numbers such as the commutative and associative properties, the hyperreal and surreal numbers, infinitesimal numbers represented by epsilon, the philosophy of why numbers are useful, fractions vs. decimals, irrational numbers like pi, multiple representations of numbers in bases, and more...
Disclaimer: Do NOT copy any dangerous-seeming actions (which were actually performed in a careful way) involving fire, tools, or other chaotic activities you may see in Combo Class episodes. This is for education and entertainment.

[@ComboClass]

This extra-long episode is my presentation about if/when/how 0.999 (repeating) equals 1. Most of this episode is mathematical demonstrations, but there is also a philosophical edge to this topic, so leave a comment letting me know your personal opinions/beliefs about this "number" (hopefully after watching this whole episode to see all of the misconceptions I cover). And/or leave a comment if you can count how many squirrels appear in this episode haha.

[@donaverboxwood]

Pardon if this is a stupid question, but in regards to infinite strings of digits in decimals, would it be fair to say they are a different kind of string than finite strings? (I mean, obviously yes, but let me explain) What I mean is, it would be completely incorrect to have a number like 0.000...(infinite 0s)...0001, where the infinite string of digits is not the last string overall, right? So there has to be a difference between what a finite string is and what an infinite string is, despite being made of the same thing (digits). I guess what I'm asking is, would it be more accurate to say that decimals can have infinite strings of digits only if the infinite component is the smallest (rightmost when written out) component? Again, sorry if this is complete nonsense I'm saying. I am by no means "good at math".

[@TaleTN]

I counted 4.999999999... squirrel appearances.

[@diribigal]

@@donaverboxwood when you're talking about real numbers in their standard decimal forms or "strings" in most other senses, yes, an infinite string like these cannot have a right endpoint. However, that doesn't mean the idea is inconceivable. If an infinite string is normally like "there's a first character and a second character, and similarly a character for every counting number", then you could certainly make up something like a super-string which has a character for every counting number, and then three extra characters which are considered to come after all of the others. This is getting very close to the mathematical idea of "ordinals".

[@MrDannyDetail]

@@donaverboxwood Maths is the study of patterns, not the study of numbers. If you mean you are not good at manually performing additions, subtractions, multiplications or divisions where the numbers are not trivially small and easy to work with then it is arithmetic you are not good at, rather than mathematics (and in any case you're probably better than you think at arithmetic). What you demonstrated in your original comment is the ability to see the range of patterns already exisiting in a mathematical system and then concevie an entirely new way of extending that system with new patterns that build on the exising system, rather than merely replacing it with a whole new system. Being able to conceive of ways of extending patterns beyond what is 'normally' done in maths classes is actually being very good at maths. Having a play with what happens if you put a finite rightmost digit (or digits) beyond a infinite string of digits on the righthand side of the decimal point could lead to all manner of interesting conclusions, to new ways of viewing existing open maths problems etc so the ability to have 'outside the box' thoughts like this about mathematical systems is what enables mathematicians to keep pushing the boundaries, finding out new things and making new theories. It's a shame that school systems in many parts of the world leave a lot of their pupils thinking that maths is just about doing hard additions, subtractions, mutilpications and divisions and similar other things like square roots and so on when really that is just arithmetic, which is merely a mathematician's basic tools for doing actual maths, and for which we have extremely good calculators and software these day anyway, whilst true mathematics almost always requires human inquisitiveness, inutittion and creativity which a machine cannot really replicate.

[@mesplin3]

Let x = 9 + 90 + 900 + 9000 +... I like that 0.999... = -1 * x.

[@GreyJaguar725]

This guy spent an infinite amount of time writing an infinite amount of "9"s after "0." for a video. Respect.

[@sirfzavers8634]

Would’ve been cool if he’d shown them all… but then the video would be infinitely long and he wouldn’t get any full views. ☹️ Edit: At least it’ll keep that fire fueled infinitely (we’ve done it boys; we’ve prevented the heat death of the universe).

[@silver6054]

@greyjaguar725 Don't think he did. He spent 1 second writing the first 9, half a second writing the next, a quarter of a second writing the next and so on (practice makes perfect). So he did the whole thing in 2 seconds, which really does earn respect.

[@sirfzavers8634]

@@silver6054 that could explain the lack of the time machine needed for our theory… 🤔

[@GreyJaguar725]

@@sirfzavers8634 I think he's referring to the sum to infinity of the geometroc series:1 +1/2 +1/4+1/8+... Which is a/(1-r) where a is the first term and r is the ratio (next term /previous term) So plugging in the numbers we get: 1/(1-[1/2])=1/(1/2)=2 so it takes 2 seconds to write infinite 9s.

[@tacobell2009]

​@@sirfzavers8634 Maybe he did, but the video is still uploading...

[@first_m2999]

Domotro has mastery over squirrels and numbers. If he would only learn to control fire, he would be unstoppable.

[@hkayakh]

And a way to prevent things from falling over

[@kamikeserpentail3778]

Squirrel Girl just needs squirrels to beat Thanos, Dr. Doom, Galactus, whomever. He's got this.

[@nitehawk86]

There's a reason why Squirrel Girl is invulnerable and all of the fire based superheroes are not.

[@MawdyDev]

He's dual classing Wizard and Druid, it might be hard for him to continue leveling if he adds Sorcerer to that list

[@peppermann]

😃🤣❤️👏👍

[@nitehawk86]

I just wanna way the camera work, on this episode is particularly fantastic. Capturing the disaster just as it happens without taking away from the lecture.

[@ComboClass]

Thanks. Shout out to my main camera guy Carlo (who’s in the credits). Although I “direct” the episodes, he has some freedom behind the camera and helps capture all the rarities :)

[@kylebowles9820]

😂 that's such a perfect way to describe this channel in general; love the chaos

[@Rhiannon_Autumn]

This is the best video explanation I have ever seen talking about this phenomena in our arithmetic system. Thank you. You're a great teacher.

[@ComboClass]

Glad you enjoyed and it helped you learn, thanks for the compliment :)

[@strangedivine]

You’re right, he’s a great teacher. I sometimes struggled with math and it was usually because of the teacher/prof’s approach in teaching.

[@TheBalthassar]

In arguments like this I always like the engineers answer "It's close enough, it fits the spec."

[@willo7734]

It’s within the tolerance.

[@stickmandaninacan]

Somehow the chaotic constantly interrupted style of presentation in combo class is actually really effective at keeping the attention of my adhd brain, it feels soothing 🧠 cute squirrel

[@Fire_Axus]

your feelings are irrational

[@samueldeandrade8535]

​@@Fire_Axus haha. More than irrational, transcendental.

[@jackputnam4273]

So glad i clicked on the first combo class vid that was recommended to me. I was immediately hooked by domotro’s style and it just keeps getting better! Such an amazing channel and it deserves a lot more attention :)

[@ComboClass]

Thanks, glad you’ve been enjoying! :)

[@rmdodsonbills]

I believe it's been proven that the infinite series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 ... (and so on) equals 1. Has to be equal to 1. In binary that's represented by 0.11111111.... (and so on). Seems like a similar logic would work for 0.9999999999... (and so on).

[@MuffinsAPlenty]

"I believe it's been proven that the infinite series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 ... (and so on) equals 1. *_Has to_* be equal to 1." (emphasis added) It _has to_ be equal to 1 in the same sense that Domotro talked about in the video. It doesn't really _have to._ There's no _a priori_ reason that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 ... should have any value at all. However, if we impose upon ourselves the restrictions that it _should_ have a value, and that value should be consistent with certain arithmetic properties working in a reasonable way, then we have no other option but to recognize 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 ... as being equal to 1. However, this conclusion relies on self-impositions, not on some universal truth or "nature" or anything like that.

[@mattiviljanen8109]

(Edit: a few minutes later just this was covered in the video!) As a kid when I learned about 0.999... = 1, the mind-opener thought was that there is more than one way to represent a number, e.g. 1.5 = 3/2. As for the usual 1 / 3 = 0.333... --> 0.333 * 3 = 0.999... --> 1 - 0.999... = 0.000... counter-argument, the trick is to get infinity right. In order for the 0.000... to ever end, there would need to be a final non-zero digit. But as per definition, 0.000... does _not_ have a final digit, hence it must be all zeros, and be exactly equal to zero.

[@diribigal]

I teach things like the surreals and the hyperreals and I'm very pleased with how you handled things here. My one real quibble is that around 17:12 when you defined the archimedean property, I wish you'd said/written "integer n" instead of "number n". Excellent work giving a fair and clear presentation that doesn't go too far into irrelevant detail!

[@ComboClass]

Thanks for the feedback, and I’m glad you enjoyed! :)

[@pedrogarcia8706]

the archimedian principle is not just true for integers though.

[@diribigal]

@@pedrogarcia8706 In a nonarchimedean ordered field like Robinson's Hyperreals, you can always find a "number" n. If a and b are positive, then certainly (2b/a)×a is larger than b, even if a is infinitesimal and b isn't, for instance.

[@pvzpokra8602]

why does this guy speak in 0.75x speed

[@samueldeandrade8535]

Sorry to tell you,he doesn't. It is you. you hear in 0.75 speed.

[@Ethan13371]

Since his speech contains 25% more info per word than normal, he slows it down for us plebeians

[@WillToWinvlog]

When I was a kid we used to assume it was wrong but we used to troll each other saying if 1/3 = 333... then 3/3 = 999... I guess our intuitions were correct!

[@Ninja20704]

While I never really doubted it, the most convincing argument to me is that theres no number you could fit strictly between 0.99… and 1. Most people can see that intuitively, but there are also rigorous ways to show that. Usually i just tend to say if we don’t accept it, we cannot accpet any fraction with a non-terminating decimal, like 1/7 or 1/11 as well. I appreciate you adressing topics like this, maybe do more of them.

[@BlackBull.]

0.99...95

[@pepebriguglio6125]

Per my intuition, I agree because an infinite string of decimal 9's WILL get literally infinitely close to the number 1, and only 1 can be 'infinitely close to' 1. BUT technically, I don't find it difficult to find an infinite amount of real numbers between 0.999... and 1. Of course I must be overlooking something. But here it is: 0.999... = S(9/(10^n)) for n=1->inf. But this is an infinite sum with ordered place holders (n=1, n=2, etc.). So let's construct an infinite sum, which approaches 1, say 11/10, times faster, which would be: S(99/(100^n)), n=1->inf. Normally we would say that this is just an alternative way to describe 0.999..., because the decimal places would then simply be occupied pair-wise, instead of one by one. But still it stands to reason, that for every value of n, the number grows by 11/10 more than in the case of S(9/(10^n)). Another way to look at it, could be in base100. Here we have 0.99;99;99;..., which again would approach 1 by a factor of 11/10 faster than 0.999... in base10 would. So I suppose the question is, whether 'faster than' implies 'bigger than', when it comes to infinite sums.

[@pyropulseIXXI]

There is a number, and it is 0.99... + ε, where ε is an infinitesimal. You can literally prove 0.99... = 1 with infinitesimals, so idk why he said introducing them messes things up; it doesn't. The hyperreals are an ordered field with all the same properties as the reals, so associativity and commutativity holds. The reals are a subfield of the hyperreals, just as the rationals are a subfield of the reals. What you meant to say is that there is no *_real number_* that fits in-between 0.99... and 1. Also, your intuition says that 0.9... = 1 is obvious; or at least my intuition does. It is so far beyond obvious, but saying "it is hard to prove, therefore the intuition is wrong!" is pure absurdity. Proving 1 + 1 = 2 rigorously is also quite hard, for the non-math initiated, but we don't say our intuition of 1 + 1 = 2 is wrong because of a challenging proof

[@martind2520]

@@pyropulseIXXI No, you are incorrect. 0.999... + ε is not a number between 0.999... and 1. 0.999... _is_ 1, they are the _same number_ so there can be no number between them. 0.999... + ε is equal to 1 + ε, which is a number slightly higher than 1.

[@martind2520]

@@BlackBull. You number ends, it ends at ...95. The number 0.999... _doesn't end_ and so is larger than your number.

[@EpicMathTime]

You can get as close as you want to 1 with _finitely_ many 9s, and 0.999... is greater than all of those numbers. The common objection that continuing to add 9's will "never reach" 1 does not make sense as an objection, because such a process never reaches 0.999... either.

[@agentofforce3467]

Its probably not possible to add an infinite amount of numbers.

[@MuffinsAPlenty]

Perfectly explanation for a common misconception, like always.

[@isaacbruner65]

@@agentofforce3467 it's absolutely possible. Just look at any convergent infinite series, for example.

[@howdy832]

In Knuth's base 1+i, any gaussian integer is represented as a + b(1+i) +c(1+I)²… wher the coefficients are 0, ±1, or ±i. Each integer has 4 representations, where leading non-zero coefficient is each option. E.g. 1 is either 1, -i +(1+i), i -(1+i), or -1+(1+i)² -(1+i)³

[@howdy832]

You can do this with eisenstein integers too: the digits become 0, 1, w=-½+sqrt3/2, & z=w², allowing multiples of ±1, while the base is b=1-z. Now you can write 1 as 1, b+z, or w - w*b

[@samueldeandrade8535]

Nice ... very nice.

[@nnnnick]

some day i will understand why this man has so many clocks

[@Programmable_Rook]

Assuming a clock ticks exactly once every second, a clock that ticks normally has slight variation from other clocks, meaning it could be wrong at every point of the day. A clock that doesn’t tick is exactly right twice a day. So if you have many clocks that don’t tick, they will be exactly right more often than a normal clock. His clocks are more likely to tell you the exact time of day than a normal one. The real puzzle is why he doesn’t make them tick backward, then they’d be exactly right 4 times a day.

[@MarloTheBlueberry]

How is a backward.clock right four times a day?

[@AaronHollander314]

He hates to be late

[@StevenLubick]

You still have time. 😀😀😀😀

[@MarloTheBlueberry]

Bro.........@@StevenLubick

[@JonBrase]

The binary version of this gets really interesting. Two's complement is used to represent signed integers in computers, but some early machines used one's complement. But if you allow an infinite number of digits on either side of the radix point, two's complement and one's complement are equivalent.

[@CassandraComar]

these are the 2-adic numbers. some of the other p-adics can even represent i (ie solutions to x^2 + 1 = 0).

[@JonBrase]

@@CassandraComar Not exactly. The 2-adics don't include digits to the right of the radix point.

[@CassandraComar]

@@JonBrase the 2-adic integers don't but the 2-adic rationals do. they represent fractions with power of 2 denominators.

[@JonBrase]

@@CassandraComar I'm still a bit nervous about saying that what I'm talking about is too closely related to the p-adics, because there's some weird topological stuff going on with the p-adics that I don't understand and I'm not sure if it's intrinsic to all digit sequences extending infinitely to the left in a positional number system, or if it's just a useful topology to define on top of such digit sequences for the type of problems the p-adics have been used as a tool for. I think there may be multiple concepts in that space that are related to the p-adics in terms of their representations in a positional number system, but quite distinct in their deeper structure.

[@hughobyrne2588]

For years after I learned about 1s complement and 2s complement, I had this nagging feeling that the extra '+1' step of 2s complement was... hiding something. It took me a long time, but I came to the same conclusion as you did, including all the digits after the 'point' makes it all harmonious.

[@consciouscode8150]

After watching, I think I would consider it a notational quirk which emerges from the imprecision of what is meant by overbar, ellipses, etc. Rather it's probably better to think of real numbers represented using base-10 notation constructively, such that 1 approximately equals 0.9, 0.99, 0.999, etc but this notation alone can't ever equal 1. As soon as you say some variation of "and so on" however, what you've effectively done is taken the limit of the pattern - so it becomes almost obvious that it would be exactly equal to 1, because notationally it's essentially the same as an explicit limit. But maybe that doesn't feel so obvious because we think of the overbar or ellipses as being part of the base-10 notation itself rather than an implicit operation.

[@kazedcat]

All real numbers are defined by an infinite sequence of rational numbers. So when the domain is in the set of real numbers then it is by definition a limit.

[@maxerboi20]

Combo class be comboling my brain

[@mrmistmonster]

I uhh didn't prove but demonstrated this to myself with Zeno's paradox shenanigans a month ago. If Achilles starts 90 meters behind the Hare and moves at 10 m/s while the Hare moves 1 m/s. If you go through it you get to Achilles passing the Hare at 9.99999999 etc meters past the Hare's starting point. But if you just solve the equation you'll get 10 meters.

[@matematicke_morce]

18:43 Here we see Domotro and the squirrel, a failed version of Achilles and the tortoise where the animal actually runs off to infinity

[@erwinmulder1338]

Can you repeat part of that? I got distracted by a squirrel.

[@Kyanzes]

Is nonary 0.88888... the same as decimal 0.99999... ?

[@Chris-5318]

Yes. More generally, if b is a natural number, then 0.bbb... (base b+1) = 1.

[@strangedivine]

Math was not my best subject in school, especially post-secondary math, but damn you make it fascinating!

[@CatherineKimport]

The argument that finally got me *comfortable* with the idea that 0.99999.... = 1 was one about how there isn't anything special about base ten. So, like, assume that 0.99999.... was some number infintessimally smaller than one. Then, shouldn't hexadecimal 0.FFFFF...... ALSO be some number infintessimally smaller than one? Would it be the SAME number as decimal 0.99999....? That seems weird, because 0.9 and 0.F are not the same, nor 0.99 and 0.FF, nor 0.999 and 0.FFF, and so on. So if 0.99999... and 0.FFFFF... represented DIFFERENT numbers, then that would mean that every base had a unique set of numbers it could possibly represent, and had a whole bunch of gaps about numbers that it COULDN'T represent. But if 0.99999.... and 0.FFFFF... both secretly equal 1, then those gaps go away. And the latter just felt less uncomfortable than the former.

[@BenHebert-no4qp]

I believe base i has an infinite amount of decimal representations for numbers, as the values for each digit position repeat every four positions. For example, i^8 = i^4 = i^0 = i^-4, therefore 10000000 = 10000 = 1 = 0.0001, which would all represent the number 1. Despite having infinite representations for real and imaginary integers, base i has no representations for non-integer numbers.

[@TankorSmash]

This was both greatly educational and yet greatly uncomfortable. Looking forward to more!

[@francevenezia]

But say this was an engineering problem and you had ten walls @ .99999. You couldn't dismiss the 10 (1/10's) but would have to calculate that into your plans. I think this might apply (calculating .99 as 1) when you're just doing theoretical math, but in say, engineering for example, the 1/10000, 1/1000, 1/100 and 1/10 need to be accounted for. If it's just theoretical math, then yes, it is safe to say .9999 (repeating) = 1.

[@briangronberg6507]

I’m thrilled I found your channel! This was a really solid presentation and I appreciated the reference to the p-adics and the small taste of the idea that there exist number systems/algebras that may not satisfy commutativity or even associativity like the quaternions or octonions.

[@cmilkau]

Did you know 1/99 = 0.01010101..., 1/999 = 0.001001001...? Stumbled over this (in fact the general geometric series limit) on my own in middle school and turned it into a popular little school calculator program that could recover arbitrary fractions from their infinite decimal representation.

[@mrosskne]

so, the frac > dec button that every calculator already has?

[@MuffinsAPlenty]

It's very nice to be able to figure out a pattern like that at your own, especially at such a young age!

[@sakesaurus]

philosophy is math. When philosophy doesn't invlove any math, it's bad philosophy. Number theory is just a continuation of formal logic.

[@kqawiyy]

I used to deny that the two were equal, but now I see just how wrong I was in *many* different avenues. Thx

[@gonegahgah]

You were right the first time.

[@AlexanderScott66]

No, you were right. What people leave out is two fold. One, the infinite sum series specifies that it's the limit. Two, limits do not mean the function(in tis case 9/10^n)has to equal anything, rather, it simply approaches it, getting closer and closer. It explains why there's no number between(although, if we were to talk about just integers with no decimals for illustration purposes, there is no number between 1 and 2, so is 1 equal to 2? No, because being as close as possible doesn't mean it's exact), it explains why it mentions limit in the sum of an infinite series, it explains why both sides think the way they do. But no. People have to argue with baseless facts, like saying 0.333... or 0.999... is even defined at all, despite an infinite sequence inherently being unable to be defined, which is where Wiki nerds get it wrong. Why there's no number in between? There is. You'll use a number line and say plot it, but what about plotting based on precision? Plot 0.9, zoom in, then 0.99, then zoom in again and 0.999, so on and so on: 0.9999, 5 9s, 6 9s, 60 9s, 100 9s, 9 novemdecillion 9s. Tell me when you mathematically can't zoom in and plot again. TLDR people forget the beautiful thing called limits and how they work.

[@АлёшаИнкогнитов]

3:27 How to become a believer in a god of math. "something off" and paper of infinity immediately drop off.

[@eiman2498]

I love watching this channel. You always upload interesting content that never fails to enlighten others (including me) !

[@AsterothPrime]

"1" is just a symbol for its infinitely accurate version: 0.999999.. Like if you looked at the number 1 under a microscope. The only issue is, if "2" is twice that of 0.999999... then logically it would need to end in an 8, but it would never be realised. Yet you would know that the universe owes you that little bit extra 😂

[@Chris-5318]

2 * 0.999... = 1.999... = 2 OTOH 1.999...8//2 = 0.999...9 but then neither is an infinite decimal and so are less than 2 and 1 respectively.

[@AsterothPrime]

@@Chris-5318 Exactly, 1.9999.. only stops being equal to 2 the moment you stop putting more nines and cap it off with the required '8', because that would mean making it a finite number. So yes, technically the universe doesn't owe you anything, since these are infinitely accurate, with an infinite amount of 9's.

[@Chris-5318]

@@AsterothPrime You said, " The only issue is, if "2" is twice that of 0.999999... then logically it would need to end in an 8" That is wrong. Logically it never ends. i.e. it does not have a last digit. 1.999... is a [representation of a] finite number - it is 2. An infinite decimal is one that does not have a lasty digit. That doesn't make it be (i.e. represent) an infinite number.

[@ilikemitchhedberg]

Are you surreal right meow?

[@ComboClass]

I often live in semi-surreality

[@Tubluer]

That was a really good video, but how does it relate to Magic the Gathering?

[@orterves]

You're right - I think something that maybe isn't taught enough in school are the constraints of the maths people are taught. There is confusion about the answers to questions like this because people don't realise that the answer depends on the rules of the system they are working with. I think that this also applies to many disagreements in life, people argue about some question not realising the question doesn't even apply given the constraints of the topic. Perhaps in general we should spend more time figuring out where we really are before arguing about where we want to get to.

[@arcturuslight_]

I remember back in school opening an algebra textbook on a small print "conventions" section, where one of the points is "for the purposes of this book we will define 0.9...=1"

[@TerranIV]

This reminds me of Fourier trigonometric series that basically shows that you can make any shape out of an infinite number of smaller and smaller cosine and sine waves. Its like everything is made of an infinite series of waves, but we just mostly interact with things that have a harmonic form.

[@pedrogarcia8706]

I don't think it's fair to say that the people who say ".99 repeating infinitely must equal 1" aren't 100% correct, because when we have conversation about what numbers equal, we're coming at them from the understanding that we're all talking about the same number system. Base 10, Archimedean, real numbers. yeah, in base 11, .9 repeating infinitely has a different meaning, but we don't use base 11, and when we talk about numbers in different bases, we specify that we're using different bases.

[@ComboClass]

I think you and I agree (I make similar points later in this episode) and are just using words slightly differently. By the people who say it “must” be equal, I was referring to people who say that it fundamentally equals that in any sense (which some people say when they learn a “proof” of the concept but don’t know the whole context) and who don’t understand the things you mentioned

[@adamswierczynski]

I tried to explain that numbers have infinite names for the same identity (due to fractions behaving as you explained) in 7th grade honors math and the class laughed at me. Even the teacher treated me like I was crazy.

[@mokey345]

Similarly in my 7th grade honors class, my teacher convinced most of my class that “of” in word problems means “divided by” instead of “times”

[@Faroshkas]

What about base 1? I was thinking a while back about that, and if a base 1 could exist, all of it's decimal representations would just be .000000000000... and not describe anything in particular.

[@chipichipichapachapaWHY]

Tally marks

[@ilikemitchhedberg]

0.99999...st!

[@ComboClass]

I sometimes nickname it “zero point ninefinity” haha, but I didn’t say that nickname in this episode because I thought it might add confusion

[@ilikemitchhedberg]

@@ComboClass thank you for the lovely lecture!

[@tyruskarmesin5418]

0.99999st!

[@ilikemitchhedberg]

@@tyruskarmesin5418 yeah, that's what I should have said

[@DaveyL2013]

I find it pretty simple: 1/3rd = 0.3̅, 2/3rds = 0.6̅ , so 9̅ must = 3/3rds, and 3/3rds = 1!

[@DaveyL2013]

After finishing the video I can say this seems slightly more complicated, but I still stand by my reasoning

[@gumenski]

Since when was this controversial? We're stuck back in the 1700's again?

[@ComboClass]

Maybe you define controversial differently, but if you look at the comments of any video like this, you will see that people still have a wide variety of different opinions on this question

[@MuffinsAPlenty]

Yeah, I can understand both sides of this. The equality 0.999... = 1 is absolutely not controversial among experts in mathematics, but it is controversial among the general public, and any online discussion of 0.999... will reveal that. However, at the same time, I don't know how many people would be defending a video which claims that anthropogenic climate change is controversial, even if there is a sizeable portion of the general public which denies it, since there is no controversy among the experts. To be fair, the equality 0.999... = 1 won't have as direct of an impact on most people's lives as climate change will, but I do think the comparison gives me pause to completely agree with Domotro here.

[@gumenski]

I didn't know mathematics was opinion-based. Would you consider flat-earth vs the regular known globe earth model to also be a controversy since there are many unintelligent people rooting for us living under a dome that god made?@@ComboClass

[@danieldover3745]

I had already been convinced that 0.9999... was 1, but the explanation that helped me really understand what was really going on, and why my initial repulsion to it was also correct, was the concept of a limit. At no specific, definable point does 0.999... equal 1, it just approximates 1 and approaches the limit of 1 if the sequence is taken to infinity.

[@Chris-5318]

Your faulty reasoning is revealed by your, " At no specific, definable point does 0.999... equal 1" and your "approaches the limit of 1 if the sequence is taken to infinity". What you don't realise is that 0.999... is constant/unchanging/fixed/static and so cannot approach anything. It doesn't approach a limit, it's value IS a limit. You are confusing the series 0.999... (= 0.9 + 0.09 + 0.009 + ...) with the sequence 0.9, 0.99, 0.999, .... It's that sequence that approaches 1 as you step through. Here's the thing, it also approaches [the value of] 0.999.... The n th term of that sequence is 0.999...9 (n 9s), and that is easily seen to be 1 - 1/10^n. In fact, the value of 0.999... := lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1. The " := " means is equal by definition. The last equality follows from the definition of limit. I suggest that you look up "geometric series". The Wiki is especially relevant.

[@AaronALAI]

It's the same as 1 because you would need to add 0.000 repeating with a "1" at the end which is infinity small; for 0.999 repeating to equal 1.

[@wiggles7976]

How can a real number have a digit after an infinite amount of digits to the right past the decimal point? The number 0.123 has a 1 in the 10^-1 place, a 2 in the 10^-2 place, and a 3 in the 10^-3 place. In you number, 0.000...1, you say 1 is in the 10^n place. What is n?

[@marvinmallette6795]

@@wiggles7976 "n" is an unsolvable. Because you can't convert 1/3 into Base 10, you also cannot get the final component of 0.999... to reverse the operation. 0.333... is not a finite number from which to perform inerrant calculations upon. All subsequent calculations are based on an unfinished calculations and are therefore incorrect. By graphing the "limit" of 0.999... it makes it obvious in the abstract, but Aaron's statement is also an observation in the abstract. He understands the problem. 0.999... is incorrect, but the margin of error is infinitely small to the point of meaninglessness.

[@wiggles7976]

@@marvinmallette6795 It seems like you and Aaron accept that 0.999... is equal to 1. You do accept that 0.999... is exactly equal to 1?

[@AaronALAI]

@@wiggles7976 Yes, 0.999 repeating is exactly equal to 1, "0.999" by itself is not equal to 1 because you could add 0.001. If I add 0.0000 repeating with a 1 at the end to any number, the sum does not change because 0.0000...1 is infinitely small. I think ""n" is an unsolvable" is the correct response, but consider this to your original question, "you say 1 is in the 10^n place. What is n?" What if n were inf then it would be 10^-inf * 1 which is 0

[@wiggles7976]

@@AaronALAI OK, you are right about repeating decimals; 0.999... = 1. However, this idea of putting a 1 after infinitely many 0s does not make sense for real numbers. I don't know if some exotic number set could be defined using ordinals instead of integers for the powers of 10 that each get scaled by some digit from 0 to 9. In the real numbers however, integers are used for the exponents. When we have 10^n, n is an integer, not an ordinal or something else. Infinity is not an integer. Thus, it does not make sense to talk about the digit in the "infinitieths place" of a real number. What you are writing as "0.000...1" is just a haphazard way of describing something exactly equal to 0.

[@dananichols349]

Just a thought... If I travel at 0.999(repeating) the speed of light, would I be traveling at the speed of light???

[@gabrielgabi543]

Haha still need infinite energy

[@caspermadlener4191]

I find it really refreshing how you acknowledge that it is not possible to give a satisfying proof of this. Axioms aren't as important as their direct consequences, those shaped the axioms in the first place.

[@mrosskne]

There's nothing to prove. A decimal expansion is just another way of writing the same number.

[@wooftubeyt]

It makes sense when you realize 0.99999... /3 = 0.333333333... which is a 3rd and 1/3*3 = 1

[@martind2520]

For some weird reason people would prefer to deny that 0.333... = 1/3 rather than agree that 0.999... = 1. It's crazy.

[@otonanoC]

He fed squirrels, and burned a guitar. After the smoke cleared, we concluded 0.9999.. = 1.0

[@sandordugalin8951]

Wouldn't 1 - 0.9 repeating equal 0.1 repeating?

[@Chris-5318]

Of course not. 1 - 0.999... = 0.000... = 0

[@MegaLokopo]

This would be solved if the base number of a specific number would simply increase over time atleast when it comes to decimal points. 0.12345 the one should be in base one, the two should be in base two, and so on.

[@cleanerben9636]

ehhhh if you squint it could be a 1

[@bgg-ji8dc]

Different systems of numbers are more or less useful in different situations. If you need to count how many people are on board a bus, either the integers or the whole numbers are natural choices for that purpose. If you want to accurately measure the weight of an object in kilograms, the real number system is a better fit. Complex numbers can model real world phenomena directly a la quantum wave functions or electrical circuits, but can also be used in an abstract setting to assist in proof or calculation, even if what you're actually interested in is better modeled by real numbers. Differential calculus can be, and indeed historically *was* defined via the use of infinitesimal numbers. The surreal number system can be useful when analyzing certain infinite two-person games in game theory. There are many other number systems of theoretical interest such as finite fields or general linear groups. Alternate number systems are not some attempt to make a better system of numbers so as to replace the system we have, nor are they some exercise in imagining how our mathematics might have developed differently if we had adopted strange or foreign conventions. They are their own tools with their own uses - Tools that nobody can learn to use properly as long as they continue to believe that the convenient properties held by ℝ are fundamental truths of the universe. Even with complex numbers, which extremely useful even among non-mathematicians, there is this stigma against them that they are fictitious or that they are a result of "breaking the normal rules of math" - even among the highly educated. We need to get away from the idea that the "real numbers" are the only "real" numbers.

[@Salted_Pizza]

What's interesting to me is that 0.999... is EXACTLY equal to 1.000...1

[@martind2520]

No, 1.000...1 has a finite number of digits, so it is factually 10^-(n+1) greater than 1. (Where n is the number of 0s in your number 1.000...1.)

[@trien30]

.111111111 = 1/9, .999999999 = 9/9, but 9/9 doesn't ~= 1. Maybe it'll be equal in Chinese numerals, Hebrew numerals, Babylonian numerals, Roman numerals, French numerals or Dutch numerals. Just kidding.

[@pyropulseIXXI]

Everything that holds in the reals also holds in the hyperreals via the transfer principle, so associativity and commutativity also holds in the hyperreals. I am confused as to why you said those properties do not hold if we introduce infinitesimals. The hyperreals are an ordered field that have all the same properties as the reals. The reals are a subfield of the hyperreals, just as the rationals are a subfield of the reals. You can literally prove 0.9 repeating = 1 with infinitesimals with relative ease. If you don't want to use limits, you can use hyperreal infinitesimals to prove it.

[@ComboClass]

I didn't mean to imply that the commutative and associative properties were specifically lost in those particular systems (I just wanted to give a few examples of recognizable properties that people often take for granted about our system), although I can see how it may have been unclear. Despite the transfer property, the hyperreals are a non-Archimedian field, which is the main difference i wanted to point out. I added a clarification to the description, and might add more details if I make a video about those systems in the future.

[@pyropulseIXXI]

@@ComboClass Ah, sorry. I didn't understand what you were saying exactly. Also, a future video going into such systems would be super interesting, just to explore them. I knew I was so excited when I first learned about hyperreals and non-standard analysis in detail

[@RandomNPC-sy6gj]

I feel less stupid for having watched this now.

[@MxGx]

Woah so trippy lolol

[@busomite]

I’d seen all the explanations but the one about points on a number line, that’s one I hadn’t considered before. Somehow that lands well with me, it says they need to be the same point. Very cool, thanks!

[@dopefish86]

proof by sqirrel!

[@oddlyspecificmath]

I've had terrible luck lately developing anything discussion-worthy, so I'm just going to kludge out the method I prefer. _There's no need to fully fill a division slot_ ... i.e., 4 / 2 is 2...sure, but that "fills" the slot. You can say instead that 4/2 is 1 remainder 2. Then 2/2 is 0.9 remainder 0.2, and so on...so you get 1.999999999.... with a remainder of 0.00000000...2 until you decide the "limit" has been reached and you fully "fill" the last division, then (under conversion from carryless arithmetic slots to based-digit slots) finally carry and end up with 2.000000.... as your answer. While I did develop this on my own, I found extensive references / wasn't first so suspect "delayed completion" isn't crazy.

[@natepolidoro4565]

Where even is bro?

[@Kopiovastaava]

I saw a squirrel, some blue tetrominoes, burning stuff and there might have been some numbers somewhere along the way. Lovely stuff as always.

[@random6033]

0.999999... = lim{n->infinity}(1-1/n) = 1 - 0 = 1 here, the entire vid is useless and it's the case unless you want to argue that lim{n->infinity}(1/n) isn't 0, which would like... break math also 1/3 wouldn't have infinite digits if we used base 12 but 1/5 would for example, say that we have a system that uses digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a ,b, then 1/3 would be equal to 0.4 and 1/5 would be 0.249724972497... Also in base 12 number 0.bbbbbbbbbbbb... would be equal to 0.99999999999... in base 10 and also equal to 1

[@justsomeguy5628]

This video is great. Btw, floating point arithmetic (of any arbitrarily large but value) is an example of a non-archemedian system, as floatingpoint +0 is the smallest number.

[@ijpthegreat]

It’s sorta funny because in other bases, like 6, 1/3 is just .2

[@Sluppie]

Short answer: 0.999 repeating = 9/9 = 1. It also = 999/999 = 1. In the same way, 0.123123123 repeating = 123/999. So yes, beyond a doubt, 0.999999 repeating = 1

[@ehxolotl4194]

17:10 As written, "Archimedian property" should be spelt "Archimedean property", and it would be inaccurate, pick x=0, y=1 and we have no number n such that nx>y. If x, y are restricted to the positive reals (and n to a positive integer), this would work.

[@ComboClass]

You're right. I misspelled it, and also didn't clarify the positive/integer restrictions, so I added a clarification to the description.

[@johnrichardson7629]

Your video re arbitrarily large numbers vs infinity is relevant. You can represent numbers arbitrarily close to 1 by writing 0 + an arbitrarily large number of 9s after the decimal point. But 0.999... isn't a case of that. It is 0 + a truly INFINITE number of 9s.

[@JerusalemStrayCat]

I am reminded of the bijective base notation system - I don't remember whether it was covered on the channel yet. The idea is that instead of having numerals from 0 to b-1 (for base b), there would be numerals from 1 to b. This prevents quirks like 0.999...=1, but also cannot represent 0, among other drawbacks.

[@Phryj]

Is there any equation x / y that gives the answer .999... ? The only one that seems to kind of work is 8.999... / 9.

[@wiggles7976]

x / y is not an equation, it is an expression. Expressions don't have answers, they have other expressions that are equal.

[@mrosskne]

Yes. 1 / 1.

[@AllenKnutson]

I can't bear to watch 29sec much less 29min about this, so I'm just going to say, if you can handle the fact that 2/4 = 3/6 then you should goddamned well get over the idea that a number should have only one name. (At which point you're ready to handle 0 = -0, too.) On Quora somebody asked, quote, "Which is the most divisive number?" I said "Zero. Is it a natural number, or isn't it? This is (unfortunately) divisive."

[@tomgooch1422]

Excellent solution!!! (The fire, I mean.) Dr. John Gustafson, of Gustafson's law, has written an excellent book, The End of Error, dealing superbly with this maddening ambiguity. It will end the era of heating computer rooms with wasted compute cycles seeking false precision when adopted.

[@domlesombred6844]

What if we assume : infinity x epsilon = x with x element of R ? Because the only problem i see see with 1= 0.999... + epsilon, is it's more or less impossible to go from a number to another if infinity x epsilon = 0.000.....1 = epsilon? Personnaly i think the real question is not 1=0.999... yes or no but instead what's the value of infinity x epsilon (should be 1 if epsilon = 1 (or x) / infinity) I don't know if i am really clear, anyway just my 2cp thought on the subject, love the videos :)

[@cinnamoncat8950]

In the first half I was somewhat hating the video cause it just kept repeating the same "proofs" everyone would use to say 1=0.999... but right after halfway the explanation of epsilon and the infinitesimals helped me realize how to describe the fundamental disagreement I have with this argument. The fundamental thing I disagree on is that I think that 1/3 does NOT equal 0.333..and π does NOT equal 3.1415... I think they are APPROXIMATIONS for values that do not work in our system. They are approximations that functionally have no difference compared to the actual number in the real world but one that exists mathematically, which is why the epsilon now helps me know the difference. It is an infinitesimally small difference but just like actual infinity, it is something that can not be represented through numbers in this system. Thanks for coming to my Ted talk. Also I appreciate the video for exploring deeper than most :)

[@TheRealFOSFOR]

So, I guess 0.999... is the same as 1 - 0.000...1 And because 0.000...1 will never actually get to the 1 at the end, because there is an infinite amount of zeros before it, that means 0.000...1 will actually be 0 for all eternity. Therefore, 1 - 0.000...1 = 0.999... but at the same time 1.

[@SteinGauslaaStrindhaug]

28:23 two plus two could equal bleem if Professor Ersheim was right... ;P (See "The Secret Number" a short story by Igor Teper and a short film directed by Colin Levy, both are available free online. It's really absurd but really good)

[@arraymac227]

'It's not describing something infinitely large.' It literally _is_ something infinitely large. My take: a countable infinity of 9s, so limit argument does not apply. Induction proves it .9...n... is < 1 for all n.

[@cursedrago]

if 1/3 is 0.33333333... and 3/3 is 1 then 0.99999999... is also 1, simple so idk why people bend their heads from such a simple question smh

[@phyarth8082]

Simple computer software has built-in functions of approximation of numbers: Floor and Ceiling functions, rounding function and truncation function of numbers. Controversial matter when simple computer software has 4 dedicated functions to approximate number. Plus in physics we have capital Greek letter ∆ - difference of terms, in computer science we have δ value very small but finite it used to do numerical integration (summation loop) operation on digital computer. And in mathematics we have  (epsilon) infinite small value that is basis of calculus which nobody proven that that it exist or define value of this small value, Zeno paradox is space continuous and uniform. Exist computer software that can perform symbolical integration but it is done not numerically but using logic, Wolfram alpha can make symbolic integration and give answer in letters x, y ..., etc. But that is not real numerical calculation. Mathematics calculus and epsilon is still abstract value hanging on trust that calculus always match observable Nature.

[@jawstrock2215]

Infinity always causes problem in math.. always. Get rid of them!

[@AlaaBanna]

Perfect video, thanks. Although I thought you'd be the perfect person to mention this, but here's how I look at it, The issue with those infinate 3s of (1/3) or infinate 9s .. etc, NOT a real one, but language related one (the base system we're using). For example, if we're using base-3 to calcualte 1/3 we'll simply have 0.1 (with a terminating 1, not repeating), And there in base-3: 0.1x3 (actually it's 10 in base-3, because only 0,1,2 are allowed) = 1.0 simple and clear. Of course, even in base-3 (or all other base systems as mentioned in the video), we'll find infinately repeating digits for other ratios (Let alone irrational numbers), but ALL rational numbers can have a terminating representation in some other system that fits them. So, finding the proper base for representing the scenario you'll find that 0.999... WILL actually actually mean 1.

[@Chris-5318]

Bases have nothing to do with languages. If b is a natural number, then 0.bbb... (base b+1) = 1. There is nothing special about base 10.

[@AlaaBanna]

@@Chris-5318 You are correct and that's what I'm saying. I just meant specifically for the 1=0.999 conflict could simply be resolved in the base-3 system as it will clearly be 1 and no-one would even think otherwise, that's why I mentioned it's a language (or the base) used. Similarly, ALL 0.bbb (related to rational numbers or ratios) can have a clear exact answer if another base, of course base-10 is not special here.

[@Chris-5318]

@@AlaaBanna I have no idea what your reply (or your original post) is supposed to be achieving. What conflict are you referring to, and what is resolved by using base 3? FWIW 0.222... (base 3) = 1. There is nothing special about base 3, or any base.

[@AlaaBanna]

@@Chris-5318OK let me put it in another way :), - Main conflict the OP (the video we're commenting) is: "Is 1 = 0.99999", right? - He offers many proofs or alternative points of views, to show us how it's normal we can find other representations of any number, and that 1, can be expressed as 0.9999.. - I did the same, offering another point of view, to indicate that if we looked at same numbers, from base-3 system, we'll find that conflict is resolved by itself, that because 1/3 will be represented as 0.1 in base-3 and not (0.333 in base-10), 2/3 will be represented as 0.2 base-3, and 3/3 will be 1 (without these 0.3333 * 2 = 0.66666 and *3 = 0.99999), that with base-3 for this problem specifically, we will not even have an issue. And that is not an special case with base-3 of course, nor something special with base-10 (agreeing with you here), but some bases are better for specific numbers to help us avoid inaccuracies of other bases. I hope my point is clear this time 🥲

[@Chris-5318]

​@@AlaaBanna You: "- Main conflict the OP (the video we're commenting) is: "Is 1 = 0.99999", right?" Me: First it's 0.999... = 1, not 0.99999 = 1, and second there is no conflict. What is it supposed to be conflicting with? I'll ignore the fact that you are not using the correct ... notation for now. You: "- He offers many proofs or alternative points of views, to show us how it's normal we can find other representations of any number, and that 1, can be expressed as 0.9999.." Me: So what? You: "- I did the same, offering another point of view, ..." Me: No you didn't. You: "... to indicate that if we looked at same numbers, from base-3 system, we'll find that conflict is resolved by itself, that because 1/3 will be represented as 0.1 in base-3 ..." Me: You resolved nothing and I have no idea what you think needs to be resolved. 0.1 (base 3) = 0.0222... (base 3) and 1 = 0.222... (base 3). You: "... and not (0.333 in base-10), 2/3 will be represented as 0.2 base-3, and 3/3 will be 1 (without these 0.3333 * 2 = 0.66666 and *3 = 0.99999), that with base-3 for this problem specifically, we will not even have an issue." Me: 1/3 = 0.333... (base 10) = 0.1 (base 3) = 0.0222... (base 3). 2/3 = 0.666... (base 10) = 0.2 (base 3) = 0.1222... (base 3). 1 = 0.999... (base 10) = 0.222... (base 3). You: "And that is not an special case with base-3 of course, nor something special with base-10 (agreeing with you here), but some bases are better for specific numbers to help us avoid inaccuracies of other bases. Me: The only inaccuracies I see are when you write, e.g., 0.99999 instead of 0.999.... 0.bbb... (base b+1) = 1 precisely in every natural number base - there is no inaccuracy. Every real number can be represented in every natural been with perfect precision. No base is more accurate than another. You: "I hope my point is clear this time" Me: I have no idea what this "point" is that you think that you are making.

[@sumdumbmick]

every Real has an infinite number of 0s to the left of the decimal point. just as the Real number 1 has an infinite number of zeroes to the right of the decimal point. and you should note that by definition, Integer 1 has no zeroes to the right of its decimal point, because it doesn't have a decimal point. but does still have an infinite number of zeroes to the left. which means that the Integers are not actually a subset of the Reals, because Integer 1 and Real 1 are entirely distinct things. this distinction is only amplified by the canonical acceptance of Dedekind Cuts defining the Completeness of the Reals, since Dedekind cuts render the Reals non-specific, such that 0.9R might be the same as 1, or it might not be. which only emphasizes the point that Real 1 is an entirely different sort of beast from Integer 1. beyond this, Natural 1 is not the same as Integer 1, since Integer 1 is positive by definition, and thus has a direction, making it a vector, but Natural 1 is unsigned and thus a bare magnitude. this means the Naturals are not a subset of the Integers, in the same way that the Integers are not a subset of the Reals. this renders equations like |-3| = 3 false, since the default assumption is that when an unmarked 3 is written the meaning is +3, which means it's Integer 3. but we know that Integer 3 is different from Natural 3, and specifically that the absolute value operation yields an unsigned magnitude, meaning that if we're getting out of the absolute value a whole number, then that must be a Natural. unfortunately, there is no category for explaining what the result of |pi| is, but the reason is specifically that the people who established Axiomatic Set Theory didn't do things correctly.

[@815TypeSirius]

If you actually have infinite .9 then it not only equals one. It equals all numbers. But, how you going to write infinite 9s? You are not. Anyone who @'s me and argues gets nuked.

[@jaybingham3711]

Squirrel ain't up for being played as fool. He rolled out the sniff test. The bite test. And the lick test. "Noice. But ffs the same can't be said for that lab coat!"

[@tuqann]

the controversy (imho) is based on the erroneous conflation of two different types of math; today in binary we denote numbers by filling bits from the starting point so for example an 8bits integer of 1 would be O#00000001 now the inverse of this process so that to count to the first number you subtract bits from a full 8bits then you have O#11111110. Now imagine being able to write a number an infinite number of bits using this way; the first number would O#11111111.... endlessly since we can never arrive at the slot where the value is not the 1 bit. since in binary a one bit flips to a zero with decimal system a 9 digit flips into a zero, and suddenly 0.99999...endlessly is how you write the first number (i.e. 1)

[@alebarojr]

If every number had his own symbol, you would immediatly shut up. Also, math literally depends on philosophy (did you ever study something that is not just a line of numbers?)

[@mikumikudice]

personally, I don't consider this a glitch. it's a consequence of what math is. a representation of the nature; anything that represents something but isn't really that thing, will end up with multiple ways to represent it. many ways you can represent a car (a drawing, a word, etc). I think it's beautiful that one of the most complex things we ever invented is not a science, but a language that represents the really and events on it

[@userpd4ze3kt4u]

Is there a number system that orders all the numbers between zero and one and would it have any practical use? Zero point zero recurring one would be the smallest number after zero, so it would be called one. Zero point nine recurring would be the second biggest number after one so it would be called infinity minus one. Half could just be infinity over two.

[@SirWilliamKidney]

I went from liking this channel to loving this channel @18:31 haha

[@jamesbusald7097]

You painted the same siding on your house they same color I painted that siding on my house, in2008. I feel at home here.

[@geminirox8635]

1 - 0.9 repeat would be what? 0.0 repeating. Would anyone disagree that 0.0 repeat does not equal zero?

[@RoBFE-fe5dh]

All those 9's in a row look like the stick figure version of the unemployment line. Just looking down, depressed.

[@cuomostan]

So what would happen if I multiplied .99999… by n, where n is any real number? Would that be equal to n?

[@tozainamboku]

I gave a squirrel 🐿0.99999... (repeating infinitely) nuts and he bit me.😮

[@randus7053]

What about dx though? Calculus seems to assume it isn't zero. I do agree though that at least in algebra, 0.9 repeating is 1.