The Truth About the Most Controversial "Number"
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- Опубликовано: 7 сен 2024
- 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:
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Domotro
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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.
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.
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".
I counted 4.999999999... squirrel appearances.
@@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".
@@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.
Let x = 9 + 90 + 900 + 9000 +...
I like that 0.999... = -1 * x.
This guy spent an infinite amount of time writing an infinite amount of "9"s after "0." for a video. Respect.
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).
@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.
@@silver6054 that could explain the lack of the time machine needed for our theory… 🤔
@@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.
@@sirfzavers8634 Maybe he did, but the video is still uploading...
Domotro has mastery over squirrels and numbers. If he would only learn to control fire, he would be unstoppable.
And a way to prevent things from falling over
Squirrel Girl just needs squirrels to beat Thanos, Dr. Doom, Galactus, whomever.
He's got this.
There's a reason why Squirrel Girl is invulnerable and all of the fire based superheroes are not.
He's dual classing Wizard and Druid, it might be hard for him to continue leveling if he adds Sorcerer to that list
😃🤣❤️👏👍
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.
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 :)
😂 that's such a perfect way to describe this channel in general; love the chaos
In arguments like this I always like the engineers answer "It's close enough, it fits the spec."
It’s within the tolerance.
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.
Glad you enjoyed and it helped you learn, thanks for the compliment :)
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.
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
your feelings are irrational
@@Fire_Axus haha. More than irrational, transcendental.
(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.
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 :)
Thanks, glad you’ve been enjoying! :)
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!
Thanks for the feedback, and I’m glad you enjoyed! :)
the archimedian principle is not just true for integers though.
@@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.
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).
"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.
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.
0.99...95
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 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.
@@BlackBull. You number ends, it ends at ...95. The number 0.999... _doesn't end_ and so is larger than your number.
@@martind2520 it was supposed to be a joke🤕
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
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
Nice ... very nice.
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!
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.
these are the 2-adic numbers. some of the other p-adics can even represent i (ie solutions to x^2 + 1 = 0).
@@CassandraComar Not exactly. The 2-adics don't include digits to the right of the radix point.
@@JonBrase the 2-adic integers don't but the 2-adic rationals do. they represent fractions with power of 2 denominators.
@@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.
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.
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"
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
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.
Its probably not possible to add an infinite amount of numbers.
Perfectly explanation for a common misconception, like always.
@@agentofforce3467 it's absolutely possible. Just look at any convergent infinite series, for example.
why does this guy speak in 0.75x speed
Sorry to tell you,he doesn't. It is you. you hear in 0.75 speed.
Since his speech contains 25% more info per word than normal, he slows it down for us plebeians
He fed squirrels, and burned a guitar. After the smoke cleared, we concluded 0.9999.. = 1.0
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.
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.
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.
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.
I saw a squirrel, some blue tetrominoes, burning stuff and there might have been some numbers somewhere along the way. Lovely stuff as always.
some day i will understand why this man has so many clocks
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.
How is a backward.clock right four times a day?
He hates to be late
You still have time. 😀😀😀😀
Bro.........@@StevenLubick
This was both greatly educational and yet greatly uncomfortable. Looking forward to more!
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.
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.
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.
Combo class be comboling my brain
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!
I used to deny that the two were equal, but now I see just how wrong I was in *many* different avenues. Thx
You were right the first time.
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.
Subscribed! You have given this the most thorough, intuitive explanation I've ever seen. You are exceptional as a teacher. Also, I love the chaos of your approach and the set lol It's a great schtick that keeps the vids entertaining. If you keep going you're going to hit 100,000 subs and beyond in no time. Keep up the great work.
I love watching this channel. You always upload interesting content that never fails to enlighten others (including me) !
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.
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.
I realized that what bothered me about it when I first learned this is it seemed like there could be a way to make it so these could be considered different. Now I realized the concept I was sniffing was the infinitesimals and hyper real numbers that can be used to define nonstandard analysis.
Except that in the hyper-reals 0.999... is _still_ exactly equal to 1.
@@martind2520 In spirit though, it was neat to find out that the idea I had a long time ago before I knew much math had some merit to it even if it didn’t apply directly.
@@martind2520I'd like to see that proof. I could learn a lot.
@@johnlabonte-ch5ul Karen, you have proven that you are incapable of learning any math. You don't even know that if a number can be written as p/q where p and q are natural numbers, that it is a rational number. You don't even know how to write 0.999.... In the surreals and the hyperreals, 0.999... = 1.
Can you repeat part of that? I got distracted by a squirrel.
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 :)
Math was not my best subject in school, especially post-secondary math, but damn you make it fascinating!
3:27 How to become a believer in a god of math.
"something off" and paper of infinity immediately drop off.
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.
Finally, a video that matches how people tend to see me when i tell them that there's a number between 3 and 4 which has no decimal representation because its a new kind of value when we consider that numbers are pixels, meaning there are numbers that are right at the sides of each one
Your comment is pure nonsense.
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.
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.
You're right. I misspelled it, and also didn't clarify the positive/integer restrictions, so I added a clarification to the description.
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.
so, the frac > dec button that every calculator already has?
It's very nice to be able to figure out a pattern like that at your own, especially at such a young age!
I went from liking this channel to loving this channel @18:31 haha
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.
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”
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.
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.
There's nothing to prove. A decimal expansion is just another way of writing the same number.
Are you surreal right meow?
I often live in semi-surreality
you are late on the topic! science gang has covered this months ago.
yk, we love your content! well done ❤
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.
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?
@@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.
@@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?
@@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
@@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.
The squirrel running up and down " Yggdrasil" was awesome!😂🐿️
0.99999...st!
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
@@ComboClass thank you for the lovely lecture!
0.99999st!
@@tyruskarmesin5418 yeah, that's what I should have said
9:42: Wow. While I admit I may have caught a glance, I wasn't looking at the screen but I instantly RECOGNIZED the sound of what had just fallen over. It's been 20 years since we threw that thing away after finally being too damaged.
Since when was this controversial? We're stuck back in the 1700's again?
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
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.
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
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
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!"
Came for the math, stayed for the squirrel. 18:41
This is the most definitive discussion on this subject as far as I'm concerned. Bravo!
This video made me go through all 5 stages of grief
That was a really good video, but how does it relate to Magic the Gathering?
Squirrel is the protagonist we didn't know we needed
Your presentation style is great. I prefer to include infinitesimals as numbers and though you may not, I'm glad you acknowledged that it is whatever we define it to be rather than stated that .999... = 1.
You don't need to throw out the assumptions we are using for everyday math. You just need to treat the equals sign we are using as if it has an asterisk where things are not truly equal, but instead, are within a given range of each other. Then, you can continue on using our same symbolic rules we are used to and simultaneously not say it means .999... is literally 1.
Do you want the world to switch to base 6? I think we should go for 12 or 30. We already learn multiplication tables up to 12x12 in school anyway, at least in the US. 10 is not great because 3 is a better prime number to evenly divide than 5 because it gives us more frequent terminating divisions. But with 6, numbers would start to require more digits to write out. I know memorizing isn't fun, but you only need to do it once. And even multiplications up to 30x30 are well within the amount of information people can retain. People that rarely do math may not know them that well, but for the people who do math a lot, it would be a benefit. I think we should make the tool specialized for the people who do the job. They do the job, so their needs should come first. We don't need to take specialized systems within every discipline of knowledge known to mankind and dumb them down for the lay person. And we don't need to here either.
For the argument that goes:
x=0.999...
10x=9.999...
the .999... in the first expression shouldn't be considered the same as the .999... in the second expression. It will have 1 less 9 in it. People say often say some infinities are the same size, but that's a conflation between terms. What they mean is they are the same cardinality. I also reject the notion that linear bijection reflects what 'size' is.
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.
As far as avoiding glitches goes, I think p-adic numbers with representations that terminate on the right avoid them... as well as the ability to represent most irrational numbers, so.
Don't you regard ...999... = -1 as being a "glitch" in the same ways 0.999... = 1 is?
Seems like the ship of Theseus paradox. If we lose a chunk of our skin through an injury, are we still ourselves? We would be more different for from the infinitesimal difference of .9 recurring and 1. I love your point about the numbers we create as being representational. Love your work 🙂
1 - 0.999... = 0. There is no difference, even if infinitesimals are allowed.
I wish you had showed the entire infinite string of 9s so we could fully appreciate the level of effort you put into your videos 😢
23:42 "Any number that has a terminating decimal expansion ... will have another form [with infinite digit string]"
I think you mean any number other than 0.
I could be wrong, but i cannot see how to represent 0 with a digit string terminating in infinite 9s. I think to generate a second decimal representation of a terminating decimal number, you have to consider which size of zero your number is on, so you can take the least digit down (toward zero) by one before appending the infinite string of 9s. In zero, you cannot take the least digit down toward zero by one. It's the same type of singularity that occurs with a compass at a magnetic pole, you are already at "zero" so any step you take can only go in the wrong direction.
True I should have said non-zero there. I’ll add a clarification to the description
I forgot to mention in my other comment that I really liked the squirrel cameos!
I'm a Pythagorean cultist. I don't believe in irrational numbers - not in any concrete real world way (irrationals only exist in a fictional albeit consistent & useful imaginary abstract world). But even in my philosophy 0.999... == 0.(9) == 1. It is rational & it is the unit one, from which all other real (read: rational (Pythagorean, remember)) numbers are defined. Excellent job explaining why here.
This was something I struggled with when younger, going through a few different phases of my understanding of this difficult concept. I remember at one point feeling frustrated, and thinking it basically didn't really matter - at the time I was still wanting to trust my instinct & felt like 0.(9) != 1, but I was willing to (grudgingly - I was also in a pretty contrarian phase at the time) concede that the difference was immaterial. But it does actually matter, and I think you also did a good job explaining why it matters in this video. Ceci n'est pas une pipe, the map is not the territory, and symbols are not what they symbolize - but it is still very important to have a correct understanding of the symbolism; any flaws in understanding the fundamental symbolism can lead to flaws in the overall understanding. And it is perfectly acceptable to have multiple different symbols representing the same concept.
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.
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.
@@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.
@@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.
@@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 🥲
@@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.
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.
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)
I find it pretty simple: 1/3rd = 0.3̅, 2/3rds = 0.6̅ , so 9̅ must = 3/3rds, and 3/3rds = 1!
After finishing the video I can say this seems slightly more complicated, but I still stand by my reasoning
Me: wants to sleep
The shark biologist from Jaws in a backyard of clocks slowly losing sanity: I don't think so
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.
Amazing episode. Thank you Domotro.
You may appreciate a poem I once wrote:
"I have seen where the one mad God lives
Far from here, yet, just a heirs breadth away
I saw him whisper into his own ear
'The world is not made,' He said
'It is Mad'
A bit of a weird fellow."
This is an amazing video! So educational and entertaining. That squirrel is hilarious! :)
What's interesting to me is that 0.999... is EXACTLY equal to 1.000...1
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.)
I like to think about it in this way:
Point is a shape without dimensions. Like it has not any, so it's 0-dimensional shape. However, we can draw a line using infinite amount of points. This leads us to paradox where 0 * infinity = C where C is some number larger than 0. Which is not acceptable. To solve this paradox we can consider point has size and its size is 1/infinity. Then our paradox will be "solved", because some number that differs from 0 can be increased or decreased by multiplying and dividing. Now if we look at the line with length of 1 we can "count" how many points creates this line.I know it may sound crazy but it's not a big problem, since there can be infinite points anyway, because we have different infinities. Then we look at the number 0.(9) In this situation we can get that number by removing one point out of our line with 1 length. I am not saying that you're or other mathematicians wrong or something. I'm not professional after all. I just gave my point to this problem. Also, this representation of point explains why if 0.(9)=1 then 0.(9) is not equal to 0.(9)8. Thank you, for reading my long comment.
The size of a point is precisely 0. Infinity, oo, is not a number. The convention is that 1/oo = lim n->oo 1/n and that is 0. You CANNOT count the number of points there are in a line because there are more points in a line than there are natural numbers (and that there are more real numbers than natural numbers).
To get some insight as to how complex the issue is, see "Number Line - Numberphile".
@@Chris_5318 I know. I just assumed that there can be. We created whole imaginary numbers in this way.
@@FantyPegasus I just noticed that I hadn't read all of your original post. You have more mistakes. For instance, you said, "Then we look at the number 0.(9) In this situation we can get that number by removing one point out of our line with 1 length". That is wrong. There isn't a missing point. You seem to think that 0.999... is a point adjacent to 1. There is no such thing as two adjacent points. Either 0.999... is the same point as 1 (on the real number line), or there are uncountably infinitely many points between them. Consider the infinite union of closed intervals:
[0, 0.9] U [0, 0.99] U [0, 0.999] U ... = [0, 0.999...) = [0, 1)
That's because none of the intervals contains 0.999... or 1. (Skipping a few obvious details).
You seem to think that [0, 0.999...] = [0, 1) and that is not possible. An open interval is fundamentally different to a closed interval. NB The length of [a, b] = length [a, b) and least upper bound / supremum of [a, b] is the same as for [a, b).
Whatever 0 * oo is undefined because oo is not a number. However, using the convention that whenever you see infinity being used as if it is a number, then I'd say that 0 * oo means lim n->oo 0 * n = 0 is the only sensible interpretation. The 0 is an actual 0, not a number arbitrarily close to 0. 0 * oo is NOT an indeterminate form as many people say. It is only a label for a type of indeterminate form.
0.(9)8 is not equal to 0.(9) because 0.(9)8 is not a valid decimal. You can't have and endless string of 9s with an 8 at the non-existent end.
I get the feeling that you are not familiar with the aleph numbers (cardinalities).
This has nothing to do with imaginary numbers. I see no analogy at all.
Your post is not particularly long.
This strikes me as being a limit problem. Limits are contrived to permit mathematicians to have it both ways. They can claim both that a limit converges on a particular number so it is exactly equal to that number and also that it isn't exactly equal to that number.
That is absolutely _not_ what limits are for.
Limits were created precisely ot _avoid_ that problem. They were invented to very precisely define concepts that were previously using very imprecise infinitesimal.
They are not mathematicians having things "both ways", they are mathematicians being _very careful_ to _not_ have things "both ways".
welp you conviced me pretty quickly and to be honest your explanations are very intuitive. the thing that got it for me was limits. thank you as always for your phenomenal vids
Loved that intro, infinite 9's!
Thanks! I had fun making the extra long paper string of 9’s with carlo to add some thematic/surreal vibes haha
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.
Tally marks
Excellent camera work
Shout out to my main camera guy Carlo! And a few other friends who helped me film some of the title cards, who are named in the credits/description :)
19:40 Max Planck: "Halt' ma' mein Bier!" (hold my beer)
My usual way of demonstrating that the number system people like to work in must require .999999... = 1 is the Intermediate Value Theorem, (IVT).
Given a continuous function that goes between points f(a) and f(b), if there exists a point c that lies between f(a) and f(b), there must be some value such that f(x) = c, such that a < x < b. Much easier explained with a visual, and drawing a horizontal line between f(a) and f(b), demonstrating that you cannot draw any continuous line between f(a) and f(b) that does not cross the horizontal line.
However, if you allow infinitismals into your number system, then the notion that is used to prove the IVT doesn't work. Quickly stated, it's usually proved using an epsilon-delta construction, where upon as the selected points a and b get closer and closer together, (within an epsilon), then the distance between f(a) and f(b) must also get closer, (to within a delta), such that the distance eventually becomes zero, which actually defines continuity. Then by symmetry, we can do this for any point between a and b, and thus the function must achieve every point between f(a) and f(b) on the range (a,b).
But with infinitismals, while we still can have a notion of continuity, that doesn't mean we achieve any particular point. A limit approaching zero no longer means that it IS zero at infinity, it could just be an infinitismal, call it Iota.
So a function would not betray continuity by defining it as being an iota less than 3 from the left, and an iota more than 3 from the right, but never actually returning "3". (You can define such a function more rigourously if desired).
This demonstration isn't really any different than the argument that the limit of 0.99999... approaches 1, but it implicitly shows people that they like dealing in a system where that is true, and infinitismals don't exist. The downside, of course, is that you need to define continuity and introduce the IVT to people who will probably never approach Real Analysis ever in their life, but I only consider it a downside for *them*. It's an up-side for me :)
I like the presentation style.. kinda cool
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.
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
"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 😂
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.
@@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.
@@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.
I feel like on the left of the decimal 9-bar.9-bar works as an equivalent of Infinity, for the same reason 0.9-bar is equal to 1. For the left side of the decimal, that is the only number which seems to have any meaning, but I'd argue it follows the same number system.
The problem I see with this is you should define that as dividing by one less than the base, so 1-bar.1-bar + 8-bar.8-bar = 9-bar.9-bar, but defining 1-bar or 8-bar isn't definable in the same way which 0.9-bar is definable as 1/9 + 8/9... So it is more difficult to define.
One of the other strange properties that this reveals is that 9-bar.9-bar is infinitesimaly smaller than Infinity just like 0.9-bar is infinitesimaly smaller than 1, which suggests that 9-bar.9-bar might be argued to be 2*Epsilon smaller than Infinity, but in the same way I think it actually reinforces the notion that 9-bar.9-bar is Infinity and 0.9-bar is 1, because Epsilon would be equal to 2*Epsilon for the same reason Infinity is equal to 2*Infinity.
No. Infinite 9s on both sides actually ends up being equal to 0, not infinity. (It is also a mathematical mess of a concept, but still ends up as 0.)
And _no_ 0.999... is _not_ "infinitesimally smaller" than 1. They are _exactly_ equal. There are a multitude of proofs to this fact.
@@martind2520 you'll have to explain to me how an infinite number of nines on the left side of the decimal point is equal to 0. As for infinite nines to the right, the infinitesimally close to 1, but not quite reaching it was a point of view discussed in the video. I don't question that it is, myself, but I was trying to relate the concept to 0.9-bar as was discussed.
This video has given me much respect for the word “infinitesimal”
Thank you Master Domo
Not gonna argue both sides. I was fermly on the "different" side at the start. because of my computer science mind (where you never test equalities for floating points numbers)
But now I'm on the "Equal" side. And could defend it pretty well. Nice video.
Still got chills over all the destroyed material though..
It’s sorta funny because in other bases, like 6, 1/3 is just .2
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.
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.
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.
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. Where 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.
It doesn't. Your sequence is just the equivalent of 0.999... in base 100.
In binary the equivalent is 0.111... = 1/2 + 1/4 + 1/8 + ... = 1.
Ternary: 0.222... = 2/3 + 2/9 + 2/27 + ... = 1.
Etc.
@@martind2520
Yes, I know. And said so myself. "Normally we would say that this is just an alternative way to describe 0.999...". But it doesn't explain why faster and denser convergence doesn't imply a bigger sum with greater approximation to the limit value, i.e. 1.
Of course, if you have already established that 0.999...=1, then it's self-explanatory because then you have defined the difference to be 0, and
(10/11)×0 = (11/10)×0 = 0.
But I think the answer lies in the convergence. If the sums didn't converge, then the greatness of each step would matter. But when they converge to a finite value, then you can chop up or put together steps as you please without changing the value of the infinite sum.
On the Q that was posed, a system with a non-integer base would allow numbers to be written in more than two ways (unless you insist on allowing only a canonical form, but that also deals with decimal 0.999..).
Like, in base phi, the number one equals
1.0000000..
or
0.1100000..
or
0.1011000..
or
0.1010101..
and many more.
(I suspect *uncountably infinitely* many more even!)
I think it's still countable, just map them to the natural numbers following the pattern
1-> 1
2->0.11
3-> .1011
n -> n-1 representation but change the last 1 to an 011
of course you still have the infinite string of .01010101... forever but that's still just ONE number you can easily map by mapping 0 to it (or whatever trick you want to account for it.) Where are the uncountable representations coming from?
@@HopUpOutDaBed I was thinking about non-canonical forms that feature multiple pairs of 1s or series of more than two 1s in a row, but those indeed never seem to add to 1.000.. exactly. Have to give it some thought whether the number one is special in that sense, or even all integers are special.
My reasoning was that when you are forming the digit expansion and you have a remainder that is just a tiny bit over the value of the next digit, then you can always choose to continue either ..100.. or ..011.. (which have the same value base phi). Another way to put it is that you can replace any ..100.. in an expansion with a ..011.. These latter forms are not canonical, but they are expressable (just like 0.999.. is not a canonical form for 1.000..). I surmised that for a typical 'random' number, such opportunities will occur arbitrarily often, i.e. there will be arbitrary many places where the expansion would have a 1 followed by two 0s, so you get a countably infinite number of places where you can choose between two representations, and 2^countably_inf equals uncountably_inf.
On second thought, that might only hold for 'random' numbers (not exactly sure what the requirement is, but the numbers with arbitrary many ..100.. in base phi should be dense on the real line, I bet).
So maybe I should rephrase it more narrowly that *with the exception of only a countably infinite number of reals* (including the integers as these exceptions), base phi allows uncountably infinitely many different expansions of a number. Or, alternatively, the numbers that can be written in uncountably infinitely many different ways in base phi are dense on R.