Matt meets Jordan Ellenberg: 0.999999... = ?
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- เผยแพร่เมื่อ 13 มิ.ย. 2015
- 0.999999... = ?
Matt and Jordan meet-up in the Penguin offices for a chat about profound but simple mathematics and adding infinitely many things.
Jordan Ellenberg's book: www.jordanellenberg.com/how-no...
Matt Parker's book: makeanddo4d.com/
Watch the BONUS FOOTAGE: • Matt meets Jordan Elle...
Music by Howard Carter
Design by Simon Wright
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
New book: makeanddo4D.com/
Nerdy maths toys: mathsgear.co.uk/ - บันเทิง
As a science guy I'm confused. Me and my engineer friends are pretty sure 0.9=1, but if that's not, then 0.99 certainly does
Even engineers aren't that bad.
and let's just round Pi to 4!
@@alejotassile6441 Oh poor Pi, why would you do this.
@@alejotassile6441 About 4. It can be different, and often is.
My bank card number is .999... But whenever I press 1 at the ATM, I can't access my money.
+pattystomper1 lol. You need to go teach them math then. :)
+pattystomper1 wow, then you must have quite a long bank card number!
+Deejay Latchuman That must take forever to type in the numbers
Forty Two literally!
+pattystomper1 That's because it is not .99 REPEATING.
I subscribed to this channel to get math standing up.
Your sitting down.
D:
Apparently YOU'RE not here for the grammar.
Unsubscribe
He is not math hence don't unsubscribe yet. With this video he made math go crazy
What I sometimes like to do is move from decimal to base 12. Then, 1/3 equals 0.4, and 3/3 = 0.4*3 = 1, no problem at all!
+Schindlabua and then you can say that 0.4 dozenal = 0.333... decimal, which people cannot disagree with - and then you say that 0.4*3 doz = 0.333...*3 dec = 0.999... = 1.
But then what happens to say 1/5
Rishabh Daga That's true, but I guess the point to make here is that numbers with recurring digits aren't special in any way, but rather a weird (and way too common) edge case arising from our choice of base.
Some time ago I found this profound comment in some random TH-cam comment section that stuck with me ever since:
"Numbers are not their decimal expansions."
+Schindlabua Base 12: Dozenal.
+Rishabh Daga
0.24 doz I believe.
It's not that pretty but at least it's not an infinitesimal xD
I came to see what proof you both thought was the best, but I was suprised and delighted to see this become a great discussion of mathematics! You touched on so many interesting ideas- I think next time someone tells me maths is boring, I will direct them here.
Well, that's what you expect from a "mathecomeditcian" !
Does that in any way seem to be analogous to how we are interpreting particle behavior phenomenon...
Here we too are somewhat "making stories that work" there are no guiding fundamentals.
What do you think?
Yeah, I suppose math and physics do have that surprising analogy. Even though they are both very different epistemologically (sorry for using such a horrendous word), they both have an element of 'taste' to them.
I like this new interview / chatting format.
Just started his book and I've been loving it! I'll have to read yours next
This was an amazingly interesting discussion! I've always had a certain proclivity for the idea of infinitesimals, nice to see some solid discussion about it :)
LOL "Very efficient use of a stroke!"
If you’re just gonna quote the video rather than coming up with your own comment, at least quote it correctly
I could sure go for some brown paper.
Great vid. I've read both your books and loved them.
I have something to add! 1/9 = 0.111... 2/9 = 0.222... 3/9 = 0.333... 4/9 = 0.444... etc. When you get to 9/9, completing the pattern gives you 0.999..., however we know the answer is actually 1.
Similarly, 1/1111 = 0.000900090009..., 1/111 = 0.009009009..., 1/11 = 0.090909..., so 1/1 = 0.999... = 1
1/3 = 0.3333
3/3 = 0.9999 = 1
This is nothing to add lol, its the exact same argument to 1/3=0.333.. which was said in the Video
1/3= 0.3333333...
(1/3)*3 = 0.99999... = 1
didn‘t he talk about that at 3:00 ?
I had a lecturer give a good argument that 1=0.9999..., which was: "If they were different, and thus 1 > 0.99..., what number would be between them?"
This is built on the fact that there are infinitely many real numbers between any two numbers a>b.
He also gave the argument, "0.999... = \sum_{n=0}^{\infty}9/10^n, which is just a geometric series (with sum 1).
0.9 + .1 = 1, 0.99 + 0.01 = 1, 0.999 + 0.001 = 1, so it seems logical to argue that 0.99...9 + 0.00...1 = 1. If you accept that 0.99...9 = 0.999..., then 0.00...1 is the number between it and 1.
Chris G, If you accept that 0.999...9 = 0.999... then you have to accept that
0.999...9 - 0.999... = 0 = 0.000...9 = 9 * 0.000...1 and so 0.000...1 = 0
So 1 = 0.999...9 + 0.0001 = 0.999... + 0 = 0.999...
I'm really not sure if you were serious. Whatever, you cannot use induction over the naturals to get to infinity. That's because there is no natural number such that n + 1 = oo.
"Infinite" means "endless", so 0.999...9 doesn't have infinitely many 9s (It just has an unspecified finite number of 9s). So 0.999...9 < 0.999...
BTW "between" is ambiguous, surely 0.9995 is between 0.999 and 1. 0.001 is the difference (or *gap between* )
How about if I amend my argument. 0.999... = 0.999...0. 0.999...0 + 0.000...1 =0.999...1 > 0.999...0
Chris G. Just to fill time, what are 10 * 0.999...0 and 0.1 * 0.999...0 as decimals?
Hmm. I guess they have to be 9.999...0 and 0.0999...0 respectively. You shift the decimal point right or left. That would be the case if the digit(s) on the right after the dots was a 0 or anything else.
Also an interestingly easy way of thinking about it: Whenever you want to simplify an infinitesimal as a division, you divide the repetitive bit by the same amount of nines as there are digits in the repetitive bit. For example 0.345345...=345/999. So if you suddenly wanted to simplify infinitesimal nines to a division, you'd do 0.999...=9/9=1. Don't know if this has been noted anywhere before but I just came up with it.
Very cool. Btw, I just found out on his Wiki page that he cameos (as a math professor) in the movie 'Gifted', which gives him a Bacon number of 2, and an Erdos-Bacon number of 5.
He also has an imdb entry.
Wonderful philosophical discussion!
The two methods for demonstrating that 0.999... = 1 are the 3 * 1/3 method, demonstrated in this video, and the following:
Let x = 0.999... and y = 1
If x, y are distinct real numbers, then there must exist a real number z such that x < z < y. ie: z = (x + y) / 2
In other words, there must be a number greater than 0.9999... but less than 1.000...
This is clearly impossible because an infinite string of 9s after the decimal point is the highest possible value in base 10 before reaching the next integer.
Contradiction. Therefore x and y are NOT distinct.
If you make an exception to the rule, and say that x < y with no intermediate z, then either:
A: (x + y) / 2 = x
B: (x + y) / 2 = y
There are no other alternatives. But...
A: (x + y) / 2 = x => x + y = 2x => y = x
B: (x + y) / 2 = y => x + y = 2y => x = y
still don't work because of contradictions (x < y).
Therefore, there is no hope. 0.999... = 1.
+nychold Well, I, respectfully disagree.
Following your logic:
1 = 0.99999...
then it would be logical to assume:
0.9 = 0.89999...
0.1 = 0.09999...
etc.
Now following this logic:
0.99999...999 = 0.99999...998
0.99999...998 = 0.99999...997
Etc.
0.00000...001 = 0.00000...000
So what you are saying that error is so small that we can assume that two values are equal. I get what you are saying.
The problem in my eyes is that convenience is chosen over precision and I don't like that in math.
It is OK to say that PI is equal to 3.14, but it's not OK to say that it's exact value. It's an approximation which can be OK in some cases, but completely wrong in other.
0.99999... approximately IS equal to 1, but it's not exact value.
Kestutis Tauckela "Well, I, respectfully disagree.
Following your logic:
1 = 0.99999...
then it would be logical to assume:
0.9 = 0.89999...
0.1 = 0.09999..."
This is correct.
"Now following this logic:
0.99999...999 = 0.99999...998
0.99999...998 = 0.99999...997"
Wrong. There is no last digit, like you are assuming here. To even fathom the idea that 0.999... = 0.999...998 is to make the assumption that 0.999... does not continue indefinitely, and therefore ends, which makes it a number wholly different from 1.
"So what you are saying that error is so small that we can assume that two values are equal."
No, that's not the statement at all. The statement is that 1 - 0.999... must equal some finite value greater than 0 if they are distinct. And that value (call it D) must then be part of the field (the real numbers). But if D exists, then so does D/2, which must also be distinct. This means there must be a number closer to 1 than 0.999... because 0.999... + D/2 cannot possibly equal 1. In short, there IS no error.
There are a lot of different ways to say it, but I find that's the most simple way.
"...and I don't like that in math."
This is the primary root of all fundamental problems in understanding math. The "I don't like it" excuse is just that...an excuse. Do you think that I like that there is no general algebraic solution to quintic equations? I tried for decades to find one, and until I learned about Calois Groups, I assumed people just hadn't been clever enough to find a solution. But they're right...it's impossible, thanks to S4 not being Abelian. There's a lot of math behind it, and I won't bore you with it, but the entire reason we still have people denying that 0.999... = 1 is because "I don't like it" sounds like a valid refutation. And it isn't. It's just arrogance.
"It is OK to say that PI is equal to 3.14..."
Not quite, but I get your point.
"0.99999... approximately IS equal to 1, but it's not exact value."
And this is entirely false because, as I stated earlier and several times throughout this discussion, in order for them to be distinct values, they must have a definable difference greater than 0. For example:
pi - pi_approximate =
3.14159265358979... - 3.14 =
0.00159265358979...
0.00159265358979... / 2 = 0.00079632679...
And so on.
nychold Thank you so much for your explicit reply. I really appreciate you taking time to do so. I also believe you have way more knowledge about math than I do.
Everything in math I do understand - are logical and intuitive. The things that are not intuitive indicate that it's either I do not understand that or that is just wrong. I'm sorry if this way of thinking comes across as arrogant - I just refuse to nod and accept something I don't understand as being true.
One more point to understand my perspective:
When someone writes and answer, a value, a real number and I verify that answer to the one I know being correct - the moment I see a wrong digit - I know that the answer is wrong, regardless of subsequent digits. So if someone writes PI being equal to: 3.15 - I know the answer is wrong, regardless of the other numbers, because 5 is wrong. So when I see 1.00000, being written as = 0.(anything) my brain refuses to accept that as a correct answer, because 0 is already a wrong digit.
I especially like the part of your explanation about "definable difference greater than 0". I do not have valid arguments to counter that. I guess that definition of real numbers is exploitable and the statement "0.(9) = 1" exploits that.
I will quote the definition from other comment that my brain likes a lot:
0.(9) + 0.(0)1 = 1
Even if this goes outside of standard definition of real numbers - it makes sense to me.
Have a nice day.
Kestutis Tauckela "I'm sorry if this way of thinking comes across as arrogant - I just refuse to nod and accept something I don't understand as being true."
This may come as a shock to you, but I wholeheartedly agree. The issue I have with some people in this regard is that is where they stop. Rather than try to disprove it, or prove they are correct, they plug their ears and say "I don't believe/understand it therefore it is wrong."
"I will quote the definition from other comment that my brain likes a lot:
0.(9) + 0.(0)1 = 1"
This is what's called amateur mathematics. There can be nothing beyond an ellipsis (...) because that is saying "ad infinitum" or "to infinity". Unlike what Buzz Lightyear says in Toy Story, there is no beyond infinity. Infinity + 1 is nonsensical, which is what 0.(0)1 is saying. The one is in the "infinitith-plus-one" column, or however it would be spelled.
There are, however, number groups, rings, and fields for which infinitesimals exist. The hyperreals and surreals both have infinitesimals, but I rather think you'll hate them worse than the real numbers because, in the hyperreals (a subset of the surreals), you have things like ...999 = -1. Here's a number that seems to be infinitely large but actually equals a negative number. It's far from intuitive, but it's still pretty cool.
+nychold I've seen some really great proofs for this (one recent proof I discovered was the infinite sum approach, which rather aggravatingly was being taught to me by a teacher who does not accept that 0.999...=1) but this distinct real numbers argument really pleases me.
Awesome :D
I've read both of those guys books - they're great!
OMG THANK YOU! You are the first mathematicians (are you?) that gave a valid answer to this. I had to derive this idea of infinitesimals myself before finding out that's what they were and why we don't use them.
I have seen a dozen videos on this and argued the point as many times and all I ever get are what you see in the comments here, more and more answers just handwaving away the infinitesimals or using circular reasoning (such as the argument with that real numbers theorem). You guys dug straight to the real issue and gave a proper answer for it!
I don't agree with 11:20 that you have to give up algebra, but I finally know the consequences of saying we can't throw out infinitesimals and where I'd need to start reading up if I want to keep them. (That would be on decimal expansion, the derivation of algebra, and most importantly, the historical work of other mathematicians on attempting to keep them.)
Again, I can't thank you enough, this is by far the best explanation, and it has annoyed me for decades since I first learned it as a kid. I can now confidently and happily accept and assume that 0.999... = 1 and work from there.
I like to go the analysis route and say if two numbers are different, then their average exists, and watch students' brains implode.
+andrewxc1335 Two numbers which are the same also have an average...
Okay: their average is distinct from them.
That assumes both are real numbers, though
Sure. And they are real. Numbers that are simply a decimal followed by standard digits are absolutely real.
+Joshu. My bad, I usually clearly state that a and b are real numbers. However, I'm pretty sure that it holds with hyperreals and the surreals. So as usual, your point is that you have no point.
If I had had a math teacher half as good as either of these gentlemen, I wouldn't be struggling in my career right now. I always saw math as this strict & rigid rule book and have never had any interest in such things so I was never good at math. But now that I have seen how much room for Parker Squares and things there is to it, I've been able to start to get into it and understand things. The only issue is now my brain is all old and dusty and it's much harder to learn things than it would've been if someone had gotten the message to me at a young age.
I did love this, I noticed that as soon as I heard Jordan's voice I thought I was listening to a podcast :p
Infinitesimals are useful in certain situations, like describing a cone that is basically another cone without the sides actually being the same, which I can immediately state is used in explaining synchronous events in the universe when separated by immense distance.
The way I think about it is that each value is a "unique size" shared by no other individual value.
For instance look at the equation 3+5=8
"3", "5", and "8" are different values, HOWEVER "3+5" has the same "effect" as the "value" of "8" alone.
Following this logic, I would say that it's possible that "0.999..." and "1" could be different "values" with the same "effect". The inclusion of an infinite series in "0.999...", for me, is an indication that it is possibly multiple things representing an "effect" that is equal to the value of "1" alone.
It seems to be an absurdity that arises from the clash of ideas VS practicality.
+Pseudo Lain They are the same value, different notation.
+Pseudo Lain yea, you are right. But a more scientific way to name your concept is mentioned by BoredDan
*****
Ah, I see now. Thanks
i cannot fathom why people are so quick to say that 0.99999999... is not equal to one. any number 1 through 9 is itself repeating infinitely after the decimal. my 8the grade math teacher was especially dumb when saying why she thought it was false. she said that she doesn't think 99 cents is equal to a dollar. it's blatantly obvious that 0.9999999... is not the same as 0.99 and this made me laugh so much in 8the grade.
.9999 Approximates one ... but does not equal one ... it is as close to one as it can be without being 1 ... you can put an orange coloured apple beside an orange and its not an orange ... but it still approximates an orange ....
.
think about that ... if you approximate something as close as you can then what you are approximating has every property of the value except one ... the actual value ... and if you dont have the actual value then it cannot be the same and if it is not the same then it cannot be equal ...
therefore ... 1 != 0.99999 but 0.999999 Approximates 1 for all purposes BUT equality
@@0623kaboom an approximation implies that our total it is only finitely close to a number. By definition, 0.9999 repeating approximates 1 to an infinite precision. This means it must equal one, as it is impossible to be infinitely precise in an approximation and not reach the number you are approximating
It gets tied up in the concept of infinity. If a decimal is two digits long, or three, or 5337478426 digits long there's an end, there's a gap. But infinity is not a number, it cannot be handled in the same way. However, infinity is found around numbers, is sometimes casually used as a stand-in for "arbitrarily large number" (same with infinitesimal and "arbitrarily small number") so people get confused and try to treat it as just another number.
It's all so clear, now! Pi is exactly 3!
+Daniele Bonadeo
Let me guess, a knock-knock joke is more your speed.
+Daniele Bonadeo knock knock
slowest knock knock joke ever, the person left :p
You mean that pi = 4, surely?
I know which are the next 2 books I will buy.
I think the main reason people are confused is that they are doing the (0.33333... * 3) operation wrong. If I were to simply ask you to multiply (0.33333.. * 4) instead you'd see the mistake. When you do multiplication, you have to start from the end, otherwise you'd have to keep going back to add the carry. And the thing 0.33333... doesn't end. So no matter how far down you start, you'd eventually have to convert the 0....33333.... part to 0.....1/3, and multiplying it by 3 yields a 1 to carry back to the next higher decimal place, and so on all the way back to the units place, so (0.33333... * 3) = 1.
"I think the main reason people are confused is that they are doing the (0.33333... * 3) operation wrong. [...] When you do multiplication, you have to start from the end, otherwise you'd have to keep going back to add the carry."
I think you're confusing _efficiency_ for _correctness_.
Can I not still compute 0.333 * 4 (finitely many 3's here) by starting in the first decimal position and then carrying? Will I not still get the correct answer? Sure, it's less _efficient_ to do it that way, but it is still correct and will always produce the correct result.
chinareds54 You should know that 1/3*4=(1*4)/3. Therefore, 1/3*4=4/3.
(Keep 4/3 rational. Don't convert because this might make this argument/conversation super long...)
@@Mars8765 But then you have to assume that 0.333... to infinite precision is equal to 1/3 to its own infinite precision. That then begs the question of, are those two infinite precisions equal? Because real numbers do have different sized infinities, say the set of real numbers vs the set of natural numbers.
I can't see any reason to mystify 0.999... It's more like: "Do we accept infinity as a reality?" 1= 0,999... simply states that the sum of an infinite series has the value 1. Just two ways to express the same numerical value. To me the identity 1.000... = 0.999... shows no more inconsistency than 5 = 2 + 3.
+Sigurjon Myrdal
"Do we accept infinity as a reality".
My argument is...no, as reality involves time and time by it's very nature is finite. Dealing with ideas/concepts of infinities is, of course, useful, but that doesn't mean that the concept can be called a quantity.
+LucisFerre1 Well, it's all about infinite series. There are two options. Either we accept it as a valid entity or not. If not, this is it, and we have nothing to discuss. Case closed! On the other hand, if we accept infinite series as an idea worth looking into, let's do so. Firstly, 0.999... means the sum of the infinite series 9/10 + 9/100 + 9/1000 + ... Secondly, you mention that you are are not satisfied with the idea of calling such a sum a quantity. To address this, let's compare this sum with the infinite series 1/2 + 1/4 + 1/8 + ... To visualise that series and its sum, draw a square with the side length 1 unit length. That square then has an area of 1 unit. Divide the square in half and proceed to divide the other half into two equal parts and so on. That is one way to visualise the infinite series 1/2 + 1/4 + 1/8 + ... and its sum (or limit if you like). Maybe this is somewhat more likeable than the algebraic procedure to show that 1/2 + 1/4 + 1/8 + ... = 1. You simply start with the sum (limit) value (a quantity!) and use it to build the infinite series. Our first series 9/10 + 9/100 + 9/1000 + ... is completely comparable.
+Laurelindo Yes, it's natural to have issues with 1 = 0.999... at first sight at least. What I have to say about this is covered in my answer to LucisFerre1 above.
Sigurjon Myrdal
I would say that inifitesimals approximate reality. They are very useful, but that doesn't mean they represent reality.
+LucisFerre1 It is very likely that infinitesimals do not represent any physical reality. But there is more to reality than physical reality, don't you agree? We shouldn't let the physical world limit our way of doing thought experiments. To doubt this "0.999... = 1", is equivalent to throwing away the idea of infinite series. If there are no infinite series, there are no sums of infinite series :)
"Math is a journey", seems to tell to us Kerouac's book in the background.
The thing that irked me most when I was learning calculus was that constant of integration that ate up any other constants that happened onto its path. I was like, "You're disappearing numbers!"
If A != B, then there exist a number C, A
Are you replying to the right post?
You can't ever choose C = (B+A)/2 with infinite length numbers, if i asked you the exact average of 3 and Pi you would have to give me some number that's not correct but close with rounding and in the exact same manner if you ask for the average between 1 and 0.999... id say 1 after rounding it up.
But if C is just a value between A and B id say its 1 minus an infinitesimal. Just like you're trying to use the concept of infinity existing like the length of 0.999... there also must be something being infinitely small and they mention infinitesimals and limits in the video, 1 is just the limit 0.99... is converging towards, and the difference of that limit is an infinitesimal.
1- 0.9 = 0.1
1- 0.99 = 0.01
1 - 0.99...9 = 0.00...1
Or another example using infinity is Gabriel's horn. The side you blow into MUST be infinitesimaly small otherwise infinity doesnt work and neither does 0.99... as a concept.
*plugs ears with fingers* lalalalalalala I can't hear you lalalalalala physics ...... planck constants lalalalalalala
damn you mathematicians and your algebra!! :P
Why did you watch the video then?
+BoomerBoxerReal If he/she knows what planck constants are then I am sure he/she was joking.
+Ziquafty Nny maybe he's a physicist?
+Madhur Agrawal physics is applied 99% maths
JoQeZzZ they always pretend to have that enmity towards each other
I remember mentally struggling with this in elementary and middle school ever since I first learned how to use decimals and long division.
Jordan, where were you when I was learning math at school during childhood? Now 0.9999... makes perfect sense to me.
I am starting to understand why math doesn't always get along with base 10.
It isn't better with other bases
Base has nothing to do with this
base has a lot to do with this, of done in base 12 this problem isn't a problem (though others arise)
in base 12
1/3=0.4 ∴ 3/3=3*0.4=1
In base 12 (using B for the eleventh digit) 1/3 = 0.4 = 0.3BBB... and 3 * 0.3BBB = 0.BBB... = 1
Damned if I can see how that helped. How about 1 = 1.000... or 0.25 = 0.24999...?
There's nowt so queer as folk.
Any base number representation system will generate repeating sequences in some circumstances in fact 12/11 is a good example that does it in both base 10 and base 12.
Base 10: 12/11 = 1.0909....
Base 12: 10/B = 1.1111...
There's 100/3 dislikes. If we get triple the views, will there be 100 dislikes?
Perhaps. If you read Ellenberg's book, he explains in depth why thinking linearly is oft incorrect. (Not all curves are lines) You are thinking linearly.
After 5 years here is still only 95 dislikes.
Matt, I saw you on this TV show where you compared an A380 to a fighter jet. What a parker show
The idea that math is not that SOLID thing with solid unchangeable thing was really a small revelation for me! Making choices and then dealing with the consequences... That guy really changed my view, wow.
The whole point of an infinitesimal is that we can't tell the difference between it and 0.
The whole point of an infinitesimal is that we can... but it doesn't always help. BTW, in the surreal/hyperreal/superreal numbers, 0.9999... still equals to 1.
@@dlevi67 I came across a paper which seems to disagree. It suggests a theory of hyperreals and natural numbers, consistent with Peano Arithmetic, in which there are non-standard natural numbers identified with infinite hyperreals, so their reciprocals are infinitesimals. Now consider .999... to be the infinite sum over all naturals n > 0 of 9/10^n. Then the sum splits into a sum over standard naturals and non-standard ones. The standard sum equals 1 as usual, and the non-standard part is an infinitesimal. The combined value of .999... becomes the sum 1 + D, where D is a positive infinitesimal, so .999... is a little larger than 1.
@@tommyrjensen It would be interesting to read it - I can't see anyone abandoning the "most standard" version of non-standard analysis where however convergent series behave like in standard analysis for this, but the counterintuitive result would be fun to understand!
@@dlevi67 I posted the link in another answer: arxiv.org/pdf/1007.3018.pdf
I am not sure that it is interesting to read. The authors seem confused and unwilling to present proofs. Actually they try to argue the opposite point: with the change of model, the limit ends up less than 1 by an infinitesimal amount.
@@tommyrjensen Thank you for reposting the link.
I am not an expert in non-standard analysis, but I think the authors are writing with a completely different intent than that of rigorous demonstration; it's a paper on teaching. FWIW, although I agree that their "Answer 3.3" (and several others!) is phrased confusingly, I don't see any claim that 0.999... could be greater than 1, even in the hyperreals (or the surreals) - I do see the repeated (but not demonstrated, though they do give the sources/references) claim that:
"Question 6.12. Why didn’t Lighstone write down the strict inequality?
Answer. Lightstone could have made the point that all but one extended expansions
starting with 999 . . . give a hyperreal value *strictly less* than 1. Instead, he explicitly reproduces only the expansion equal to 1. In addition, he explicitly mentions an additional expansion-and explains why it does not exist!" (my emphasis)
I may be completely wrong - as I said, I'm not (at all!) an expert in non-standard analysis, but Katz & Katz's main "beef" (if you pardon my adding to the zoo) seems to be the possible ambiguity in 0.999... being interpreted as an infinite terminating decimal in hyperreal notations, where it corresponds to a whole class of numbers - whereas no similar ambiguity exists when using the language/notation of standard analysis and "Cauchy/Dedekind" Reals. Perhaps inappropriately (and incomprehensibly to those that don't know what Telemark skiing is), but what comes to mind is the old canard: "Free the heel, free the mind..." - to which the reply is: "Fix the heel, fix the problem!"
(apologies for the edit - nothing changed other than this note and the paragraph spacing!)
so the entire point of this video is to say that "Mathematicians play around with axioms". Right?
+Geek37, economic systems do fail completely.
They fail somewhat, not completely.
Not with axioms, but with definitions. That is different and legitimate
4:07 Matt changes '1' to 'not(1)'. Equation corrected!
6: 00 wow, using deconstruction to discuss math. I didn't think that those two things went together
Algebra isn't the problem, it's division in base ten that's the problem. 1/3 isn't a problem in base 12, but 1/5 suddenly is... which may explain why the Bablyonian math geeks (or maybe even Sumarian math geeks?) went to base 60 and the clock is divided into 12 hours of 5 minutes each, because even back then, enough people wouldn't compitulate with the idea of 0.9999999... = 1. We get 360 degs in a circle, which is just 60 times the next whole number, 6 and is why doing geometry in radians is so much cooler than doing it in Cartesian, even with the PI/TAO mistake.
So did the Babylonians not know about prime numbers or did they use a mystical correlation bias and stop at 6 because 360 matched fairly closly to the number of days in a year and an underlying truism was falsely thought to be understood about the nature of the universe? Obviously, this was set in mayhematical stone before the Bablyonians conqured Israel or else they would have known that God created the world in 7 days (the "rested" day counts!), and then have multiplied 60 by 7, which would have given us a proper 420 degree circle, making things like septagons not impossible to clasically draw. Game of Thrones aside, there were enough septagons and 7 pointed stars floating around in the ancient world to make one believe that there were secret sects of mathematical anarchists running around who were using something akin to a 420 degree circle. And since 7 (days) x 4 (weeks) x 13 (months) = 364 days, much closer to the actual days of the year, and some cultures did do this, but other cultures didn't because they were afraid of the number 13 long before the Knights Templar were killed on a Friday the 13th.
We could have drawn a nice star pattern within the 420 deg circle to represent a precession of years. This could have all worked out nicely.
But sadly, this possibly proposed bit of mathematical innovation never caught on :(
(or maybe a 420 degree circle is the secret knowdege that makes the Iluminatti different from the rest of us?)
It does make one wonder, though if any infinite decimal can be represented finitely given the appropriate counting base?
Okay, the above was the result of a two hour stream of consciousness poundering on the 0.999... = 1 controversy and is a good example of why these sorts of oddities should never be locked down one way or the other. They make you think and that thinking could lead to completely different ideas, some trash, some just for fun, but some really interesting. The trick is not to stop at a trash idea, but to put the pencil down, get some sleep, and pick it up again and pick it up again at the next appropriate time.
It's how we write numbers that's the problem, and the cause of all this issue. I know some math nuts calculating pi to a million decimal places, like it matters, and caring when someone in a song got it wrong. But rounding 0.9999999999999999999999999999999999(etc.) to 1 (which is essentially what they're doing) is fine. I think a better explanation is: as long as you're writing numbers down, and displaying numbers in a way we can read, you're going to get a small amount of fuzziness. Get over it.
It's generally people interested in theoretical maths that do that sort of thing, and sometimes theoretical math leads to useful concepts. If I remember correctly, that's what happened with integrals, which are immensely useful. Also, by the various proofs being used, 0.999... isn't rounded to 1, it's exactly equal to 1.
Oh yeah, I get that's what they're saying, don't get me wrong. I probably didn't explain what I getting at very well. What I'm saying is the fuzziness that's inherent in writing down numbers as digits in some sort of number system like decimal or base-ten, causes 0.99999999(etc.) to equal 1. These are really quite metaphysical concepts that we're trying to make somehow concrete, so there's going to be some wiggle room.
Again, I don't know if I explained my brain properly.
Talking of bases, I'm intrigued by the possibility that the bigger the base n, the closer 0.(n-1) is to 1. So for example 0.F in base 16 (where F is the biggest single digit) might be proportionately closer to 1 than 0.9 in base 10. Would that make 0.FFF...(16) also closer to 1 than 0.999...(10) is? Or are they simply both exactly equal to 1, no gap at all so no bigger or smaller gap? If there is such a gap, then maybe there's enough room for another number, provided it's expressed in a bigger number base.
If 0.9 repeating did not equal 1, not just algebra but calculus would break as well, which would be unfortunate.
+Brady Pomerleau Unless we jump to hyperreals!
but it doesnt Equal 1 ... it APPROXIMATES 1 ... it has all the features of being 1 without being 1 ....
.
1 - 0.99999 ... does not equal 0 but some very small number .... 2 x 0.9999 does not equal 2 it is equal to 1.88888888 ...
.
0.99999 is only equal to 1 when the difference between 1 and 0.999999 is so small you can truly disregard it ....
.
if you take processed cheese melt it in a high hydrogen atmosphere ... you get plastic ... not cheese ... yet it is still called cheese without being cheese .. it approximates cheese in every respect without being cheese
@@0623kaboom did you really Just say 0.9999 is equal to 1.8888? Please trink about it again, 99*2=198, 9999*2=19998, and the pattern continues to infitely many 9's, so its equal to 2
There are no infinitely close numbers, if there is no number between two numbers, they are the same
Goddamn. I love you guys.
Sometimes you can't use infinity without using infinitesimals though... for example if you wanted to make a regular infinite sided shape to create a circle for instance, you need to accept the lengths of the sides are infinitesimal. Otherwise if you say all infinitesimals are equal to 0 then that would make the circumference of the circle equal to 0 which would bypass the c=pi*d rule.
I might be an ignorant pleb but why does 0.999... have to be anything other than 0.9 recurring? Why can't 1=1 and 0.9 recurring = 0.9 recurring?
+HOLyPumpgun | Gaming Because if algebra is correct (that is, if we follow the rules of algebra that we created and have used reliably via the axioms of set theory), then it's inevitable that those two are the same number: that is, '1' has two different decimal expansions (1.0000... and 0.999999...). Basically, the maths that we have inevitably concludes that these two expressions have the same value
+HOLyPumpgun | Gaming A basic rule of number theory is that if 2 numbers are not the same, then there always exists another number inbetween them. So if 0.9999......and 1 are 2 different numbers, then there has to exist a third number inbetween 0.99999...... and 1. So, what would that number be? Unfortunatly no such number exists, and since there is no number inbetween 0.99999...... and 1 they must be the same number.
+HOLyPumpgun | Gaming That's sort of like asking why 1/2 can't just equal 1/2 and 0.5 can't just equal 0.5. Sure, these are both true statements, but the rules of mathematics show us that these two very different expressions actually represent the exact same value, just like 1 and 0.9 recurring.
***** I think the hard bit to wrap one's head around is that 1/2 has a single decimal representation (0.5), while 1 has two different decimal expansions (1.000... and 0.999...), and since we expect different decimal expansions for different numbers, we conclude that they must be different numbers because they have different expansions.
We'd be _wrong_, of course, but it's easy to see why converting a fraction into a decimal may not be the most satisfying counter-explanation.
Natasha Taylor Yes, I was just hoping to give him an idea of how two different expressions can be equivalent. Although 1/2 actually _does_ have two decimal expanions: 0.5000... and 0.4999...
So 2x 0.99999.... = 2 ?
Yes.
Yes,exactly. since there wont be an 8 at the end it is 1.999999.... and that is simply 1+ 0.99999, since that is 1+1 it is indeed 2. This is easier to see when you take 10 times 0.999999. You simply shift the dot to the right and get. 9.99999.... which is by definition 10.
See i love math and other sciences. One of the things that really anoys me is the disconnection from reality sometimes. We all know that there is an 8 in the end. But since the numerical rules that we have invented doesnt show it its disregarded. Its like that kid that plays D&D and uses the Warchain rules that allows for free attacks everytime you get close. It doesnt work that way in real life...but the rules say it does.
+Tharos Gröning First of all I'm writing this on mobile so please disregard any spelling mistakes. We have taken a path away from reality when we invented the negative numbers. What does -1 Apple look like? the thing is it is still useful. We need these numbers. They are essential for mathatics and even well our real life. I would personally say that the biggest part of maths finds no use in our everyday life, I'd go even further and say it has no use in reality whatsoever. Infinity for example. Nothing is infinite. ( Well space os probably but that is more like edgeless). This is the difference between Platonism and (forgot the other name) the believe that we made up Mathematics.
we all know there is an 8 in the end?? i sure dont... and how in the world do you know theres an 8 instead of a 4 or a 3 or a 7 or anything other than 9, when you were just told its always just 9?
0.999... those dots in the end mean, theres always just 9s. nothing else. where is your feeling of an 8 coming from? what is 1/9? are you happy with it being 0.111...? or do you also feel there is a 2 in the end somewhere?
0.111... is not aaaaaaaalmost 1/9. it is exactly 1/9. it's just written down differently. 1/9 times 9 is not aaaaaaaaaaaaaalmost 1, it is exactly 1.
Wait a second... I recognize this guy! He spoke at my school!! What a pleasant surprise!
It's always fun when this is brought up. So much angst and argument about something that should be pretty clear. 1 = 0.9... due to the nature of our representation of real numbers, and it does not depend on the basis. We would also, obviously, have things like 0.5 = 0.4999... and in base 2, 1 = 0.111... Those three little dots change the game because they turn a short expression that looks like a normal number into an infinite series.
I know this is an old post, but the base does matter in some ways actually.
I disagree with Jordan very strongly at 6:28. We have already signed meaning to numbers and decimals. So based off of that, there should already be a pre existing meaning in .99999.. even if no one has discovered it, only because we have already assigned meaning to all of those symbols.
Like we have assigned a meaning to the number 3, and 5. Therefore we cannot make up a meaning for 3 + 5. Everything has to be consistent.
Visionary Universe Visionary Universe Well, can't we make up what "+" means? Couldn't we, in theory, let that be any binary function? Say plus(x,y) is the function that is just the normal definition of addition (I think Wikipedia goes into detail about how it's defined; it's not important). Say we then define the new function newPlus(x,y) to be equal to plus(x,y), except when the inputs are x=3 and y=5, in which the output is 42. plus and newPlus are both perfectly valid mathematical functions. plus(x,y) is much more useful than newPlus(x,y) (e.g., plus(x,y) has nice properties like being commutative, associative, and not being made up on the spot to make a point), but the key is that there is nothing that makes plus (x,y) more "correct" than newPlus(x,y); both functions exist. If by "x+y" we mean newPlus(x,y), then 3+5=42. This is an incredibly stupid and useless definition, which is why "x+y" is defined the same as "plus(x,y)," but both conventions are equally mathematically valid.
The same thing is happening with 0.9999.... I alluded above to the strict standard definition of addition (plus(x,y), that is; not newPlus!). The definition is only for two inputs, so "3+4+5" technically doesn't meaning anything if all you have is the bare-bones definition of addition. Since addition is associative, we can extend our notation so that by "x+y+z" we mean "x+(y+z)," the latter of which only uses binary addition, so the bare bones definition can take over from there. This works for any finite number of additions; "1+2+3+...+99+100" is defined to be shorthand for a whole series of binary operations, which again is what addition is at the lowest level. We run into a problem at infinity, however; what do we mean by, say, "0.9 + 0.09 + 0.009 + 0.0009 + ..."? This cannot be broken down into a bunch of binary operations in first order logic. So, in order to make the above make mathematical sense, we extend the definition of addition to include a provision for infinitely many addends. Some definitional extensions are more sensible and useful than others, but they are all mathematically valid. The most obvious way to give meaning to an infinite sum is through limits; this definition assigns "0.9 + 0.09 + 0.009 + 0.0009 + ..." a value of 1, and 0.9999... is defined to be an abbreviated form of the above infinite sum.
You made some good point. But I still hold to the idea that 0.999.. is based of of concepts and operations that have already been predefined to match logic so there is already an existing meaning in it without having to define something new.
Visionary Universe you created 0.99.. in the first place, but it. could have multiple meanings or even contradictory ones.
is it a natural? a rational? a real? does it satisfy all the properties of the fields? is it some different number like √-1 that requires a new definition like (√-1)²=-1, with √-1 € somewhere?
those aren't simple questions. Even though Cantor answered them with his theorems and today everybody accepts that 0.999..=1 , but whatever
Maybe it's just me but I haven't found any contradictory or absurdly new meanings in it.
I think the missing thing is the axiom of infinity and the definition that says that an infinite sum is equal to the limit of the infinite sum.
Algebra does not rely on 0.99... being equal to 1, and I'm a little miffed that it's the conclusion of the video. The weak link in the 10T argument is not the algebra itself, but the part where you go "10 * 0.99.. = 9.99..". That operation singlehandedly takes the infinitesimal out of 0.99.., effectively skipping the controversial part before algebra is applied.
+Triggerfisk I don't believe they said algebra relies on 0.999.. being equal to 1. It's quite the opposite, their point is that whether 0.999... = 1 or not depends on whether you choose to follow the rules of algebra.
What is 10 * 0.999...? Well if you follow the rules of algebra, you must conclude that it equals 9 + 0.999... because that's the only way algebra can deal with numbers. If you start going outside of standard algebra (the part where they talked about nonstandard analysis) then you can make it equal something different with infinitesimals, but it doesn't work in algebra.
+Triggerfisk How do they take the infinitesimal out? It is still 9 + original value.
How about we make this a geometric series with first term a = 9/10 and common ratio r = 1/10? The infinite sum is undoubtedly 0.9999... just by looking at it (0.9 + 0.09 + 0.009 + ...), and yet the usual formula of a/(1 - r) yields a value of 1. No shady or sketchy moves with this method as far as I can tell (I'm not a maths major sooo).
+Sullivan Muse
It's the same thing as saying infinty - 1 = infinity and thereby stating that 1 = 0.
+Jerry Nilsson Not at all, as anyone working with set theory would preach to you about.
Wow, this guy is great! Can you bring him on more? Or drag Brady out to feature him on Numberphile? He's excellent!
This guy was my professor in college!
What is the counter to the idea that you can't multiple 0.9999... by ten because moving the decimal also includes adding a zero to the end. which your not doing with a repeating number.
isgdre Multiplying by 10 doesn't necessarily need to add a zero to the end. Moving the decimal place is fine.
isgdre Well they are, but before the 0, there is an infinite amount of 9's, so in a way, it cancels out the 0.
Jonas It in no way cancels it out. Are you implying that just because it's too hard (impossible) we should just not bother.
DonkeyCore No. it always requires a zero to be added. We just don't bother to write all the ending zeros after the decimal.
isgdre There is no such thing as ''the end'' of 9's. There are infinitely many.
so 0.24999..... rounded to the nearest tenth is 0.3. got it. smh
Well that explains a lot
( damn the shit you've to deal with for a single comment~ )
no, it's 0.25....stop shaking your head, and read a maths textbook
Michele Zappano ROUNDED TO THE NEAREST TENTH. Did you even read my comment?
does anyone here know what rounding means??
Okay, here we go. We round according to the convention that if *everything* to the right of the digit we're rounding to (in this case tenths) is *less* than 0.5 x 10^-x (in this case x is 1, because we're rounding to the tenths digit), the rounded digit remains the same. If it's equal to or greater than 0.5 x 10^-x, the digit rounds up by one. In the case of 0.24999..., we're looking at 0.04999... As the video explains (using the case of 0.999...), 0.04999... is equal to 0.05. You don't have to like that system, but it's the one that's been agreed upon because it maintains the functionality of algebra for infinite sums, which is what 0.24999... or 0.04999... really is. Thus, because 0.04999... is in fact equal to 0.05 and is NOT less than 0.05, 0.24999... rounded to the tenths digit is 0.3.
So even though we get used to just looking at the single digit to the right of the digit we want to round to, when we have a number like 0.24999... we have to be a little more rigorous.
This is the exact equation I came up with long time ago :D
I like Conway’s numbers in this context as an alternative.
I like to just use this as an example for what infinity does to math, sort of like how we can never understand what it is
2:42 i never thought of a repeating decimal as an infinite sum but that's totally what they are we are writing a little program that's a loop that can be expalnded infinitely
@SubZee not symbolically
5:53 On the 1= .999... thing, Jordan Ellenberg says it all right there. Besides paradoxes which conflict with ordinary notions, there are contradictions which can, and do, occur when using an infinite process, because you can sometimes arrange two different processes involving the same numbers to get two different values, if the process never ends. That can happen because the final reckoning can be forever postponed, whereas with a finite process the final reckoning never is.
The idea of a limit helps because it at least confines what processes will be accepted. The idea of a limit is more trustworthy because it never counts on knowing what infinity means. (And infinity means different things at different times.) But the idea of a limit does not entirely take care of all problems with with two different infinitely long procedures, because you sometimes can still alternatively arrange the reckoning to approach different values.
A computer programmer would never presume that two programs involving the same numbers, but with rearranged instructions, would always produce the same results.
+Kenneth Florek The sequence 0.9+0.09+0.009+0.0009.... is a convergent series though. What you're thinking of is a divergent series which brings forth a lot of other fun mathematics ;-)
I don't understand the computer programmer analogy in this case I mean a program of all "0's" is going to have only one result nothing, and a program that continually loops on "1's" is never going to end. 0.999... doesn't have different values in arranged in different ways it has one value that being equivalent to 1. This is one value, perhaps with different ways of expressing it but still the same value. No calculator is going to get different results for 1+0 and 0+1, yet these are arranged differently. Neither should mathematics get different results for 0.999... or 1.
I think convergent sums are fine to stick with, but for divergent, stick to Ramanujan sums, because they make the most sense, even though it sounds weird to add infinitely many positive numbers to get a negative.
The contradiction stems from the decimal (any base is problematic) expansions not exactly representing fractions (or radicals for that matter).
There is no contradiction here. And decimals do perfectly represent fractions. We just can't visually depict that representation. What we can write does not define how math operates.
Hey Matt, I really like your videos but i have two questions;
Why do you always put a decimal-point so that it looks as a multiplication dot?
And why do you make your x-symbols in such a strange fashion? (I've seen others do this, it looks more like )( than x )
Not sure why this was suggested to me today but about a month ago I posted this riddle to someone and it got me pondering and I think I know the problem. I remember when I was younger that if you entered in a problem on a calculator with an infinitely repeating answer it would always make the last digit 1 larger, for example if you entered 1/3 it would say 0.33333334e and I always wondered why it ended with a 4.
Of course now I realize that at the end of the infinite 3's there would have to be an infinitesimally small 4 at the end, otherwise it wouldn't add back up to 1, but we're so used to being told '3 repeating' that we never think about the end of that repeating 3, so it lets us slip in riddles such as this.
Your calculator should have rounded to the nearest digit. I should have given a 3 not a 4. There is no 4 at the end of 0.333 . . . because there is no end. In the current context, "infinite" means "endless". Also, there are no infinitesimally small decimal places. The nth decimal place is a multiple of 1/10^n. There is no natural number that makes that be an infinitesimal. Although there are infinitely many natural numbers, every one of them is finite (ie. there is always a next one).
@@Chris_5318 After the infinite 3's there would be a 4, otherwise they'd never add back up to 1. It would always be less.
@@TheNdoki Wakey, wakey, *there is no "after the infinite 3s".* That would imply that the 3s come to an end. In which case there aren't infinitely (AKA endlessly) many 3s. in the current context, infinite means without end. It gets much more sophisticated than that. The infinity here is the simplest one of all of them.
Now try this: 3 * 0.3334 = 1.0002
So if you were right, you'd have something like 3 * 0.333 . . . 4 = 1.000 . . . 2 and that is greater than 1.
BTW 3 * 0.3333 = 0.9999 = 1 - 0.0001 and is closer to 1 than 1.0002 is. So your 4 should be a 3. Better still, it should be 33. Better still it should be 333, . . . Better still, it should be 333 . . . where the 3s do not come to an end.
Also try this:
10 * 0.333 . . . = 3.333 . . .
So 9 * 0.333 . . . + 0.333 . . . = 3 + 0.333 . . .
So 9 * 0.333 . . .= 3
So 0.333 . . . = 3/3 = 1
This is eighth grade math that you are flunking.
It is only possible for the question 'Is 1.00... = 0.99..?' (i.e. infinitely precise real numbers) to be answered if the definitions of =, and != (i.e. equals, less than, more than, not equals) are defined first.
The = is asserted, but the definitions of and != are undefined. Therefore the question is undefined also.
For precise real numbers there is no problems, just the infinitely precise ones.
Where can i find the piece of music that plays at the end? i cant find anything on Howard Carter???
I was hoping Matt had said "and my book...let me just pull it out of the fourth dimension...here it is!"
yayyyyy Jordan Ellenberg!
0:41 "Matt-Meets-Jor-Dan-Ellenberg!"
Thank you for this video :D
Finally no video that's rubbing it in: '9.999... = 10!' or the opposite, but a video that explains both sides. I like infinitesimals!
I'm fine with dropping algebra for infinite sums :P
9.999... is _NOT_ equal to 10! ;)
EDIT: 9.999... is equal to 10 though ;)
***** Fixed. Would've thought people would get the message but apparently not. Oh well :)
haha, I get it.
9.999... ≠ 3,628,800
10:14 for Gazelle shadow puppet
In more layman's terms, this is how I would prove 0.999... = 1
Consider the fact that 1/9 is equal to 0.111...
2/9 = 0.222...
3/9 = 0.333...
...
7/9 = 0.777...
8/9 = 0.888...
9/9 = 1
After 8/9 = 0.888... shouldn't we have 9/9 = 0.999...?
Anyway, I thought up a similar proof:
1/1111 = 0.000900090009...
1/111 = 0.009009009...
1/11 = 0. 090909...
1/1 = 0.999... = 1
which I think's pretty neat. Seen it before?
The music sounds like very reminiscent of the Super Hexagon game music. At least to me it does.
What is actually a convergent series it is a dieing wave. It can have any kind of frequency and shape. Mostly waves have to sustain frequency rather shape. That is the reason why we don't have a true number.
You can do a phase shift to add for converging series.
They should do something similar to pi versus tau on numberphile with points and stuff
Is that Danny Baranowsky's music at the end? :O
non-standard analysis is a completely fine way of doing things.
The problem lies in the fact that there is no unique way to right a number. There's no reason to argue that 4/2=2 we know that for sure. But thats also true for decimal representations of numbers as well.
I *love* "How Not to be Wrong"!!
They only get to the crux of the matter at about 7:20 - the decimal expansion is only a representation of the underlying number, and decimal representations are not always unique (one to one). For example, pi is represented by the symbol for pi uniquely and exactly, but can also be represented by an infinite, non-repeating string of decimal digits, or in binary, etc. There is only one pi, but different representations. In decimal, 1.0 and the infinite series of digits .999... represent the same number.
I was blown away by the proof they showed here when I was in 7th grade, and it always bugged me until I took real analysis in college and saw diagonalization arguments, etc with the realization that the decimal number is only a representation of the underlying idea of the number itself.
The Calculus proof is simply that 0.9 repeating is a geometric series which looks like 0.9 + 0.09 + 0.009 + ... with 'a' = 9/10 and 'r' = 1/10. 'a' is the starting term and 'r' is the number multiplied to get the next number in the sequence.
(ex 0.9 * 0.1 = 0.09, 0.09 * 0.1 = 0.009 and so on)
The sum of a geometric series is equal to a/(1-r) so thats (9/10)/(1-(1/10)) or (9/10)/(9/10) which is 1.
This is also how you get 1/2 + 1/4 + 1/8 + 1/16 + ... is equal to 1 as 'a' = 1/2 and 'r' = 1/2. (1/2)/(1-(1/2)) = 1.
There are infinity many ways to get 1 as the answer to a geometric series sum as long as 'a' and 'r' add to 1.
Now that is simple, elegant, and make much more sense than almost anything anyone else has said in this entire comment section. I mean this tackles the issue at the true heart of the "problem" and doesn't do anything questionable.
The building permit branch in honolulu requires that the space between stair railings be less than 4" and NOT 4" max. Because, you know, a kid may get his head stuck in railings set with 4" spaces, but won't get stuck in railing spaced at 3.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
That's because your number is less than 3.999 . . . (= 4). They should make it illegal to have a head less that 4" in diameter.
The real numbers between 0 and 1 are abstractions for the probabilities of events in the universe. Finite values result from observation.
Is it just me that wants to see a book about the Shallow and Complicated quadrant?
1:57 where should we put Banach-tarski?
I like to think that the universe's compiler tried to yell at the big programmer in the sky about possible precision loss, and that infinitesimal precision loss is what leads to the uncertainty principle of quantum mechanics.
Totally agree .99... is equivalent to 1 but not equal. There is that pesky infitesimal which is also trivial.
The circular "proof " does not break algebra, any whole, or whatever, number such as x=anything. 10x=10anything. Subtracting. 9x=9anything. Divide by 9 x=anything. Then e=1, pi=1, etc...
There is no infinitesimal involved. NB Jordan Ellenberg did hint at that possibility, though.
The proof was not circular. We started with x = 0.999... and ended with = 1 and so by your logic we conclude that 0.999... = 1.
If 0.999... and 1 differed by an infinitesimal, they would be neither equivalent nor equal.
The sequence of partial sums of 0.999... and 1 are in the same Cauchy equivalence class, hence 0.999... = 1.
Karen, what is this twaddle you wrote about " 10x=10anything. Subtracting. 9x=9anything. Divide by 9 x=anything. Then e=1, pi=1, etc"? In particular, how did yo conclude that e = 1 and Pi = 1? Elsewhere you used the same argument to "prove" (LOL) that 0.999... != 1. Your proof was as follows, except that you used "0.999..." instead of "2/5 + 3/5".
x = 2/5 + 3/5
10x = 10(2/5 + 3/5)
Subtracting the first from the second
9x = 9(2/5 + 3/5)
Divide by 9
x = 2/5 + 3/5
Therefore 2/5 + 3/5 != 1. Classic.
Base 12 and 3 are fine.
Base 12: 3(1/3=.4) you get 1
Base 3: 3(1/3=.1) you get 1
1 = 0.BBB... (base 12) and 1 = 0.222 . . . (base 3). Also 1/3 = 0.3BBB . . . (base 12) and 1/3 = 0.0222 . . . (base 3).
why can it not just be that the infinite series 9/10^n reaches a limit of 1? so for all common use they are the same
he has the first and last name as the south park creators
Casting shadow puppets while discussing calculus at 10:13
At 8:00
My thinking was as follows:
0.999... is a decimal expansion:
9/10+9/100+9/1000 = sum over natural n of 9/10^n
of course 0.000...q where q is the last digit is a thing that, in mathematics, we have nothing other than "infinitesimal" to describe... But since we are already working with infinite sums why don't we just use another infinite sum to describe it?
For instance:
1 - 1/2 - 1/4 - 1/8 - ...
Alternatively if you want to stick with 0.000...1 then:
1/10 - 9/100-9/1000-9/10000... = 1/10 - sum over natural n of 9/10^(n+1)
Ultimately you get something fairly interesting that:
1/10 - sum over natural n of 9/10^(n+1) + sum over natural n of 9/10^n = 1
or rather simplified:
sum n=1 to inf of {99/10^(n+1)} - 1/10 = 1
I must have made a few mistakes with my earlier calculation?
infinitesimals are good
Xeno's Paradox:
2 = 1 + 1/2 + 1/4 + 1/8...
If you do the calculations, you'll see that a process keeps repeating, as if a "machine" was moving forwards, leaving 9's behind it.
So, if we do it infinitelly, it would be equal to 1.999...
Therefore, 2 = 1.999...
2 - 1 = 1.999... - 1
1 = 0.999...
I just got into high-school, so, I probably messed up a ton of things. If anyone can correct me, that would be great. It's just I haven't ever seen anyone bring this point up.
Now this is deep...
Where did they film this? Vsauce made a video there about cruel bombs if I'm not mistaken.
Can you please answer this question if 0.9 recurring equals 1 that would mean 1 over infinity 0 and infinity over infinity 0 making the algibric rule of x/x=1 fake
Infinity, oo, is not a number. By convention, 1/oo means lim n->oo 1/n and that is 0.
I have no ide how you concluded that x/x = 1 would be a fake. BTW that is not true if x = 0, because 0/0 is undefined.
Look up "improper integrals" for another example of using such a convention.
This is always my argument: Let's assume .9999.... is different than 1. Two numbers can be defined to be different iff there exists another number that separates them. What value separates .9999... and 1?
It's .999.. to infinity, plus one extra 9 at the end. It's infinity + 1 trailing nines.