Indeterminate: the hidden power of 0 divided by 0
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You've all been indoctrinated into accepting that you cannot divide by zero. Find out about the beautiful mathematics that results when you do it anyway in calculus. Featuring some of the most notorious "forbidden" expressions like 0/0 and 1^∞ as well as Apple's Siri and Sir Isaac Newton.
In his book “Yearning for the impossible” one my favourite authors John Stillwell says “…mathematics is a story of close encounters with the impossible and all its great discoveries are close encounters with the impossible.” What we talk about in this video and quite a few other Mathologer videos are great examples of these sort of close encounters.
For those of you desperate to get hold of the t-shirt check out this link: shirt.woot.com/offers/how-natu...
Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Enjoy :)
Great video, I rate it 0/0. Full marks!
lol
nice
Good job.
The True Fizz So, do you hate it or like it?
Raffy Tabingo I'm afraid The True Fizz 's opinion on the video would be indeterminate. Unless maybe he gave us some more context on the functions that are approaching 0.
this man sort of comes across as a bond villain but is friendly enough so that I think he would be the assistant to the bond villain and would end up somehow disarming the nukes of the villain as a sort of double agent. these are the things I thought about in college. and I wonder why my degree didn't work out.
Definitely worthy of his own Mini Me
Shouldve taken writing. Everyone's useful somewhere.
Because of his accent? Hmmmmmm. Whos the villain?lol
> and I wonder why my degree didn't work out. Maybe your teachers were not this good?
Sounds like AD(H?)D. Not diagnosing anyone over KZhead, but that sounds classic GT/ADD daydreaming, where your brain takes you down all sort of imaginative rabbit holes without your knowledge or consent (in the moment), all of which objectively and infinitely more fascinating than anything happening in the classroom. That almost always corrollates with intelligence, even though tired/ego-driven teachers often get their panties in a bunch over it and can make the student feel like they are stupid. They are not. ❤️
Now I can sneak up on zero
Lol
Sneaking up on Zero already sounds like a book title
4:09 "but as long as it's staying off 0..." Nice! You'd be surprised how rare people explain this important piece of information when they explain derivatives
It's not explaining then.
Seriously they miss that it's so important lol
The real reason that you are advised to avoid indeterminate forms is that your must invoke L'Hôpital's Law -- which you will not be able to pronounce to to everybody's satisfaction.
Tim Harig I have a solution for this problem: Taylor series!
Or just learn French.
Nah, let's stay reasonable!
It's alright, you could just pronounce it "Johann Bernoulli." :) Aw yeah, hmu for more 320 year old math jokes.
Isn't it spelled with an s?
I asked Siri what 0 divided by 0 is, and it broke my heart. Siri why are you so cold!!
Have to admit that I was more disappointed that Siri did not have any heartbreaking answers for any of the other indeterminate forms :)
Mathologer Yes, I was disappointed as well. At least it consulted Wolfram Alpha!
Google Assistant said "it's undefined. What a mystery"
Siri is a woman..
@@johnraina4828 what's your point? lol
"Noone would know Isaac Newton. That would be really sad, right?" I bet Leibniz wouldn't agree.
Lmfao
well, actually archimedes discovered the basics of calculus before either of them. look it up, he wrote in a book called "the method"
Shut up toxic math student. Even Leibniz was a fan of Newton. One incident. Christian Huygens faced an unsolved problem in mathematics, he brought it to one of the greatest mathematician on the planet at the time, Leibniz. Leibniz tried very hard, multiple attempts, but couldn't solve it. He said to Huygens to take the problem to 'Isaac' for he would solve it for him. Huygens took it to Newton, he solved the problem in a few minutes and moved on. That's Newton, the god of science.
@@maxwellsequation4887 source please, because I can find none on the internet also they weren't really being toxic. that only applies if the statement was serious or intended to express inferiority : )
The person who invented 0 gave nothing to mathematics
Ah, yes, the zero paradox.
Very droll, David. :)
Genius
Nothing is valuable
0 is the most important thing of mathematics. BTW you gave nothing to mathematics.
Newton should be getting more credit because the term he used for derivatives/velocities was cooler: _fluxions_.
But Leibniz's notation has been integral to the foundations of modern calculus.
Cooler only until you read about the various meanings of flux... ;)
+Oscar Smith But who derived those foundations? How exactly does one differentiate between the two notations?
Leibniz, since he derived calculus as an operator/transform, rather than a function. Since it's an operator, one can freely switch between differentiation and integration, rather than continuously write functions within functions (a.k.a. chain rule, Laplace, etc.) Nothing much in real calculus, but a lot when writing it. EDIT: I meant to say is, there is a lot of detail left out when writing in Newtonian notation, such as limits of integration and independent variables
Yeah but Newtons notation was insane and much more confusing than modern day notation
My takeaway from this is that, because "0/0" is undefined or indeterminant, it can be anything -- and *thus* we have to look at it in context to see what value it makes sense to be (if sense can indeed be made). I've never thought of this that way, but it makes sense! And it makes sense not just in calculus, but linear algebra, too, where the determinant of a matrix being 0 means it has multiple possible values for an inverse as well. Heck, this even puts kernels of homeomorphisms in abstract algebra into context, as well, where you can describe the spaces of things that go to 0!
Could you please make an example of a matrix with determinant 0 which has multiple "inverts"? Because as far as I know A*B = 1 has no solutions when det(A) = 0 because det(A*B) = det(A)*det(B)
"of things that go to 1" what do you mean by that?
I literally have no idea why everybody is going so insane over 0÷0 Let’s say you have 0 pizza, and you divide 0 slices from it, THE ANSWER IS 0 There is no pizza or slices to begin with, so its nothing. Aka its 0 IF YOU HAVE NOTHING, AND TAKE AWAY NOTHING, THE ANSWER- IS NOTHING. So 0 divided by 0, IS 0
@@jessie_daily but 0/0 would also equal 1, as anything divided by itself would equal 1 so the answer is an infinite amount of numbers, like how tangent lines always touch therefore, 0/0 isnt indefinite, but rather every answer, like an infinity of sorts
@@AvgCooki hmmm, true
Best T Shirt Ever.
This cracks me up
@@angelinebena9675 yp
@@codexryder8781 yp
@@TheLeorex123 yp
No, #Migos song is!
You're a fantastic teacher. In less than a minute I went from not being sure why dividing by zero doesn't actually work to completely getting it.
Poor Leibniz never gets any credit
Not really.He start infinitesimal math.Read The Decline of the West vy Oswald Spengler
he gets for his notation
Leibniz is the true inventor of Calculus. Long live Leibniz.
Funny how a German (Mathologer) credits Newton and the English/Americans credit Leibniz.
Or Bernoulli. L'Hôpital's rule is probably his.
As an aeronautical engineering student I find it extremely satisfying to see stuff I am learning at the university.
How'd it go?
Just want to say that your explanation that both 1 and infinity are both functions just cleared up a lot of confusion about infinity for me and opened my mind to a totally new way of thinking about numbers. Thanks!
5:49 *THAT'S FUN*
Man, I really wish someone showed me this video in high school. I kind of figured it out on my own, but it always really bothered me and made me feel like I didn't understand math. This is so simply expressed and explained. I suspect the reason why most people struggle with explaining why you can't divide by zero and related is because they don't actually know themselves. They just memorized that it causes paradoxes.
Agreed! People just 'take it on faith' and don't dare ask the questions you did! If you figured it out correctly, what was lacking was confidence in your own ability! You were actually doing math research, so far as your own understanding goes. Excellent!
I often got that kind of experience at school. I realized that this guy, the teacher, who appears to know what he is talking about and can rate my own understanding on final exam is actually blindly following the note of another teacher, in some case the original creator when the school added this subject for the first time. That other teacher did understand and would have been much more interesting to listen to.. A good teacher will try to remember his own questioning when he learned and will highlight discretely these points by encouraging brief exchanges of "who know the answer to this...". Sometime, one student happen to know and find the words that his fellow classmate are more familiars, so his answer help more people to finally "get it".
The reciprocal of infinity is infinitesimal. Think about it. The opposite of ∞/1 isn't 1/0 it's 1/∞. Infinitely small, infinitesimal. You can't divide by zero because you can't do something with nothing. You can't divide anything into no pieces. The only reasonable argument that can be made is that if you divide something into no pieces you have obliterated it completely leaving you with nothing. Hence, if any argument is to be made it is that dividing by 0 gives you 0.
Dividing by zero is attempting to multiply by infinity. In real life, it just mean that we are on the rising edge of a square wave. Suppose you have a variable gain amplifier/attenuator. Looking at the division: y = x / 10 We are saying: y receive 10% of some input (electric current fluid, mechanical force, etc) y = x / 5 : 20% y = x / 2 : 50% y = x / 1 : 100% For the dividend, any value from 1.000 and up actually mean that the output receive a fraction of the input. Now, when we cross the threshold from 1.0 to any lower value (0.999... to 0.00000...1) we suddenly need an amplifier instead of an attenuator In real life, an amplifier always need a source of energy and a command as input. The output is a scaled up version of the command. for example, y = x / 0.5 means that the output y is twice as big as the input x. y = x / 0.1 The output is 10x y = x / infinity, The output is raised to the maximum allowed by the source of energy. All these example are actually considering a system where the command is just controlling how much energy (or material) goes from a source to a destination.
Students today are very lucky to have good 'internet' teachers. in the 1960s many teachers couldn't explain things very well. There is no excuse now for students not being good a maths.
Before I started calculus I was determined that 0 divided by 0 was 0. When I was younger I had it explained to me that with x/y = z, z is the answer for how many times you need to subtract y from x to get to 0. And with 0/0... how many times do you need to subtract 0 from 0 to get to 0? Uh... 0 times, right? That's what I thought, but when we started doing limits I realized that it would create crazy jumps in otherwise continuous graphs, so I gave up on it.
Noah Fence Or you could subtract 0 19,463 times and it would still work. Any number :P
Right, and of course, with 0/0 you can subtract 0 from 0 as many times as you want and you'll always get 0. So just like the with the algebraic description, the answer is arbitrary.
0/0 should equal 1 if you ask me. similar to what you said, except given x/y=z, z should be the amount of times y must be added to *absolute* zero before it reaches x. 2*0 =/= 1*0, so a constant multiplier should always be given when using 0. if none is given, assume 1 as we do with every other number. thus x/y becomes (1*0)/(1*0), which is the same as x/x, which is always 1.
0/0 is called indeterminate /because/ it depends on the situation your function is in. IT IS NOT ANY SET NUMBER LIKE 1. If you have 1/x, and you have x going to closer and closer values of 0, you get (1/(1/1000) = 1000, (1/(1/10000) = 10000, etc, until you get closer and closer to positive infinity - a very very big number. POSITIVE INFINITY IS NOT 1. If you have x/x, and you have both x's going closer and closer to values of 0, you get (1/1000 / 1/1000) = 1 , (1 / 10000 / 1/10000) = 1, the sequence trends to 1. BUT IT IS ONLY ONE BECAUSE OF THE FUNCTION, NOT BECAUSE 0/0 HAS ANY MEANING. TL;DR: 0/0 doesn't give enough information, and that's why dividing by zero gives an invalid result. One of many ways to figure out what happens in a 0/0 case is to find the limit on both sides of a function. If the limit exists, it's the limit as your sequence goes closer and closer to 0. If the limit doesn't exist, it's a jump - and there's nothing actually there at all (at least, in the real numbers)!
***** that is exactly right! thank you for clarifying
I love this representation of 0/0... It gives me a great deal of context that touches on many other ideas I find familiar. You are doing a fantastic job, and I look forward to every new video( as well I find I play the most intriguing many times over.) Thank you for your passion, inspiration, and creativity.
Thank you so much for ALL of your videos. I love your teaching style; you are clear, concise and to the point, delineating complex ideas into simple, easy to understand terms and examples and I very much appreciate it.
Zero has been my favourite number for as long as I can remember having one (sorry, 5). I love how it seems to act like a bridge between real and numbers like infinity. It's also a bit of a gem beneath our noses because the number seems so simple. I'm glad to see a mathologer video with plenty of zeros in it. Also, could you please tell me where I can get that shirt?
Spherical Square Thanks!
The reciprocal of infinity is infinitesimal. Think about it. The opposite of ∞/1 isn't 1/0 it's 1/∞. Infinitely small, infinitesimal. You can't divide by zero because you can't do something with nothing. You can't divide anything into no pieces. The only reasonable argument that can be made is that if you divide something into no pieces you have obliterated it completely leaving you with nothing. Hence, if any argument is to be made it is that dividing by 0 gives you 0.
Alexander Desilets Hi Alexander. I'm not sure who said that the reciprocal of infinity is 1/0. Some people think that the two values are equal (1/0=infinity), but I've never heard that the two values multiply to produce 1. I would argue that saying you can't divide by zero is like saying there are no square roots of negative numbers. If you can find or 'create' a solution which doesn't result in any contradictions, then it may well be a very useful way of looking at it. Picture this: you measure the speed of a car. In the time period of 0 seconds, it moves 0 meters. v=d/t =0/0. Here, the car could be moving at any speed. All real numbers are solutions. You can divide by zero. Although I agree with a lot of what you said, it's not really good practice to imagine dividing as splitting x pizza slices between y people. Instead you can think of it as the stretching of the classic number line. You can look that up if you haven't seen it before.
Alex Desilets wheel theory mate
It cant be infinity because if you approach it from the negative numbers it goes to -infinity
In his book “Yearning for the impossible” one my favourite authors John Stillwell says “…mathematics is a story of close encounters with the impossible and all its great discoveries are close encounters with the impossible.” I hope you like the examples of such close encounters in this video. I actually put up a version of this video earlier today. About three minutes later twenty of you pointed out a REALLY silly typo. Just could not live with that, hung my head in shame, pulled the video and fixed it. Here it is again. Hope you like it. One more thing, if you contribute a translation into a language other than English, could you please let me know by sending an e-mail to burkard.polster@monash.edu. KZhead is not very good at notifying me when new subtitles are waiting for me to approve. Also, please add your names at the beginning of the subtitles. A lot of people are asking about the t-shirt and the missing bits at the bottom. If you are interested have a look: shirt.woot.com/offers/how-natural-selection-works?ref=cnt_ctlg_dgn_1
Mathologer hey you are just awesome bro...
Mathologer hahaha, I liked it man! I find it also pretty funny that you talk about deviding by zero just a few days after I saud a comment about it. Did it inspire you?
Just a coincidence, really :)
although I was surprised by this mathematical nonsense, but I'll throw off "the soap"
Could you do a video on infinitessimals? One of my old workmates used to use them but I never learned what they are or why they even make sense.
This is such an awesome series. I love math! Im in diff eq right now and to learn about all these things i learned in calc and linear algebra like this is awesome
I was interested in these strange numbers for the past few days and i finally found a video that puts everything going together nicely!
Another good one Mathologer. The closing moments were the most important IMO: aspiring mathematicians and those interested should always remember that they're free to redefine expressions and loosen axioms depending on their area of work - much new material can be discovered in this way (e.g. NE Geometry). Similar arguments made for defining what 0**0 should be, and different answers depending on who you ask of course. Perhaps even video worthy :) Thanks again, have a good one.
I've dealt with a lot of indeterminate forms in calculus, but I never really understood what they meant until you went in-depth about it. Thank you.
Glad this worked for you :)
Hello! I'm a fan of your videos. It is remarkable how you manage to explain both extremely complicated and more basic (like these indeterminates) mathematical issues in an entertaining and comprehensive way. That's why I thought it would be worth a go to help to promote your videos further with what I can. I constructed Russian subtitles for this video and just finished them. Since you wrote half a year ago that KZhead does send notifications about this, and because it is almost my first experience in adding subtitles, I decided to write a comment here. Just in case. Thank you again for extremely educative and amusing videos :)
Probably will continue with the videos on Riemann's paradox and the ones about Ramanujan.
+Danil Dmitriev That's great, thank you very much. I am particularly happy about this because Russian is actually a language that I understand myself :)
Wow, it is great :) It is then even more pleasant for me to do this. Anyway, hope that it will help!
Whenever I find myself feeling too confident about my own intelligence..........I look up a video about a mathematical topic. And I am humbled almost immediately. I think I'm a fairly intelligent person. But....there are expressions of intelligence that are as far beyond me as the things I'm capable of understanding are to a cat.
But the human brain also has the potential to learn what the "person" might not even realize is possible until they tried. Calculus's difficulty in terms of grasping and understanding is blown out of proportion in terms of difficulty because it is hard to compress the time and practice it takes into a relatively short semester in secondary or post-secondary. It just takes practice and reps to master the actual calculating bits and integrate what exactly it is you're doing into your intuition. That's hard to find sufficient time for and good friends/tutors to study with when everyone everywhere is always balancing so much in life. It's lifestyle being a big balancing act as is that makes things that take a bit of time difficult. Like learning to paint, cook or play an instrument well. Different things take different amounts of time and patience, and those are the impediments for most people, not raw intelligence.
10:50 L'Hospital, your help in tough times.
Please continue with this topic. There is obviously much more to talk about here than what you covered.
I absolutely hated math growing up. It was the first class I ever got a B in in 5th grade. It then became the first class I ever got a C in 7th grade, and finally, the only class I ever got a D in Junior year of high school. I didn't study psychology in college because it required too much math. And yet in spite of that- or perhaps because of the void it left- I enjoy your work. Terrific way to make up for lost time and enjoy seeing patterns play out without the abstract jargon nor the pressure of testing.
And the Mathologer is such a great teacher. Nothing turns people off of math quite like a bad teacher does.
Man i wish this video existed before i started with calculus years ago. Students nowadays have such amazing possibilities for learning - with great tutors such as Mathologer.
You sir, give me the most intense mathgasms. Thank you!!
Now there's a word I've never encountered in all my mathematical life :)
Mathologer hahaha glad I could teach you something, for a change
Sir, you just give sense to the meaning of derivative, I used since 10 years without knowing the real sense, thanks. And congratulations for a well done and interesting KZhead channel.
Appreciate your teaching style. I have studied math for fun, in addition to formal college study. Your a good teacher. For all education levels. From those learning, or who need to brush up. New sub!
Wow! Numbers standing in for functions. That's something I learnt today 😀 Thanks for the lesson 🙂
Numbers are functions. More specifically they are constant functions.
Numbers are functions
This makes me want to crack open my old calculus books.
Michael Miller do it, you may change the world...
Not being able to know Isaac Newton is one thing, but not being subscribed to Mathloger? Tragic. Great video this one - it hits home well for me, as someone who had to repeat calculus one too many times (4 times total), and spent a fair amount of time studying limits.
I'm thankful my first calculus class professor went about introducing the subject using a similar tactic to create a sense of wonder that drove my continued interest. 0/0 was the only thing on the board on day 1. Very interesting stuff.
this video took me back to when i was to school. in our last year we did these, at this point i loved mathematics
I really like this guy's videos :) keep up the good work!
Cool t-shirt! But I have to know what on the last row is. It was cut off in the video :O
I want this T-Shirt :O
Where do you get these awesome t-shirts?
Do you have a link for purchase? I'd like to see it myself.
+McMuffin I think I got that one from a site called Woot :)
+Peter Tran That's the one :)
I love the videos. I wish we had teachers whom could describe these things as clearly as you.
:)
I always was told by my friends it is undefined but I was like, cant 0 fit into 0 1, 2, 3, 4, 5... times. I think it has an infinite number of solutions. Thanks for clearing this up.
I remember learning this in first year calc, very interesting. By the way, where did you get the shirt?
shirt.woot.com/offers/how-natural-selection-works?ref=cnt_ctlg_dgn_1
-You have no cookies and you have no friends. - That's fun.
That was insane Looking for this stuff and atlast got answer Tyvm❤️
Very well explained thank you! I'm glad Stand-up Maths and Numberphile led me to your channel. Some of your videos make sense just with memories of school maths, some of it is way over my head beyond the few university level maths concepts I can remember... but it all inspires me to keep thinking. I hope you don't mind me sharing some of those thoughts on some rather long comments! I'm surprised how much I had forgotten at every level from primary school to university. E.g. I had forgotten that the triangle of binomial coefficients is called 'Pascal's triangle'. I had forgotten the word 'quotient' as well. One of the hardest things for me to remember was the names of theorems in maths and physics and long complicated words (too many of those in chemistry!). Equations were easier to remember than names! PS a thought (or a joke) about 0/0: I suppose if someone gets 0/0 in an exam they get 100% for nothing (they had no wrong answers!)? No that makes no sense! There were no exam questions...! Unless the exam is marked by percentage of the questions attempted, so if someone attempted 60 questions out of 100 and got 57 of those correct they get 57/60=95% not 57%?! That would be nice, but unfortunately omitted questions get marked as 'wrong', not ignored. Otherwise you can get 100% for getting one question right and ignoring the rest!
wanna see the entire t-shirt..please.
Here you go: shirt.woot.com/offers/how-natural-selection-works?ref=cnt_ctlg_dgn_1
danke dir! Deine T shirts sind immer lustig. tolle show! es ist immer wieder ernüchternd, neue perspektiven auf die mathematik zugewinnen. gruss Saqib
T-shirt message makes totally sense.
5:40 So i've asked Cortana, from Windows Phone, "what is zero divided by zero", and she said: "Mathematically, it's undefined. Philosophically, it's one of those deep questions, like... how do you hold a moonbeam in your hand?". HAHAHAHAHAH Nice video and explanation!
i never would've imagined to finally understand and get "a feeling" of derivates thanks to a video on a different topic, thanks
Great video! You are very good at explaining difficult (for me at least...) things... Salutations from Italy.
Great video, thanks so much for these. Wouldn't three divided by zero be a different infinity then say 2 divided by zero?
Ontological Motivation 3/0 and 2/0 are both undefined
For example if 3 divided by 0 equals infinity and any other natural number divided by zero eg:2,3,4,5,20... are equals to infinity that means that 1=any natural number
@@olivermorrison7127 The limit 1/0 approaches to positive Infinity and another approaches to negative Infinity
Did u guys know that Infinity-infinity= infinity was also written in Vedas which is a book with no author and no one knows when it the book was written
Mathologer I would love to see you do a video about nonstandard analysis!
Sir as a Maths enthusiast, I and my sister are looking forwards towards a book on calculus where we can find the very philosophy of calculus. Problem is that we have finished the curriculum books but we together ponder why the hell limits were invented, why continuity was thought, why delta is used in differentiation and last but not the least the algebra of differential equations as it is tremendously difficult to imagine these things. And also as an emerging teachers some students in turns ask us similar question to which we have ponder a lot but we do not know the exact sequence of event in the development of the subject Calculus. We like your videos and we hope that your journey into the subject must has stopped somewhere like our. Can you suggest the book that sailed you through such questions.
Found my new favourite channel.
Iam from germany and we learn that Leipnitz found calculus, or as we say Analysis.
yea, "newton did it at the same time" , but mathloger hate leibniz
even though he uses Leibniz's "d/dx" notation instead of Newton's "f'(x)" notation.
yep
evildude109 Newton did differentiating, and Leibniz did integration. Leibniz published first, but records say Newton found it first. Also f'(x) is Lagrange notation.
mmmm you are right about the lagrange notation, but leibniz also created the chain rule for differentiation , (in my opinion newton stole Leibniz )
Thank you much for the German and French subtitles . They are very helpful .
The way it was explained to me, "undefined" means there is no valid result for the calculation, while "indeterminate" means there are multiple (possibly infinite) valid results with no clear way to choose between them. Since indeterminate forms in calculations of limits often DO lead to a single, clear, valid result, it's fine to use them (or work around them with L'Hôpital).
You really hammer it home in the second part, but the first part could have used a tad more emphasis that you're not really dividing 0/0 but instead making the claim that you can make the value arbitrarily close to the limit of the independent value that you're trying to approach. I think too often we conflate the limit with the "answer." This can be particularly true when we talk about infinite sums.
Well, these videos are always a crazy balancing act trying to be at the same time as accessible, concise, understandable, etc. as possible. Having said that, I really think (like pretty much all other mathematicians) that defining the sum of an infinite series to be the limit of its partial sum is a very natural choice. Of course there are other choices which are also explored in mathematics. I talk about different possibilities in these videos: kzhead.info/sun/nceEgquohJuMh68/bejne.html kzhead.info/sun/n8l_ldRxsKZpfZE/bejne.html
Mathologer Of course. What you do is not an easy thing to do (especially when you do it as well as you do). With that said (and I hope I wasn't too harsh in the original comment), it was a great explanation. And yes, I didn't mean to derail the topic by bringing up a (somewhat) youtube mathematical controversy. The limit of partial sums is a very intuitive definition of infinite series, but my only point was that its still a limit and not *"really"* a sum; much like how a derivative is the limit of velocity between two very small points but isn't *"really"* a velocity at all. While we call it "instantaneous" velocity, it doesn't really make much sense to call it that from out perspective. Limits are very very strange things.
Sure, in fact you are in good company. If you look at the history of calculus there is no shortage of heated debates among very smart people about things like sums of infinite series.
Mathologer Yea its always really interesting, and it actually has a lot of implications about just the nature of numbers in general. For example, I think even in your .9 repeating = 1 video, I think you point out that .9 repeating can be described as a geometric series (which equals 1). But if that series isn't an actual value (and it only means that we can make it arbitrarily close to 1) then what would I actually be saying? Its really hard to keep it all straight and think clearly about it.
Every maths/physics student should watch this video before a lecture on l'hospital's rule!
I have a lot of problems in my business, so i came here to watching this complicated problems, it makes me to think that my problems are very small and somebody in this world has bigger then me by thinking of this kind of things.
Amazing. My old friend Corey from high-school was always good at math. You could ask him a math question and he would give the right answer within seconds. Impressive!
Diavolo: it's just an arrow, what could it do? i'm still stronger! Giorno:
"Calculus, courtesy of zero divided by zero." - Mathologer 2016
Lol
I love your shirt! 🌹 Where is it from?
Where were you and the internet 30 years ago!!! Oh, damn! Thanks for these great videos, at least my daughter will enjoy them.
When I took Calculus in college, I intuited and tried out several of these 'interesting' alternatives on my teachers. They particularly did not like it when I treated infinity as a 'destination' rather than an 'endless journey'. They only gave me hard times about them, and said nothing about there being special times, places and methods where it was okay. Yes, I was making life a bit complicated for them, but that is no excuse for them to _overgeneralize the everyday rules._ Sheesh. :/
I haven't learned calculus yet, and this video makes me want to.
I wrote the paper about it. Here is in short how we can do this. (full document is 49 pages long with many pictures, graphs and examples). Sooo... By precise analysis of multiplication and division I've found out that they are both one and the same operation, which is the transformation of the pair of numbers into another pair of numbers (proof and examples in my work). It seems that talking about numbers we are ALWAYS referring to a pair of them! Then I proposed that the natural form of numbers is the ratio of the certain value and the certain base measure that this value is related to. For example saying 5 we really think 5 related to (base) 1. When we will accept this approach we can easily understand everything related with division by zero. It is not only possible, but we can easily understand that 2/0 is something different then 1/0. We should not treat 1/2 as equal to 2/4. Think about it ... If you will take 1/2 of the apple, you will have something different then, what you will have, when you will take 2/4 of the apple. If you do not believe, you can cut an apple into two parts and take one ... then cut the other apple into 4 pieces and take two :) Everything is explained and proved in my work here -> vixra.org/abs/2001.0475 For example I presented graph of the function f(x) =1/x ... without discontinuity point ! :) It can be presented for every x, and I'm also explaining why our traditional (wrong) graph has discontinuity at 0. If you really want to understand it ... you need to read it and understand all presented examples. Enjoy :)
that's a very nice video!! thank you so much for making it;
The music makes me think of Nine Inch Nails - March Of The Pigs where he says "Doesn't Make You Feel Better?" Cool Video, thank you for posting. Love learning about these concepts.
Kate Bush
mathologer, you use the leibniz notation for the derivatives but ironically you are giving all the credit of the infinitesimal calculus to newton , ¿why you hate leibniz ?
ikr? not even one mention of him! Leibniz has even written at length on this very subject!
+alejandro duarte +Rumford Chimpenstein Well, 1. I would imagine that everybody who watches this video knows that Newton and Leibniz (and a couple of other people) were responsible for the invention of calculus. 2. I only said that nobody would know Newton. I did not say anything else. 3. The only reason why I mentioned Newton at all was because I wanted to use the apple story as part of the framing of this video. :)
Why do you hate Newton ?
Mathologer if you are trying to make your videos "more accessible" as you put it in another comment, why assume all your viewers already know the history of calculus?
@Element 115 sure, but saying that 0/0 caused Newton to be remembered and then using the Leibniz definition of a derivative (Newtons fluxions don't use 0/0, IIRC), the d/dx Leibniz notation... Also the apple (if it existed) didn't cause Newton to invent derivatives or 0/0 but a theory of gravity.
this is the greatest math channel evah!
At 9:20, when he invokes e^ln(stuff), could he have chosen any matching exponent and logarithm base? 10^log base 10(stuff) for example?
"Lives in Australia, originally from Germany"... Genius and unable to be killed by creatures!
lol
Didnt know johnny sins was so good at math
He is good at everything
havent you seen his maths class video where he teaches the girl?
This is the first time someone ever explained "why can't you divide by zero" to me in a way simple enough for even me to understand. It feels so obvious in hindsight, but the way most people - even Siri - try to put it, they make it _sound_ like x÷0 _should_ equal 0 (e.g. "All of your friends get zero cookies if you try to evenly split zero cookies"), when that isn't the case (and is where the analogy of evenly splitting something physical ultimately fails).
great video! He explains things well, doesn't go too fast
Interesting coincidence. Earlier today I was thinking about a situation in which 0*inf = -1. If m and n are the slopes of perpendicular lines, m*n = -1. But what if one of the lines is vertical and the other horizontal? You either make an exception to the rule or define 0*inf to be -1 in that context.
Excellent example.
That is just bc you are using a form of that rule that gives you indeterminate value in this particular case. No exceptions need to be raised. In fact you could rearrange the rule bf calculating the limit: set m=-1/n instead of mn=-1. You just get that 0=-1/inf, which is true. The rule still holds.
in the case of perpendicular lines, shouldn't m*n=-m²=-n² ? and therefore, m/n = -1
Given two lines passing through the origin and having normal vector respectively (a,b) and (c,d) we have that their equations are =0 and =0. The condition of perpendicularity is therefore =0 that is ac+bd=0 that is -a/b=1/(c/d)=-1/-(c/d). Since the slope of the first line is by definition m=-a/b and for the second n=-c/d, we have that the two lines satisfy the condition m=-1/n. If the line doesnt pass through the origin the argument is still valid bc only a constant term is added to the equation and doesnt change the slope. The criterion ac+bd=0 always works. You should start from there and then apply more special cases (like m=-1/n or mn=-1) when possible. The thing is, this special cases cannot be always applied because they are not formulated in terms of coefficients of the lines, but n terms of slope and y-intercept form, which is weaker. Therefore the language of limits is used to make some sense out of them in these exceptional cases. But it should be avoided in rigorous mathematics just getting back to the general and deeper condition ac+bd=0. Hp to have solved your doubt.
Giordano Giambartolomei Yes. Thank you.
"And you are sad that you have no friends" "Thats fun!" :D
Where did you get that T shirt the one with three extinctions. I love your T-shirt.
I put the link in the description :)
I independently came up with a non-calculus hypothesis for 0/0=1. In short, the Identity Rule of division (anything divided by itself is 1) overrides the second clause of the zero rule of division (if zero is the denominator, you cannot solve the equation without calculus.)
Love that shirt. Great video!
There is actually some more to this t-shirt. Have a look: shirt.woot.com/offers/how-natural-selection-works?ref=cnt_ctlg_dgn_1
Wyvrn I was looking for this! Great video!!
"All things are numbers" - Pythargeuos. What would he say about 0 and infinity? What do they represent?
Tim Westchester 0 is a number Infinity is not (however there are number which are infinite)
Tim Westchester Although Pythagoras personally didn't believe root(2) should be a number.
You can represent things by saying something like 0 is apple, 1 is car, 2 is earth etc. So, can't you say, for example, infinity is 3?
If anything, 0/0 is simply every number. Because as x approaches 0 in the function x/qx, the output will approach 0/0 but will always be equal to q, and q can be set to any number.
Question: What does q mean?
@@dr.danburritoman1293 q is just a variable, it can be any number.
I wish there was KZhead in the 90s with this guy explaining stuff that I found too hard to figure out on my own
My math teacher once told me: When you ever come across with these bad-boys ( meaning the indeterminate forms), use this nuclear weapon and solve the problem. That's how I learned L'Hôpital method.
*cries inside* "That's fun!"
Would it make any sense to say that 0/0 yields the constant set? In calculus, taking the antiderivative of something yields a function + C, which is the set of the sums of all possible constants and that function. When you find 0/0 inside a limit, then you are really finding one specific value of that constant set.
Please make an introductory video discussing the other branches of mathematics that use infinity and zero as integers for rational expressions
You took the original video down because of the wrong derivatives for x^n, right?
That's right, just couldn't live with this typo :)
I didn't understand most of it... but it seems interesting, I guess I'll look at it in the future when I know more about maths
The video definitely moves quite fast and covers a lot of ground. Maybe watch a couple of times and pause every once in a while. At least the first part up to the Siri interlude should be very doable in this way :)
I usually really like your videos but I think in this one you assumed people knew way too much about calc. I also think using limit notation would have made more sense for viewers who are just now being introduced to limits and derivatives. I took calc one this year but girlfriend, who I watched the video with, took her last calc class a few years ago. She was completely lost and I had to keep pausing the video to explain what was going on. Don't get me wrong, I like your content and sub to your channel, but I think this video could have been better. I think you tried to cover too much in 12 minutes, and had to cut too much out to fit the time slot.
I actually agree to a large extent with your assessment when it comes to people who've never heard of calculus before. In fact, I actually don't expect people like this to get much beyond the Siri interlude (if they actually get everything up to that point I am more than happy). The second part is really aimed at people like you who've already seen some calculus. The video is a bit of an experiment in this way. At least in terms of overall response it's turned out to be quite a successful experiment :)
I am sorry that I am about to ask an out-of-topic question: where did you get your t-shirt from?
I have a question (and I am not a math person, so forgive me if it is obvious). If you are bringing the two points of a cutting line together (3:20) are you finding the slope of a point? I'm sorry if this is a silly question, but visually that's how it looks.
I've suspected that since learning calculus that forms like x/0 or 0/x depended to the context of the math. like when you get removable descontinuities or vertical assymptotes when taking a limit of an expression. I suspect that there is an awesome field of mathematics behind the number zero waiting to be discovered.. :)
yep...
Alright man, I'm saying all of this as a friend, and I'm not attacking you. First of all, "0/x" is always zero. Just pointing that out. X can take any value in the denominator you like, but as long as the numerator remains zero, the function is zero everywhere. There's nothing special about that function. Second of all, L'Hôpital's rule is a standard part of derivative calculus (or so called "Calc. 1"), so what do you mean you've "suspected"? Don't you mean you learned? For anyone that has taken introductory derivative calculus, this is just review. And lastly, when you say "x/0," you're confusing the entire issue here. The point of L'Hôpital's rule, and using limits in general, is that it depends on the context of the functions you're describing. When you say "x/0," you're saying that x varies while zero is is the denominator? That function itself isn't defined. If you recall from evaluating limits, you get results like 0/0 or ∞/∞, only when the functions in the numerator and denominator approach that value. So "x/0" means that you didn't reach that zero in the denominator through a limit, so it's meaningless. Your final thought that there is plenty more to discover about the properties of 0 is perfectly valid, but your reasoning tells me you need to review your derivative calculus notes.
Nathan Klassen твоя мысль понятна также как понятно, что автор ролика не понимает о чем говорит. Чтобы убрать эти парадоксы наше понимание должно стать другим - нам следует поменять философию математики.
Святослав Глуздов Я согласен, что математика может извлечь выгоду из новых идей, но, как и любой математик (или физик в моем случае) скажет вам, что эта тема обсуждалась в этом видео математически звук и очень хорошо понимали.
+Nathan Klassen You are clearly quite confused :) I happen to teach this stuff at university and I assure you that everything I say in this video is by the book. Maybe have another listen to what I really say and not what you imagine I say :)
0:36 My math teacher said not too long ago that when someone tells him that 3/8 is equal to *green* (yes, green) he knows for sure he's on drugs. I'm dying XD
Where can I find your t shirt? I love it!
Link is in the description :)
Mathologer thank you!