The Opposite of Infinity - Numberphile
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- Опубликовано: 27 сен 2024
- Continuing to talk Infinitesimals, this time with Dr James Grime.
See last week's video: • The Infinitesimal Mona...
More links & stuff in full description below ↓↓↓
Dividing by Zero: • Problems with Zero - N...
James Grime: singingbanana.com
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![DDG & G Herbo - Nosey [Official Video]](http://i.ytimg.com/vi/I9Ra0Zc2NlQ/mqdefault.jpg)
I hated math in school but now i realize after watching these videos that it wasn't math I hated, it was the class.
+
+
Yeah, many people can't see the forest past the trees.
I always thought that if we learned math not from other or from a book, but invented math ourselves under proper guidance, than we could all understand math very easily.
But imagine if everyone had to be a genius to create 1000's of years of math progress. That system doesn't work, just like our own system.
I love the way James says "noooomba."
I love the way he says everything.
also 'area'
noomba sounds like a kind of goomba in a spooky level.
aaaaaaaarea
aaaawea
I love how all the people in these videos are so excited to talk about these things.
xD
+Jackie I'm always excited to tell my family this stuff but they look at me like, "AHHHHH!"
Mafioso Math haha! xP
+Jackie ditto.
so YOU HAVE A LOT OF this infiniTESimals...
And then the engineer comes along and says
"Eh it's within 10%, it's fine..."
buhahhahaha! XD
Pi=3
e=3
Pi=е
I'm a civil engineer 😎
@@Кирилл-д6е4р pi = 5 if it breaks it ain't my fault
@@Alguem387 just round everything to 1 or 0. Pi = 1
@@Кирилл-д6е4р You can have Euler's π but you may not eat it.
I like this sneaky way of teaching your viewers calculus without saying the scary C word.
I was really enjoying that little calculus part XD
It was all calculus
Was it? Oooo interesting!
+Vicvic W calculus is always interesting
+NowhereManForever - 6:45
The amount of theory in his brain has impinged so hard on on his motor functions that he can no longer draw shapes. Bless him for the sacrifices he has made.
Smart people usually have the worst calligraphy
F
Whoosh
I’ve never seen that word before lol
Loooool
8:58
You are a damn genius. You taught why the derivative is the inverse of the integral and what slope has to do with area in less than 30 seconds
what? he didnt say anything special.. what do you mean?
@@erdo4321 my guess is just that James described it in such a clear and intuitive way. Always love videos with James!
Ztingjammer same
He basically said that when you differentiate an integral, what you get is the function. This makes clear the fact that differentiation and integration are, by definition, opposite operations
Kakali Mukherjee 2nd Fundamental Theorem of Calculus ftw
"let's say I have a circle"...draws lopsided potato
edit: this is meant to be a humorous observation I have nothing against his theoretical circle
Let’s see you try then
😂😂😂
you aint got nothing on spongebob bro
😂😂😂
He's a mathematician, not an artist
I’m pretty confident that if I had Dr. Grimes as a teacher I wouldn’t have switched majors and gone on to Calculus II
Je
can't stop the fire
Why can't you just flatten the curve line
You can find the distance of the curved line flatten it out and set it above to the lowest height point
I had a horrible teacher but I am in calc 2. I studied myself, didnt listen teacher. İt is your failure. Blaming your teacher is not a solution.
Teachers in school go: "Area of a circle is 2pi r^2". The kids ask why, and teachers just say "because it is". Two weeks after class everybody's already forgotten what the formula was. If we were taught WHY in school (just like this three minute segment at the start) we would never forget these things.
+Tiago Seiler Most students would still forget because most humans are ignorant peasents.
I am hwoever pro-understanding education.
In this case... It kinda is. I mean Pi is a concept only understood when you understand that it is just a number derived from calculations.
+EmperorZelos ./.
Miguel Sambaan
?
+EmperorZelos I infer you are the type of person who contributes little to society, yet is in full capability to do so.
Outfinity
Haha!
Exfinity
outfoutity
@@devonqi interior:exterior infinity:exfinity
@@JamesM1994 i understand hehe.. im just makin a lil' lul bruv..
Newton: I have invented calculus
Leibniz: I have invented calculus
Newton: That sounds derivative
Leibniz: but integral to the problem...
@@BritishBeachcomber Damn I was gonna say something like this...
let's set a limit to the level they can go down fighting
@@MrParry1976 it may never end, there maybe no limit
4 AM on a work night, a video about the opposite of infinity? BRING IT
Planing at night to occupy more countries?
+ZeroSum Game Lmaooooo I see what you did there... must be Ukrainian aren't you?
lml!
+ZeroSum Game boy, that escalated quickly!!
+Masha Vasilchikova haha good luck ;D
.l.
10:00 "They discovered that Newton came up with it first. Leibniz then died..." what, like, immediately? That's harsh.
they roasted him straight into the grave
about three and a half years later, apparently
dx/100
Just correlation or causation - for some measurements we will probably never know...
My man just drew the worst circle ever and then proceeded to draw the most perfect f i have ever seen.
You can see my amazement at 7:59
I will become a mathematician, just so i can write on brown paper with green sharpies.
Manuel Pilarczyk
Lol
...As long as you do it without using random commas like the one here between "mathematician" and "just."
@@HelloKittyFanMan. They said mathematician, not an English major 😊
Did you?
When I started to watch this video I had a finite amount of brain cells. When returning to my math homework, I realized I actually had an infinitesimal amount of brain cells.
SuburbAllied Was this an intended pun?
you had to check the spelling of infinitesimal, didnt u
SuburbAllied well you have to be more specific, what are you classifying as brain cells? Just neurons? Or any cell in the brain? It's been est that the adult male human brain, at an average of 1.5 kg, has 86 billion neurons and 85 billion non-neuronal cells
hang in there, my friend. newton described himself as a little boy on the beach, that, every once in a while, found a stone that was a bit more shiny than the rest.
Dr. Grime's enthusiasm is immensely encouraging. Even the least curious among us must find his presentation engaging.
I wonder whether any single person fortunate enough to have studied under his personal direction did not succeed. Bravo!
Isn't the opposite to infinity finity
Yes, but not in mathematics. Mathematicians are very strange people.
By that reasoning it should be out finity. :)
XD
Nah, it's ytinifni
The deception of logic:
Take a wooden chopstick, with length measuring 20cm.
Break it in half into 2 and we now get 2 sticks, each measuring 10cm in length.
20 = 10 + 10
20 - (10 + 10) = 0
What is lost in-between? Absolutely nothing!
Suppose you were required to draw a line in the center of the original stick before cutting it in half.
Where would you draw the line, with nothing in-between?
To know what 3-D is exactly, first imagine 1-D and 2-D in their purest forms.
Can anything possibly exist in just a 1-dimensional or 2-dimensional form?
If 1-D and 2-D were totally imaginary, how real could a 3-D object be? 🤔
Are you able to reconcile 3-dimensional concept with reality?
Theory: The entire Universe began from a single point.
Now, what is the smallest possible point?
It is impossible to reconcile this theory with logic as well.
Can you draw a "perfect circle" using a compass, without the smallest possible point?
Question nothing, to question everything. 🙂
the opposite of infinity is my will to live
Deep.
infinitejinpachi :0
AHAHAHAHAH WHY IS THIS RELATABLE
The Golden Carpenter
Depression
gravestone quote
I love this guy. - Am I alone?
+PunktKommaNull Prof. James Grime is awesome!
You are not alone
James is like the Ainsley Harriott of mathematics.
+PunktKommaNull we are shy to say this :p
hakkihan tunbak Thanks for the hint! Didn't know that yet! :)
This is literally the basics of Calculus and I never really realized it until now
Edit: I commented literally right before he started going over calculus. Wack
Thanks for giving Leibniz some shine, he also has some very interesting philosophical works if you enjoy logic employed in a different way
Mathematicians are great philosophers
Plus: The bisquit is tasty!
It'd be cool if there were Numberphile action figures, or even just 3D printed figurines of 3D full body scans of our Numberphiles Heros. This would definitely include a "Brady" with a replaceable exploding head for every time his mind is blown.
+Fish Kungfu I feel there should be action figures for all of Brady's channel: James for Numberphile, Prof. Polyakoff for Periodic Videos, Prof. Moriarty for Sixty Symbols... Ooh, this would be so fun.
+Fernie Canto I'd definitely buy a figurine of James!
+Fernie Canto Don't forget the legendary Keith from the Royal Society from Objectivity ;)
+Fernie Canto CLIFF. STOLL.
+Fish Kungfu it would be a great idea for kickstarter project) definitely for it)))
'8' is the opposite of infinity ;-)
or it is rotation at 90deg ( counter or clockwise)
thatsthejoke.jpeg
infinity * i = 8
... ∞i (complex infinity) ? :)
Devendra S wooooooosh
With teachers like him one can never hate maths
I always loved math but not necessarily all of my teachers
Wouldn't the mathematical opposite of infinity be negative infinity? I would consider an infinitesimal to be the inverse of infinity.
+MultiWafflemaster I feel like I get what he was saying.... What is the opposite of the "biggest thing" - it is the "smallest thing" or "negative the biggest thing".
I guess it is like what is the opposite of being unimaginably rich? Is it being really poor, or being in debt?
Both arguments hold water for me.
The problem with "negative the big thing" is that it still has a kind of great magnitude in my mind.
Just one man's musings.
+MultiWafflemaster Infinity is a concept, not a number. Surely the opposite of something too big to measure is something too small to measure?
The inverse of an number depends on the group you are in. If it is addition, then the negative value is the inverse. If it is multiplication it is one divided by the value. If it is NxN matrix then you inverse the matrix. If it is MxN matrix, then you need a pseudo inverse.
+MultiWafflemaster No it wouldn't because they both have the same magnitude but in different directions. If that makes sense? Like if I asked what would be the opposite of the word huge you wouldn't say oh negative huge!
+Lachi Agnew Read +L0LWTF1337 's post. Both answers are right, due to lack of further specification.
Smallest possible number that's still bigger than 0....
Just look at my exam results
Fair
Technoultimategaming at least yours are bigger than zero
Meanwhile me :
Laughing in negative 😂
Plank distance.
@@Enter_channel_name farts on your screen
I honestly had no idea what this person was talking about for the majority of this video, but I watched whole thing because I enjoyed watching his genuine enthusiasm for the topic and for math
An interesting thing I found on Wikipedia is the projectively extended real line, where the number line is wrapped around into a circle, and the point where they meet is infinity, which is neither positive nor negative. In this system, x/0 is equal to infinity, and x/infinity is equal to zero. The coolest part is that it would also work with complex numbers if you wrapped the plane of real and complex numbers into a sphere, which is the Riemann sphere.
I always thought x/infinity would be equal to infinitesimal
@@R0llingHard and that's because infinitesimal IS equal to zero.
the usual definition of infinitesimal is "a number that's as close as possible to zero without being zero" problem is that such number can't exist since we can always get the average between zero and whatever number you believe to be infinitesimal.
but, if you define infinitesimal as "the smallest non-negative irrational" then, infinitesimal = 0.
the only difference between both definitions is the inclusion of 0 and that you can't use the average trick anymore.
the issue of the cylinder, shown in the video, having volume while it's cross sections have heigh zero can be explained because we have infinite cross sections so they add to 0*infinity (which isn't defined)
The infinitesimal is not equal to zero as the hyper reals can show. And the infinitesimal is also not a number, an all-too-common misconception.
@@DanielRossellSolanes "the usual definition of infinitesimal is "a number that's as close as possible to zero without being zero""
Usual definition in which context? I have typically seen infinitesimals described as something like "positive numbers which are smaller than every positive real number." But I may not have experienced the same contexts as you, so I'm genuinely curious!
@@edwardpotereikoWe're talking about standard real numbers system, No field extensions are related to it.
I saw what you did there at 4:22 Numberphile
+stingersplash16 watch it again, pay real close attention to the video and you'll see it!
+Jake Equilar kepler 39? the planetary system?
+Jake Equilar Who is this guy having the number 39 on his back?
+Thomas Korbacher indeed who is he? O _ o
warwick capper haha
I loved that demonstration of the fundamental theorem of calculus. Absolutely beautiful and simple and excellent!
Valera 8 No, I don't have to do anything.
No estaría 'troleando' a nadie, sólo escribiendo mal. 2 pequeñas correcciones si me permites: *I'm really sorry *Not Spanish but Spaniard. Saludos.
I recommend the book written by Keisler named Elementary Calculus. It uses infinitesimals to teach calculus.
Also great video!
I made a presentation and speech about this for school this year and seeing this made me really happy that the area is more popular than I thought.
Thanks!
The opposite of infinity is finity. The end.
Cool.
+MadaxeMunkeee story
+Leon Gerity Bro
+ImJustACowLol Sir you are a true genius.
+This Could Be You!!! Thank you, thank you. I just received word that I am nominated for the Nobel Price of Mathematics. It is the first time such a nobel price is going to be given, as prior to this date the Nobel Price for Mathematics did not exist yet. Awesome, right?
mathematics such as calculus are difficult to many because too many have been taught since they first entered grade school that math is a memorization game.
+pantheryou Not really, if you understand the principle you don't need to memorize anything.
It's not difficult. It's only difficult if you don't want to learn it
R. Rain re-read my post. what you have typed is precisely my point.
+pantheryou i absolutely agree. Math is really just logic and philosophy. If you understand the logic behind it without the numbers, then you can do the math but most people believe that math is a dark magic where stuff just gets pulled out of mathematicians hats
I prefer the circle area proof that rearranges the wedges in an alternating zig-zag to form a rectangle, with one side r and the other side pi*r. It's a cleaner proof because it doesn't skew the wedges, and the area of a rectangle is slightly more trivial than the area of a triangle.
I think I understand...
Is it:
infinitesimal = 1/∞
???
Basically
That's what I always thought, and the only reason I came to the comments!
NO
You can not divide by infinity, it is not a number
@@MsAlfred1996 you are right it can only happen in limits
Sorry for my bad English!
0,0=infinitesimal
∞=infinitely large
0=naught
ᴑ=impossible
Thank you James Grime for saying that 1/infinitesimal is infinity, which means that the other Numberphile videos about 1/infinity equals 0 are false! The good way to see infinity is that one that you just used: 1/infinity equals infinitesimal and defines infinitesimal, and 1/0 having no answer
+Thomas Godart Except that it is wrong.
+Thomas Godart lim x-->infinity 1/x=0
lim x-->0 (+) 1/x=infinity
lim x-->0 (-) 1=x= - infinity
Infinity is not a number, meaning that problems that involve it have to use limits.
+BlackSkullRacer613 One-sided limits are often useful. Since the whole infinitesimal/illimited etc. discussion has been confined to the nonnegative numbers, the limit while approaching 0 from above is relevant.
Did any one noticed the picture that got mixed with Kepler's photo in 4:22 XD
Warrick Capper, an AFL star and meme
@@robertbell2159 thanks for the clarification dude
@@meyupme9854 This isn't the only time I've seen his picture show up in an academic video. He shows up the same way in the Teaching Company series about the American Civil War I think.
A little late but I think this is also a clever reference to Kepler-39, 39 being the number on the jersey worn by Warrick Capper. Could be coincidental, tho.
In multivariable calculus one calculates with differentials as they were just ordinary variables and it all works out and is rigorous and consistent. But differentials are not numbers.
Or are they?
Vsauce music plays
Enter... the *hyperreals.*
deleuze borrows differentials from calculus
he says that they are the infinitesimal blocks of change itself
their relative magnitudes dictate the nature of encounters
think of omicron notation, two functions come together and one may overwhelm the other
and he says that the way they have been marginalised, the way "instantaneous change" has been termed an oxymoron instead of a generative paradox, is basically the attempt of state science to enforce thinking in terms of only being and identity
as opposed to becoming and difference
differentials will always be too small for state machinery, from this comes said machinery's imprecision, the inevitable "negligible" error to which pure difference has been relegated (dialecticians call this "negation")
instead of putting up with this failure, the imperfection of the world compared to the actual numerical measurements with which we seek to capture it, deleuze says we have to finally start thinking of pure difference itself, because it is what drives being
the way the derivative drives a function with infinitesimal steps
@@heartache5742 Sir this is a Wendy's
Honestly if it was explained to me this way, I would actually have understood what I was doing at uni
7:00 unknowingly makes pi
I see it!?!
@@KaliFissure what is the what?
@@KaliFissure
No. The Planck length is a fundamental metric of the dimension of length. Asking for an opposite is a non-sequitur.
How? I don't see it
@dominic twaites When he draws the two outer vertical lines, it sort of looks like the symbol for pi. :)
Who is the guy 4:21 with 39 on his back?
+Tymon0000 I think it's Capper
+ObeseYeti Kepler*
ikbeneenpop1 The guy with the 39 on his back is Warwick Capper
ObeseYeti
Do you happen to know why he is there?
+Tymon0000 Pun on the name, maybe?
I find it amazing that Newton and Leibniz both came up with Calculus independently.
This actually happens a decent amount throughout the history of science and mathematics.
Newton and Leibniz were both very intelligent people, but people today often view them as some sort of super mega geniuses who developed calculus all on their own. The state of mathematics when they both lived was ripe for the development of calculus. If Newton and Leibniz had not done it, someone else probably would have within the next 20 years or so anyway.
The idea behind integrals (the method of exhaustion, like what is shown in this video to get the area of a circle) existed for millennia before Newton and Leibniz. About 50 years before Newton, René Descartes introduced coordinate geometry, which was a fundamental step toward developing calculus. Around the same time Pierre de Fermat posed the question of how to find the tangent line to a curve at any given point. Within about 20 years before Newton, James Gregory gave the first sort of argument for the Fundamental Theorem of Calculus - it was a highly geometric argument which connected areas under curves with the tangent lines of those curves. Later, Isaac Barrow developed the tool of infinitesimals and used it to solve Fermat's tangent line problem. Barrow also gave the first rudimentary proof of the Fundamental Theorem of Calculus using his infinitesimal techniques.
Then we get to Newton. Isaac Newton was a student of Isaac Barrow and learned about infinitesimals (and how they relate to tangent lines and the big connection in the Fundamental Theorem of Calculus) from Barrow. Essentially, Newton was in exactly the right place at the right time to develop calculus. Pretty much all of the requisite tools had been developed right before he started his studies, and he learned directly how to use the last necessary tool from the very person who developed it. Newton saw how to put these tools together in a meaningful way, and more importantly, saw an application (physics).
While Leibniz doesn't have the tools handed to him on a silver platter like Newton did, Leibniz still lived in the historical context where people knew about the method of exhaustion and already had coordinate geometry. People cared about Fermat's problem, and knew about Gregory's connection between area and slope. All it takes is for Leibniz to do the same thing Barrow did and just imagine the infinitely small and then run with it.
This is what I mean when I say that the mathematics community was "ripe" for the development of calculus. The general trend of mathematical thinking and interest were moving toward calculus anyway and both Newton and Leibniz happened to be the right people living in the right places at the right time.
Math and science are rarely developed solely by lone mega geniuses.
(Another example of this phenomenon is the theory of relativity. Although we credit Einstein for the theory, he also lived in a context where people were studying and developing the same sorts of things. There are many mathematicians including Henri Poincaré and David Hilbert whose ideas about relativity were instrumental to getting the full theory. Yet science history tends to wipe away the contributions of everyone but Einstein and paint a faulty narrative of Einstein as a lone super mega genius who did everything without anyone's help. No, he lived in a context which was ripe for his ideas.)
Before watching the entire video:
∞/1 = ∞
1/∞ = [opposite of ∞]
My guess is 10 to the power of minus infinite.
Actually this is invers of infinity
I was going to say the same lol
Thats the recipicle of infinity
You can’t divide by infinity
the feeling of "it works for daily usage but somehow im not happy cause I disregarded a little fact" is what bothered me in school so much
+teekanne15 well limits get around the error by just essentially saying "i bet that you can't make that error ever give me a wrong answer because i can always draw enough triangles" hence the standard epsilon-delta proofs.
+Ben Nutley What we actually do is the equivalent of "Tell me how much error in the result you will allow, and we'll find a small enough delta (or large enough N, in some cases), that our procedure will be at least that accurate."
Happy to see a video about this!
Wrote my bachelor's thesis on this very subject. It's an interesting area of mathematics that I hope will get more visibility. Specifically we looked at Picard's theorem and how much simpler the proof is by using non standard analysis (I'm far from competent enough to understand the standard proof).
It's beautiful, massively useful and intuitive in a way that limits aren't.
That being said. Both are needed
Am I the only person who spends his days watching videos like this not knowing wtf these people are talking about but still liking them
Leonardo Acuna you are not alone, bro
Maybe theres apart of you that dose know
Leonardo Acuna same here ^•^
I like to pretend I understand.... I just enjoy watching somebody enjoy such a mad subject. He's great!
Leonardo Acuna
I'm with u
*sees f(x)* *PTSD fires up*
Blan Morrison reminds them of awful high school math classes
Unit circle fires up my ptsd
💀
2pir made me habe flashbacks
yes i was there, a suicide bomber detonated during the exams @@katyameowmeow
"...it could be thrown out from theory"
*lies back*
"...they make a comeback"
*goddammit*
"You're not fooling me Sonny...It's Turtles all the way down!!"
2 years later and I finally realize why the base of the triangle isn't infinitely long.
It has infinitesimals which is an infinite number of slices. Each slice can be sliced in two again, meaning you can never run out of slices, so why isn't the base infinitely long?
Well, every time you slice the slices, they'll be half the size, so the length of the base have not changed at all. Stacking two triangles with a base of 1 is the same as stacking one of length 2 or 4 of length 0.5, so it doesn't matter how many times you split it, the length will always be the same. So the length is the same, the height is always the radius and therefore the area will not be infinite.
the triangles go around the circle so when you add up the triangles bases it equals the circumferences of the circle
so imagine the circumference is 3
the 3 is divided infinitely so the base of each triangle is 3/(inf) now to form the triangle we want to add all the base together and because the base of each triangle is the same we can multiply by how many triangles there is which is infinity
so we get (3/inf)*inf the 2 infinitys cancel and we are left with 3
Welcome to the Super Task
Welcome to convergent infinite series. Heard of Zeno's paradox?
I yearn for the day I can say "Yo guys have you heard, infinitesimals made a comeback!" and have people look at me like I finally lost it
Try watching this in 0.50 speed! at the start he looks so DRUNK !!! XD
XD
+Niels Ponssen "the smawwwlllllest thinnnnggg you can havvvvvee"....[pinch pinch pinch] LOL
+Niels Ponssen Drunk math explination?! BRILLIANT.
Its like that Drunk in history thing, but with math!
PS watch it at 1.5 speed, he seems like he is cracked out.
haha
he is
"I have a circle"
Sorry, i would call that a blob ;)
"Now it turns into a triangle, you see that triangle"
Sorry, i would call that a blob ;)
1/(infinity)?
Please don't roast me, I am not smart.
I said you could represent pi in a fraction by saying pi/1
Infinity is not a real number, so it can't be used in equations :/ However, the number Aleph Null, the smallest infinity and is the cardinality of all real numbers, IS a number, but 1/Aleph Null is strangely still Aleph Null.
You can use non real numbers in equations. I use "i" all of the time in mathematics class.
Derrick Barnes No, I mean that infinity is not a number. At all.
***** isn't i also not a real number... Imaginary.?
Whenever im super tired or need sth to entertain my soul, i watch the clips of this channel. Thank you :-)
Because of him i changed my major. I was only 14 years old when i had to make the decision, now 3 years later iam happy i met him.
what if a rectangle had an infinite length but an infitesimal width?
Joel Tailor if my understanding works. The area would be one. But I don't know if having infinites makes it work differently.
You can't have a rectangle with infinitesimal width because it wouldn't be a rectangle anymore. However, you can take the limit of a rectangle as its length approaches infinity and its width approaches 0. Then, depending on how the limit is approached, the rectangle would have a finite area. The Dirac delta function is basically a limit of a rectangle approaching infinite length and infinitesimal width, and has an area of 1. en.wikipedia.org/wiki/Dirac_delta_function
randomwindowsstuffz I'm going to act like I understand the Wikipedia article and say thanks
Pi
Indeterminate form. Infinity times 1/infinity is equal to infinite/infinite (one of 6 indeterminate forms).
Is no body going to talk about the picture at 04:22?
The basketball player!!!
[ 4:20 + 0:02= ? ]
39?
I had to Google it, but the 39 on the guy's jersey + mentioning Kepler might be a reference to the brown dwarf star Kepler-39.
Edited to add: someone else on here ID'ed the guy as Australian footballer Warwick Capper.
Yeah whats that about....
I enjoy the work of Numberphile - Dr Grimes et al. You guys really love your work and it's infectious. Thank you.
1/infinity ?
+Dominic London is epsilon
+Dominic London Supposedly it equals 0 in the same way 0.999... is 0. Except it's what an infinitesimal is. And 0.999... isn't actually 1, as ViHart has shown mathematically, and all of the proofs basically rely on oversimplified concepts.
+Abandoned Void
The question I would pose to ViHart is: Within the strictly real numbers, what is 1 - .999...? I think you are forced to say 0 because there is no number in between the two... no difference, and therefore they are the same number when talking STRICTLY about the real numbers.
As for using infinitesimals, that may be a game changer.
noah schaefferkoetter Sure, but restricting it to only real numbers is oversimplifying math
Is it though? It's still a complete and self contained system. Answers to questions such as these should still be able to be answered. I think that's what's causing most of the debate.
This guy is awesome he literally breathes life into Mathematics for those who hate maths, now find out it was probably just the class you was in.
Even though I have been out of college since 1989 when I got my BS in chemistry, this guy might have made me change my major to math.
Am I the only person that gets chills from the idea and sound of marker on construction paper!?
So how does this change that old problem of: "Is 1 equal to 0.9999-repeating?"
If they are an infinitesimal apart then in the hyper-real system they are not equal?
they aren’t an infinitesimal apart, they are the same number. it is just an artifact of a base 10 number system.
.3 repeating is what? What about that number multiplied by 3?
@@ar_xiv never thought of it like that
Ok so we all agree that for every integer execpt zero n/n = 1. So 1/3 = 0.3 repeating. 2/3 = 0.6 repeating. So 3/3 has to be 0.9 repeating right. But every n/n = 1.... thus 3/3 = 1 = 0.9 repeating
Yea, I don't like that saying. Topologically speaking, 0.9 repeating is in the open interval (0,1), while 1 is not.
Before starting video, I was thinking, oh that's so simple. This guy's going to teach us about limits. Lim(1/X) ,X→∞
Then as the video started, "oh is it something different? Seems like he is going towards integration
by the end of the video, I'm happy, and also realized I'm rusty. Thankyou
Amazing!!
+Strength And Philosophy nice channel
+Amanda Vicious thanks!
I subbed to your channel bruv.. not really into that motivational BS, but I love your approach... guys check this guys ch out!
+johnathan1784 Thanks Johna - appreciate that!
If you alternate the orientation of the small triangles, you don't need to stretch them. You end up with a regular rectangle with height r and base pi r. Simpler and more convincing than the stretched triangles method.
"1/0 is not infinity, we would never do that"
Me studying stability of transfer functions using final values: 👁️👄👁️
Any finite value is infinitely small relative to infinity
Literally every mathamatician
MY DRAWING IS BAD
I love the fact the original mathemations decided to just ignore the curve parts
Wow, i remember watching these videos as a kid, not understanding literally anything, now im on calc 3 and this is a really nice summary!!!! It really shows your teaching abilities when you can captivate a kid with no knowledge or conception of calculus and do it again to the same person many years later!!!!! Thank you
Infinite opposite = finite
+gunup2 But what is "finite"?
+Thomas Korbacher Finite means it ends eventually
Thomas Korbacher limited
Ye i know, but you cant give "infinite" a symbol like inifinity has
+Thomas Korbacher the question is, where and when would u ever need our use one? (If you can give me an example, a LEGIT & REALISTIC one tho) I'll give you a cookie!
actually the opposite of infinity is something that ends. because infinity doesn't have an end. for example a hotel with infinity floors and a hotel with 6 floors. 6 is where it ends
***** Infinity isn't a number. infinity is unlimited, endless. So how can infinity have an end? Because if it does, it destroys the definition of infinity.
+zivanni 0 is the opposite, imagine a glass of water, an empty glass is 0 and a full glass is infinity.
+The Homunculus if it was that easy they would have know that by now
koen bruske
So simple logic is false, because it's simple?
No simple in this concext is false
This guy has the brain of The Brain, but the voice of Pinky 😂😂😂 Narf!
Ahh the pinky and the brain.
Very famous cartoon here in India during early 2000s😇
Similar physical attributes as pinky also hahaaaa
James Grime: ”2x ≠ x.”
0: ”Allow me to introduce myself.”
The entire thing I was just waiting for the next time he says “area” he holds out the a and it’s awesome
Thank you Fred Weasley, these videos are really interesting.
What the haha
I used to hate maths until I discovered your channel, thank you! ♥️
Interestingly using the slices underneath a curve in calculus is a very similar concept to how we record audio digitally. You take little rectangular chunks of the sound wave in exactly that way missing little bits at a time and convert them to bits of information. Any wonder why 8bit music sounds like that? It's because the rectangles used are really big so much of the sound is missing. Science and maths will always be best mates.
Then there is the classic:
Set n = 0.999 repetend.
Multiply both sides by 10 (10n = 9.999 repetend).
Separate the right hand side (10n = 9 + n).
Subtract by 1n (9n = 9).
Divide by 9 (n = 1 = 0.999 repetend).
1 - 0.999 repetend is zero.
Infinitesimal is zero.
+Lyle Lange At no point in that problem did you deal with infinitesimals at all.
You're forgetting that 9.999... has, however possible, one less 9. It' be more like: n=1-(1/k) as k approaches infinity. (still that .9999 number) 10n=10-(10/k)
10n=9+1-(10/k)
-(1-(1/k))
9n=9+(-9/k)
n=1-(1/k)
Nice try
George Salafatinos
No, that's just not how it works. There is no "one less 9" to an infinite decimal expansion. Besides the wonky bit about infinitesimals, this is a very standard demonstration of 0.999... = 1.
Even using your starting point and saying that n = lim(k->∞) 1 - 1/k, you get:
n = lim(k->∞) 1 - 1/k
= (lim(k->∞) 1) - (lim(k->∞) 1/k)
= 1 - 0
= 1.
If you were already convinced that 0.999... = lim(k->∞) 1 - 1/k, you don't even need the algebraic proof.
So is it true then that 1 - 0.999 repetend is not equal to zero?
Lyle Lange 1 - 0.999... is equal to zero. Your original post was correct besides saying that an infinitesimal is equal to zero. The problem is that 1 - 0.999... does not equal an infinitesimal.
I had no idea that this had been controversial. They taught us infinitesimals in school as normal math.
But how is a limit approaching zero NOT an infinitesimal?
i would argue that it is. logically speaking
@Philip Moseman its aproching infinty bro
@Philip Moseman well actually no we know for a fact sum of these numbers is 1 ( numbers don't exist in time )
cool, so this was how they calculate areas of curves before integration
This is the basis for integrals.
If you have ever proven integration you know what I mean.
+Marcello Morales This is actually how they still calculate areas of curves in most practical situations, because usually you don't have closed form equation to work with and even when you do, it's usually not easy to calculate the integral. But computers can very easily apply this to find the area to any arbitrary accuracy, and so it's often the typical approach.
+Marcello Morales Actually this is the concept of integration.
+Marcello Morales The expression they came up was de Derivate of the Primitive function - it is the basis for integration...
Here is my answer for the opposite on Infinity:
it's any value that represents half the way to the next value!
Let's say you want to cross a street and every step you take is half the way to the other sidewalk => you'll never arrive!
So the opposite of Infinity is a Regressive Infinity !
Seems like you will actually arrive. Go watch their video about Zeno's paradox, I just watched it, and it talks about how the infinite sequence which gets halfed each time is well behaved and thus actually has a real sum.
@@didibus thanks for the tip: but notice that my assumption DOES NOT envolves the Time factor; which one can use to "solve" the paradox, but instead, my proposal in a clear and democratic concept:
if Infinity exists, one should assume that a Negative Infinity is also possible.
This is what I believe.
LAOMUSIC ARTS infinity (in most of mathematics) exists only as a limit of a function. As does negative infinity. The limit of tan(x) as x approaches pi/2 is infinity, and the limit of tan(x) as x approaches 3*pi/2 is negative infinity. What you described is a limit. So the limit of 2^-x as x goes to infinity gets smaller and smaller, halving and the limit is zero.
@@samburnes9389 In Mathematics, if one has a ratio (between the diagonal and the side of a square) that is irrational, it will be the limit of an endless, nonrepeating decimal series.
Have checked also the Zermelo-Fraenkel set theory ?
Recent work suggests that Cantor’s continuum hypothesis may be false and that the true size of c may be the larger infinity ℵ2.
6:20 You can’t get an *_ABSOLUTE_* volume; but, assuming the heights are same for those 2 stacks, you can get a *_RELATIVE_* or *_COMPARATIVE_* volume (”Stack A > Stack B” / ”Stack A = Stack B” / ”Stack A < Stack B”). Actually, those 2 stacks of pennies *_DON’T_* have the same height: The straight cylinder has 19 pennies, while the jumbled-up cylinder has 16 pennies. I *_KNEW_* they weren’t quite the same.
these videos are awesome. youtube can be used to educate.
So basically, there's an infinite amount of infinitely small things, which means these infinitely small things are, collectively, infinite in size.
That's wild.
Watching this video made me go back... Back to my first semester in college, when they threw us into a Calculus class without any care. I didn't hate Calculus, but Calculus is full of concepts that aren't intuitive at all. This video does a great job at explaining why those concepts aren't intuitive.
I failed that class and, in the very next semester, I took the class again and then I aced it. I didn't suddenly got smarter, I just understood those very basic (albeit not intuitive) concepts.
"Lets say i have a circle." - Thats not a circle its a potatoe...
+Lexus33 For one, nobody can draw a 100% perfect circle; for two, you're observing his circle from an angle.
4:20 dawhat? got the trikot-number 39.
whats the hidden meaning of this? O.o
I did a google search of the image. His name is Warwick Capper, an australian football player. I guess the joke is that the names sound so similar.
well makes sense :D So you snapshotted it and trew it into googles image search? I was in a hurry before and didn't felt like i would have time to do it, so well done!
Didn't know that numberphile is placing easter eggs anyways :P
took me ~10 attemps. didnt realized i could slow the vid down then XD
I'm on mobile so I couldn't slow it down. took me my third try. I got FCKING LUCKY DIDN'T I? O.O
Got it on first attempt..Sorry :D
Absolutely fictional idea, but something that could be fascinating to think about:
I am convinced that over time it'll be possible to proof anything. Like with enough effort and time, it's possible to find systems, concepts, theories, etc. that satisfy anything you want. Mathematics is discovered, and the predictions are discovered. The fact that we discovered that 1 apple + 3 apples = 4 apples before discovering mathematics, was just a happy accident, because that discovery is just very easy to make. So I believe that anything already exists, but it's so complicated or subtle that we don't realise it until we've discovered theory that predicts them.
Well to me it seems the counter-intuitive part of using infinitesimals to calculate area isn't exactly the "error" being ignored, but that two seemingly contradictory arguments are being made. First: all the tiny infinitesimal slices add up to some finite number(not zero), but also: all the tiny infinitesimal errors added up are somehow negligible(actually zero).
"I might use a different color" 😂😂😂
i am a musician.. i love physics but i hate math.... i cry after seeing this why Area of a Circle is ( Pi x R squre )
You even find infinitesimal spaces on graphs, for example the x and y axes on a graph of y = 1/x is in an infinitesimal space that doesn't intersect the curve at any point. If you moved them even slightly, they would intersect the curve but they don't.
At the beginning of the video, I thought the question of what the opposite of infinity is, was about the concept of infinity in general, and that the answer would be rather philosophical. Like if infinity pushes the limits of quantification-no matter how big or small-it's opposite would be something like nil. Not zero, just... non-existent. The same way infinity is not a number but rather a concept.
zero is a concept of its own sorts
In that manner, the opposite of infinity would be the empty set, which is ironic given that the Axiom of Infinity from Set Theory literally creates an infinite set from the empty set, the empty set union with the set containing the empty set, and so on.
And in a way, I think that the empty set is a better answer for the opposite of infinity than infinitisemals are.
If you watched this video you can sleep in and miss your first few weeks of Calculus