almost. it means u can fill the 3D shape with water or paint, but if u slice it in haldf u get an area u cannot cover with 2D paint. I wonder if u could cover it with 3D paint tho...
@@hwytube the length of the sequence of digits necessary to express π in base-10 (excluding any terminating 0's) to the right of the decimal point (which is the most popular numeric base system, and probably the most popular method of representing arbitrary numeric quantities) is countable infinity. Even though countable infinity can be, in my opinion, represented by any number which has infinite digits before terminating zeros going from the decimal point to the left, its not the same thing as having infinite digits before terminating zeroes to the right because to the right of the decimal point the exponent of the digit decrements such that the nth digit gets raised to the 10⁻ⁿ power so it gets smaller and smaller faster then if u keep adding and adding the smaller numbers so that it converges to the number,
at 5:10 she said, the area is diverging but... if you add all the diverging areas then they converge..... then she said i think it is cool. 1/x was the best function she chose, this is in fact true. you can think of 1/r as a density fuction with a singularity but if you try to integrate it in the spherical coordinate, the volume integrand term r^2 sin (theta) goes to 0 fatser than the diverging density function, 1/r, it becomes even physically and scientifically possible. it means basically there is nothing left to integrate around r=0
@SparkleyEyes08 I LOVE CHALKBOARDS. Also, it's MIT. The chalkboards are a traditional part of MIT. I hope I get into MIT so I can see those chalkboards one day.
a segement of something infinite (or not; doesn't matter) is finite. infinity is the infinite cherry pie, 'infinity pie', a nice wedge of it is a subset. dig in.
If you're going to do that then the correct cross section to consider is PI/x^2 which is finite. Otherwise all this is saying is the object has infinite length but finite volume.
randall williams you would get 3.142857.... if you consider the fraction 22/7. pi is not exactly 22/7: it is in fact (21.98....)/7, so you would get 3.14159265... . hope that helped
right now i'm having either an epiphany or a very stupid idea : maybe we could only paint the volume of the horn that is present in the first three dimensions of space, but we can't fill the other 7 dimensions mentioned in string theory because they are infinite (i know that what i wrote is very likely some bullshit but if it's true it will change how we see the world )
agent47 no u silly it has nothing to do with the limitations of physical space. agent47 the reason that this happen is because in 3D the measuring unit is a cube, and in 2D its square, and a cube has a bigger lebesgue mesure then a square alos theres other shape that have infinte perimeter but finite area for instance Koch Snowflake. any programmer who knows fractlas and watch 3Blue1Brown on youtube knows that :)
I've seen a lot of comments on this video that are completely incorrect, and I'm worried students learning calculus for the first time will be misled. The "paradox" is that the solid formed by rotating y=1/x with x>=1 around the x-axis has finite volume but infinite surface area. This is seemingly contradictory when you consider that filling the solid up with paint will naturally paint the entire surface of the solid (i.e. using a finite amount of paint to paint an infinite area). But there is no paradox--realize this problem was solved in the 17th century, when calculus was in its infancy. The correct interpretation is that as you fill up the horn starting from the bell and working towards the mouthpiece, the diameter of the horn thins at a rate determined by the derivative of y=1/x. The rate of thinning is fast enough such that the volume of paint converges to the quantity π.
Jiangfeng Chu The way I remember it being explained was the thickness of the paint actually gets infinitesimally small to catch up with the surface area.
Evaluating Gabriel’s Horn Paradox . Area of horn= infinite square units. . Volume of horn= pi cubic units. . Paint required to cover the area= infinite × thickness approaches zero = zero cubic units. . The paint of pi cubic units will cover the inner side with zero amount of paint ( infinite × thickness approaches zero ) and fill the horn with all amount of pi. . Note: no surprising for an object has area bigger than volume, for instance, a cube with one unit length, has one volume unit and six area units. .The misunderstanding of Gabriel's Horn has existed after We had compared volume unit with area unit. We should not compare different units.
". Paint required to cover the area= infinite × thickness approaches zero = zero cubic units. . The paint of pi cubic units will cover the inner side with zero amount of paint ( infinite × thickness approaches zero ) and fill the horn with all amount of pi." Got it. No paint to paint that outer, non-zero area. And pi amount of paint to not paint the inside. So at the mouth of the cup, there is zero amount of paint, implying that I could then add more paint, I could repeat this forever, is there a hole in your horn or what?
Pi equal even floating point positioning lenkment additionally diffrencial p0 Proof π=point zeta function 0/1 Non alphabet 0 P Point One Line Point Proof P Line P Cross Proof Imunitation Logic slash zero equal moduler 0 ≡ 3 Proof
Would you be so kind Miss Breiner and explain this to me in conceptual terms. What occurs to me is…IF the surface area which is a boundary, is infinite, so must the volume it bounds, be. I don’t care what the math says. Here you have a conceptual contradiction from which you are proceeding by which I would think, you would know that the math you are applying is wrong. A boundary is a contingent phenomenon. It is the edge of a volume or area. IF you wish to claim that a boundary can be infinite but that which it “bounds” is finite as with Gabriel’s horn then you would have an infinite amount of boundary bounding nothing. This is a contradiction from which you are proceeding with your math and thus, your results cannot be but false. I have been discussing this with others who claim that there are other examples of infinite boundaries and finite volumes which I find to be pure sophistry. For example, a 1 x 1 box, subdivided 50% each time, infinitely would enclose a finite volume of 1 x 1. If you have a box and keep subdividing it by half, infinitely, the volume remains finite as the boundary internally grows to theoretical infinity by those divisions. But this is NOT even close to being the same scenario as Gabriel’s paradox. This example is NOT a boundary in infinite extension, but rather a boundary which is itself subdivided infinitely. Quite a different phenomenon. The boundary of the volume of the original 1 x 1 square is finite as is the volume within it. Then, this square is subdivided by new boundaries within the boundary of the original square which is still finite and bounds a finite volume. You are redefining the boundary over and over forever (which is materially impossible and only a theoretical consideration) each segmentation of which steals a part of the volume for itself when in fact that volume taken belongs to the original square boundary. Again, here one employs sophistry to win your argument and compare apples to elephants. Would you be so kind as to offer any thoughts?
Liyuan Boo you are wrong. If anything, the 3D analogue of PI in 2D is Infinity but still, just no. ruclips.net/video/D2xYjiL8yyE/видео.html Alos check out this proof that PI=4 get disprovedl same area as a circle but more perimeter.
Asymptote well said circumference/diameter is actually an rational number but some mathematician do consider π as irational number that's kinda mindfuck though
Professor Breiner, thank you for another well explained video.
The chalk..its so thick and easy to the eyes O_O
Don't you mean the professor thiccc
ah a man of culture.
Size matters.
this means you can fill this area with paint, but can't simply paint its sides. craziness.
almost. it means u can fill the 3D shape with water or paint, but if u slice it in haldf u get an area u cannot cover with 2D paint. I wonder if u could cover it with 3D paint tho...
Isn't pi not finite ..3.1416........
@@hwytube 😒
@@hwytube the length of the sequence of digits necessary to express π in base-10 (excluding any terminating 0's) to the right of the decimal point (which is the most popular numeric base system, and probably the most popular method of representing arbitrary numeric quantities) is countable infinity. Even though countable infinity can be, in my opinion, represented by any number which has infinite digits before terminating zeros going from the decimal point to the left, its not the same thing as having infinite digits before terminating zeroes to the right because to the right of the decimal point the exponent of the digit decrements such that the nth digit gets raised to the 10⁻ⁿ power so it gets smaller and smaller faster then if u keep adding and adding the smaller numbers so that it converges to the number,
But if you fill it with pain then the sides will have to be painted
at 5:10 she said, the area is diverging but... if you add all the diverging areas then they converge..... then she said i think it is cool. 1/x was the best function she chose,
this is in fact true. you can think of 1/r as a density fuction with a singularity but if you try to integrate it in the spherical coordinate, the volume integrand term r^2 sin (theta) goes to 0 fatser than the diverging density function, 1/r, it becomes even physically and scientifically possible. it means basically there is nothing left to integrate around r=0
@SparkleyEyes08 I LOVE CHALKBOARDS.
Also, it's MIT. The chalkboards are a traditional part of MIT. I hope I get into MIT so I can see those chalkboards one day.
Do a thought experiment. You have 200 people in lecture. The computer freezes. The projector fails. Power goes out. Chalk boards still work!
a segement of something infinite (or not; doesn't matter) is finite. infinity is the infinite cherry pie, 'infinity pie', a nice wedge of it is a subset. dig in.
Wait, they teach this course at MIT? Hasn't everyone at MIT already gotten through this during HS?
This is a course for English Lit majors at MIT :-)
Pi is finite but irrational. It lies between 3.141592653 and 3.141592654.
randall williams the digits go on forever and ever, I think it’s infinite. I respect what you are saying but I don’t agree 😝
nice tutorial :0
If you're going to do that then the correct cross section to consider is PI/x^2 which is finite. Otherwise all this is saying is the object has infinite length but finite volume.
That’s an MIT professor, who are you?
Hmm but isn’t pi an infinite decimal so it technically is not a finite volume?
@SparkleyEyes08 Their chalkboards are really nice :O
Not I
Where did you get this approximation for pi? My ten digit calculator gives 3.141592654 - not 3.142857.
randall williams you would get 3.142857.... if you consider the fraction 22/7. pi is not exactly 22/7: it is in fact (21.98....)/7, so you would get 3.14159265... . hope that helped
That is actually a better approximation than yours
right now i'm having either an epiphany or a very stupid idea : maybe we could only paint the volume of the horn that is present in the first three dimensions of space, but we can't fill the other 7 dimensions mentioned in string theory because they are infinite (i know that what i wrote is very likely some bullshit but if it's true it will change how we see the world )
agent47 no u silly it has nothing to do with the limitations of physical space. agent47 the reason that this happen is because in 3D the measuring unit is a cube, and in 2D its square, and a cube has a bigger lebesgue mesure then a square
alos theres other shape that have infinte perimeter but finite area for instance Koch Snowflake. any programmer who knows fractlas and watch 3Blue1Brown on youtube knows that :)
I've seen a lot of comments on this video that are completely incorrect, and I'm worried students learning calculus for the first time will be misled. The "paradox" is that the solid formed by rotating y=1/x with x>=1 around the x-axis has finite volume but infinite surface area. This is seemingly contradictory when you consider that filling the solid up with paint will naturally paint the entire surface of the solid (i.e. using a finite amount of paint to paint an infinite area). But there is no paradox--realize this problem was solved in the 17th century, when calculus was in its infancy. The correct interpretation is that as you fill up the horn starting from the bell and working towards the mouthpiece, the diameter of the horn thins at a rate determined by the derivative of y=1/x. The rate of thinning is fast enough such that the volume of paint converges to the quantity π.
Jiangfeng Chu The way I remember it being explained was the thickness of the paint actually gets infinitesimally small to catch up with the surface area.
Evaluating Gabriel’s Horn Paradox
. Area of horn= infinite square units.
. Volume of horn= pi cubic units.
. Paint required to cover the area= infinite × thickness approaches zero = zero cubic units.
. The paint of pi cubic units will cover the inner side with zero amount of paint ( infinite × thickness approaches zero ) and fill the horn with all amount of pi.
. Note: no surprising for an object has area bigger than volume, for instance, a cube with one unit length, has one volume unit and six area units.
.The misunderstanding of Gabriel's Horn has existed after We had compared volume unit with area unit.
We should not compare different units.
". Paint required to cover the area= infinite × thickness approaches zero = zero cubic units.
. The paint of pi cubic units will cover the inner side with zero amount of paint ( infinite × thickness approaches zero ) and fill the horn with all amount of pi."
Got it. No paint to paint that outer, non-zero area. And pi amount of paint to not paint the inside.
So at the mouth of the cup, there is zero amount of paint, implying that I could then add more paint, I could repeat this forever, is there a hole in your horn or what?
Pi equal even floating point positioning lenkment additionally diffrencial p0
Proof π=point zeta function 0/1 Non alphabet 0 P Point One Line Point Proof P Line P Cross Proof Imunitation Logic slash zero equal moduler 0 ≡ 3 Proof
Would you be so kind Miss Breiner and explain this to me in conceptual terms. What occurs to me is…IF the surface area which is a boundary, is infinite, so must the volume it bounds, be. I don’t care what the math says. Here you have a conceptual contradiction from which you are proceeding by which I would think, you would know that the math you are applying is wrong.
A boundary is a contingent phenomenon. It is the edge of a volume or area. IF you wish to claim that a boundary can be infinite but that which it “bounds” is finite as with Gabriel’s horn then you would have an infinite amount of boundary bounding nothing. This is a contradiction from which you are proceeding with your math and thus, your results cannot be but false.
I have been discussing this with others who claim that there are other examples of infinite boundaries and finite volumes which I find to be pure sophistry. For example, a 1 x 1 box, subdivided 50% each time, infinitely would enclose a finite volume of 1 x 1. If you have a box and keep subdividing it by half, infinitely, the volume remains finite as the boundary internally grows to theoretical infinity by those divisions. But this is NOT even close to being the same scenario as Gabriel’s paradox. This example is NOT a boundary in infinite extension, but rather a boundary which is itself subdivided infinitely. Quite a different phenomenon.
The boundary of the volume of the original 1 x 1 square is finite as is the volume within it. Then, this square is subdivided by new boundaries within the boundary of the original square which is still finite and bounds a finite volume. You are redefining the boundary over and over forever (which is materially impossible and only a theoretical consideration) each segmentation of which steals a part of the volume for itself when in fact that volume taken belongs to the original square boundary. Again, here one employs sophistry to win your argument and compare apples to elephants.
Would you be so kind as to offer any thoughts?
Bro you are schizo schizoing out
@@caleb.39 but you cant comment or refute what i posted? Isnt that typical.
@@jamestagge3429 please send 10k to this Bitcoin address 1J7MCJcY18my8Tuc8q2nEB8hjvdfw8oMXA I'm down bad
Ez pz
it's 2011, don't you think chalkboards are a little out-dated MIT?
Spaekley Lamp reaf the other comments??
I think Pi is infinity, so the volume is infinity.
Liyuan Boo you are wrong. If anything, the 3D analogue of PI in 2D is Infinity but still, just no. ruclips.net/video/D2xYjiL8yyE/видео.html
Alos check out this proof that PI=4 get disprovedl same area as a circle but more perimeter.
But the volume is not infinite.
Sebaka & Co. get off Facebook and go back to school.
technically Pi is infinite, ie, it never stops so volume is also infinite.....
Asymptote well said circumference/diameter is actually an rational number but some mathematician do consider π as irational number that's kinda mindfuck though
That is 100% false.
my iq dropped to -10 when I read this.
just two constants, infinite volume and infinite dwell ("time", the carbon units on this planet call it).
.... Pi itself isn't equal to infinity