Infinite Surface Area but Finite Volume!?!? *Gabriel's Horn*

Gabriels' Horn - aka Torricelli's Horn - is one of my favorite examples in Calculus. This is a region of revolution where the surface area is infinite but the volume is finite. How cool is that! In this video we recall the formulas for surface area and volume of regions of revolution and compute out the improper integrals, in one place having to apply the Comparison Test.
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[@chasereiter4760]

My favorite thing about this video is how smoothly one idea ran from the next and how simply everything was explained. Well done

[@maxlen234]

So you can paint an infinite surface with a finite amount of paint?

[@roejogan2693]

Theoretically it's possible

[@tubax926]

@@roejogan2693 It is possible if you can use fractions of atoms at some point because you will eventually get to a point where the volume of the sections are smaller than the atoms. But at that point the compound stops functioning as you split it up and loses its paint property.

[@roejogan2693]

@@tubax926 that's why I said theoretically

[@tubax926]

@@roejogan2693 yeah even then, no

[@roejogan2693]

@@tubax926 atoms don't exist in mathematics

[@aggualaqisaaqs3272]

Absolutely mind blowing! Math never ceases to amaze me This kinda makes sense if you imagine the horn looking more and more like like an electrical wire as you go out to infinity with the surface area being something like the “sheath,” the. it’s easy to conceive of an infinitely-long wire with “nothing” inside.

[@KOMATH]

Your videos are awesome! I hope I can make such quality videos in the future.

[@DrTrefor]

I hope so too!

[@rahadiannow]

I dont understand calculus, but i understand the meaning

[@ThatJosiahGuy]

Maybe this is a vast oversimplification, but the way I understand calculus (conceptually): How much space does a thing take up? → integral How much does a thing curve or bend? → derivative Are these two related somehow? → "opposites" (i.e. the fundamental theorem of calculus)

[@UTU49]

@Aaron That's not bad at all. I think a simpler, clearer, more practical description of Calculus is that it is simply about “speeds”… or generally, “rates of change”. Anything changing: distance, speed, temperature, surface area. When you are waiting to turn left at an intersection and you are watching the oncoming car and you are trying to decide if you have time to turn before the car gets to the intersection... THAT'S Calculus. You're comparing their speed with the distance they need to travel, and deciding whether you have time to safely and comfortably get through your turn. Simple examples: If you travel 10 miles in 2 hours, how fast were you going? 5 miles per hour. If you drive 30 miles/hour for 3 hours, how far did you go? 90 miles. These are both Calculus questions. The reason why most people understand these situations is because they are the simplest possible Calculus questions. Everyone knows a little bit about how speeds and distances relate, from life experience. If you have information about an object's location, or distance, (as an equation or graph) and you want to know about its speed... that's a derivative. If you have information about an object's speed (equation or graph), and you want to know information about its location (or distance), that's an integral. TLDR: I think the place to start with Calculus is looking at the relationship between how fast an object is traveling with how far it has traveled. If you have information about one (and a length of time), you can calculate information about the other.

[@tomificationness]

Great video!

[@Alienrookie]

Great job!

[@aerospaceprincess]

Thank you Sir!

[@AndrewPa]

well done mate ... will subscribe to your channel

[@JGamingness]

It kind of makes sense to me like this: the surface area will just keep going forever, getting infinitesimally small as it goes on, but as it does that the inside will also get smaller. So small to the point that not even a single quark could fit inside it, which is where the volume would converge, being as it can no longer hold any more of anything.

[@spookynoodle3919]

This is actually incorrect because this is a purely mathematical structure and it couldn't physically exist.

[@jees9179]

This is insinuating that some part of its outer surface is actually smaller than a quark which is practically not possible.

[@aryaa7334]

it's sad that i now understand this. as a middle schooler i used to be just a smarty pants who pretended to know these kinds of things. how time flies :(

[@Llaveroja27]

How do you know you're not still pretending 😑

[@tubax926]

You're till a "insert current degree of eduation" pretending who just knows other kinds of things. There's always something you don't know but you think you know.

[@aryaa7334]

@@Llaveroja27 because the idea of the video is part of A levels and i had to do a part of it.

[@ronaldmcdo5516]

You still do dumbass

[@RF-fi2pt]

I see other object, with finite value at 2 dimensions and contains an infinite at 1 dimension: circle have finite area to one given R (although the precision is given by the π decimals). One swirl line starting at center until that R have 1D Length infinite, as the line diameter is infinitesimal. The Integral from 0 to R of 2πr, gives exactly πR^2, but trying to see the integration process as that increasing swirl see the 1D line Length going to infinite. At Gabriel Horn is the same. 3D finite Contains a 2D infinite.

[@19EHF]

So its like folded space. In that case if i wanted to fit something inside like a city could you do that?

[@ThatJosiahGuy]

It's not exactly folded space... think of it like this: Say you walk into a bar and order a beer (assuming you're of age, lol). So the barkeep pours you one. Another person comes in and asks for half a glass, so she opens a 2nd bottle and pours him half of that. A different person comes in and asks for a quarter of a glass, so she pours her half of what was left. Now, say you've got an infinite line of people at the door, and each is only asking for half of what the person before them got. How many beer bottles did the bartender have to open to give everyone? Just 2, right? She gave you the first one, but each person after only got half of what was left by the previous person from that 2nd bottle. So an infinite amount of people drank a finite amount of beer, namely, a total of 2 bottles. This is just like Gabriel's horn. The infinite line of people is like the surface area, and the 2 beers are like what's inside the horn.

[@theleviathan3902]

How can it have a finite volume if the area under 1/x from 1 to infinity diverges?

[@ibrahimanaz2424]

İs there a region of revolution where the surface area is FİNİTE but the volume is İNFİNİTE? if there is not, how can we prove it?

[@user-ez2di8lv7q]

You can prove that it is impossible for any revolved continuous function on a closed set to have finite surface area with infinite volume. Hope it helps !

[@csdufoupdates5502]

Awesome 🤙 Example

[@DrTrefor]

Thanks! 😃

[@UTU49]

Chris and Pat both have a Gabriel's horn. Chris is going to fill the horn with paint. Pat is going to paint the inside surface. Both have 4 cubic units worth of paint. Chris will never run out of paint while filling their horn. Pat WILL run out of paint while painting the inner surface. These are the ramifications of this situation. I think it's a good thing that an infinite Gabriel's horn does not exist. :P _______ Note: If the numbers on the axes are cm, then the volume units will be cm^3, which happens to be the same as mL, etc.

[@toomanycharacter]

I don't think you can use cubic units as square units in pure math though. And if you try to, then you would probably get an infinite amount of square units of paint, since area is infinitely thin. Funny joke tho

[@VulcanOnWheels]

I'm sorry to say, this went over my head.

[@jamestanny849]

Gabriels horn explains when you take a divine item and places it within our time and space encapsulation in which it should not exist

[@danielebucchi8018]

Is it a regular surface?

[@abuzarmahmood96]

I have not understood why for volume the formula is integration of pi.square(f(x)).dx it should have been integration of pi.square(f(x)).sqrt(1+square(f'(x))).dx

[@avishrohil929]

what software did you use to draw the graphs?

[@avishrohil929]

@@DrTrefor omg thanks for the fast reply, you are a literal life saver

[@gridcaster]

@@avishrohil929 you won't share it with all of us??

[@avishrohil929]

@@gridcaster he deleted his comment, i forgot what he wrote, sorry. But i ended up using geogebra

[@gridcaster]

@@avishrohil929 thanks...i've been looking for a good app forever...you'd think there would be tons of them for free out there, but there aren't.

[@marshallodom1388]

You're changing directions with those formulas when you do your measuring. From a revolving plane to disks. The same reason pi as an infinite number after the decimal is also why the surface area is infinite. The ratio and the volume are infinite.

[@morbideddie]

Pi is not infinite, if I draw a circle with a 2m diameter I can’t fit North America in it.

[@riordanblackwood]

Correct pi is a finite number that is Infinite after the decimal point, meaning irrational. Because the ratio of pi never ends the walls of the horn become infinity thin. While there is an edge, making the volume finite, there's always room for a little more. Don't tell Planck though. But since it's not a real object you can't use an example of filling it with a real thing like paint either.

[@OBGynKenobi]

Pi never ends, right? So doesn't that mean that everyone digit is a new amount of volume, ad infinitum?

[@morbideddie]

It has an infinite decimal expansion but the value is finite since that sequence of digits converges. No matter now many you add it will never be 3.15, even infinite digits later. It’s like the sequences 1 + 2 + 3 +… and 1 + 1/2 + 1/4 +… both are infinitely long but the former equals infinity and the latter equals 2.

[@OBGynKenobi]

@@morbideddie ok now I see it. Thanks.

[@g-ray4088]

title is your first reaction to fractals

[@Efrain0465]

Qué programa usa ?

[@carultch]

El maestro esta en Canada.

[@shifagoyal8221]

Interesting

[@gamecoolguy619]

Ummn the area at x=1 is pi... So unless pi + {fraction of pi} = pi Which based on how your limit works is true, but logically you agree with this?

[@gamecoolguy619]

@@DrTrefor the 1 is based on your integration including it in your volume from 1 to infinity. Is that not how volume works in comparison to area of 2D shape anyway (not talking about surface area) www.wyzant.com/resources/lessons/math/calculus/integration/finding_volume (Assuming what is said here is correct) "To find the volume take the vertical slices of the solid (each dx wide and f(x) tall) and add them up" So this include the vertical slice at x = 1 so pi(1)^2 = pi (it is a circle when u rotated it 360° right?)

[@gamecoolguy619]

@@DrTrefor hmm I see

[@gamecoolguy619]

@@DrTrefor What about graphs such as e^-x or n! (but in reverse) this would be the same results but I would be curious if the answer comes out to be interns of pi or something else for especially for e^-x?

[@gamecoolguy619]

@@DrTrefor Hi I was watching this: kzbin.info/www/bejne/h32re59olMaAa7M And I was wonder why can we not just use 2pir for the surface area as all we are doing is adding circumference So: Integrate(2pif(x)) at least in this case?

[@gamecoolguy619]

@@DrTrefor Hi when you integrate a function from a to b Which result in f'(b) - f'(a) Isn't that always the same as Integrating from 0 to b subtract integrate from 0 to a. From the drawings we use that seems the case but it doesn't seem like we always follow this rule?

[@abhishekpattanshetti6771]

Why does the width in case of volume problem is dx but ds in case of surface area?

[@abhishekpattanshetti6771]

@@DrTrefor but why the different approach? Aren't u loosing some volume by choosing to chop x instead of the arc??

[@THEMATT222]

Noice 👍

[@capofigo]

Yeaaaah i got the same results with guldino let's go babyyy this is what it's all about woooooooooo

[@hgfuhgvg]

what about painting it from inside? (by filling it with paint)

[@nosuchthing8]

That's the paradox. You can fill the horn with paint. But there is not enough paint in the universe to paint the inside.

[@Alllgr]

@@nosuchthing8 And of course that makes no sense. How can the horn be full if parts of the inside aren't being touched by paint? This seems more like the kind of Reductio Ad Absurdum that tells you you have make a mistake somewhere.

[@nosuchthing8]

@@Alllgr it's not MY paradox. It's the mathematicians paradox. The only way I can make any sense of it is to say that the time it takes to fill Gabriel's horn is infinite, because the thing is infinitely long. So you cant ever see it filled to the brim nor can you ever see it painted on the inside. Maybe two types of infinities?

[@Alllgr]

@@nosuchthing8 But one isn't infinite at all, it's just pi. The horn will overflow because there is no room for more paint, but somehow parts of the interior wall haven't contacted paint yet? That is a contradiction, no matter who's paradox it is.

[@nosuchthing8]

@@Alllgr I agree with you. The problem is where is the error? Excellent post by the way.

[@henrikantonsson2460]

Gabriels horn is the reason why I dont believe integrals and approximations can lead us to 5he theory of everything.

[@aidencamilleri3372]

Is the volume really finite if pi is an irrational number? Instead of the volume being finite, as x approaches infinity, pi is just becoming more accurate and adding more digits past the decimal.

[@MuffinsAPlenty]

The fact that pi is an irrational number doesn't really have any bearing on the "paradox". Pi is finite. This statement might be surprising since a lot of people have heard that pi is infinite. But we have to be careful what we're talking about here. Yes, pi has infinitely many digits when written in decimal notation, but pi is finite _in value._ Indeed, pi is less than 4. When we're talking about the surface area being infinite here, we mean infinite _in value._ Why should we suddenly switch to using infinite _in number of digits_ when talking about the volume? We shouldn't. We should keep both things in terms of value.

[@joe.007]

yes but isn’t pi infinite as the surface area increases the volume will increase in other words when the area becomes infinite so does the volume (reaches 3.14… till infinity)

[@morbideddie]

Pi if a finite value. It has an infinite decimal expansion but it is a real number.

[@Urbanstrangler]

Im fairly sure if you built this it would be a cognitohazard

[@tubax926]

you can't build this it has an infinite length

[@gridcaster]

This paradox has been bothering me on and off for several weeks. The big sticking point for me is that the volume is equal to pi...which means the shape will be filled perfectly with exactly pi units of paint. Why does this bother me? The exterior surface area of the horn is equal to the interior surface area of the horn. So, if the exterior surface area is infinite so is the interior surface area. If I can fill the horn perfectly with a finite amount of paint then how can I not also be covering the interior surface area of the horn with paint at the same time?!?! My brain is bothered and hurt and I just want to know why....

[@isaacbruner65]

You can't fill the horn because it would take an infinite amount of time to cover an infinite surface. You'd be pouring forever.

[@user-ez2di8lv7q]

The apparent paradox could be resolved if you allow the paint to get infinitely thin. For example, imagine if you have a finite volume of paint of 1 m^3. Say, you have surface area of 1 m^2. How much of those volume do you need to use in order to paint that surface area ? Perhaps you need to use 0.1m^3 ? or 0.01 m^3 ? 0.0000000...0001 m^3 ? It really all depends on how "thin" the paint can get and still be able to cover the wall. Since we are speaking from purely mathematical point of view, you can imagine the paint can be infinitely thin and still be able to cover the area. This is implicit in the construction of our horn since as x->infinity, the diameter of the throat of the horn is getting infinitely thin. So, you see, the paint is getting "thinner" at a fast enough rate as the surface area is growing for x-> infinity. Hence, what this paradox "shows" is that it is possible to paint an infinite surface area with finite volume of paint which has a magical ability to get infinitely thin.

[@Brian.001]

@@user-ez2di8lv7q But that doesn't work. As long as pi litres of paint fills the horn, then pi litres of paint coats the entire inner surface. The paint doesn't have a specific thickness, as it fills the entire volume. This is logically impossible.

[@morbideddie]

@@Brian.001 how is it logically impossible? Gabriel’s horn demonstrates that a finite volume having an infinite surface area is possible so a finite amount of paint can cover an infinite surface area without causing any logical issues.

[@MotesTV]

The interior surface area is not equal to the exterior surface area, which gets thinner than a molecule of paint at the end and goes on forever.

[@7wardie]

if you imagine this as a sphere would it describe a black hole...?

[@BOBimusRex]

I get the math. Really, I do. I love math. But if you can fill the horn, and the horn has the same surface area inside as out, then by waybof filling the horn, you are covering an equal surface area on thebinside of the horn as you would be covering on the outside if you were trying to paint it.

[@markremi]

Isn’t Pi infinite too? We’ve only gone so many decimal places with it.

[@morbideddie]

Pi is finite in value. No matter how many positions you go to it will always be less than 4.

[@carultch]

The number pi is not infinite. It is more like "in-between-finite" than infinite. It has infinitely many digits, but the significance of these digits becomes infinitesimal as the number of digits becomes infinity.

[@mattparadisi7016]

Why can we treat the volume segments as cylinders, but we can’t treat the surface area segments as rectangles? Wouldn’t that resolve the paradox?

[@user-ez2di8lv7q]

In fact, you don't have to treat the volume segments as cylinders. You can also treat it as cones and you will still get the same answer. This is because in the case of volume, the error is in the second order. Note that in calculus, when you "multiply" two infinitesimal values, you approximate it to zero.. Now, for the line segment, error between dx and ds is dx - sqrt( 1-f'(x) )dx = [ 1-sqrt(1-f'(x)) ]dx which is in the first order. Also, when you use dx as your line segment, this will always give a lower bound to using ds.

[@songemaestronsunza4407]

The volume is not finite. It just becomes negligible as you approach infinity therefore convergong to Pi. In fact the curve never touch the y axis therefore in reality both surface area and volume have no finite value.

[@morbideddie]

As you go further along the horn the volume converges on pi, it will never be larger than pi and the final shape has a volume of pi. pi is a finite number, it will never be more than 4. Therefore the volume is finite.

[@UTU49]

@@morbideddie Exactly. You will never run out of paint. You couldn't literally fill an object of infinite length, but its volume is finite.

[@somnathsaha7391]

🙂

[@Salsaneer82]

For the volume..... why wouldn't the value, as t goes to infinity, just approach pi but never actually BE pi.... therefore being infinite. Asking for a friend.

[@morbideddie]

Being infinitely close to a number doesn’t make it infinite. The limits are what matter here and between the limits the volume is pi, we don’t approach it since the object is already formed and infinitely long.

[@iangray5407]

But in reality it has no surface area or volume since it only exists as a Platonic ideal rather than as a physical object

[@vedantkokate971]

Obviously

[@feefifofum6383]

Not with that attitude

[@cumjelly9271]

jesus christ

[@rolandkarlsson7072]

This is no paradox and have never been a paradox. Yes, the volume is finite and the surface infinite. But, as the surface has 0 thickness, the surface has no volume.

[@Simcore999]

Logic says that there is an error in the math...

[@forbiddenera]

Your mic is clipping.

[@user-rs5lf7ec3e]

i don't agree with you I don't see a problem in this, so the area is infinity aaccpetabl but the volume is pi that is not logic

[@rahadiannow]

How many angles that we need to make circle? The answer is infinity We need 0.1 degre, 0.111 degree, ...

[@ryklin1]

It would be beneficial to show some visualizations because your analytical explanation was not intuitive.

[@TheElRash]

Well i think this is all theoretical and Not practical i think if u have the gabriels Horn in real Life u could Just paint it by Filling it with paint and pouring out i think you would have to have infinite paint to pour in

[@jdrmanmusiqking]

As a man who holds math in the highest regard, Im not above my biases to point out that this is simply a fun math trick and not even close to possible in reality (or at least our slice of reality that contains classical properties) As a commentor below stated, Pi fills the entire thing.... Yet the surface area is infinite If the outside surface area is infinite then the inside surface area must be infinite. But Pi fills the the entire shape which includes the inner surface so such a shape cannot exist as the inner surface is infintie

[@taylorwoods8001]

it just doesn’t sit well with me that a positive constant greater or less than 0 over infinity converges to 0.

[@IICOR1552]

Can you repeat that but this time use english!

[@Cyb0rgd3ck3r]

This mathematical abstraction speaks to an incorrect understanding of method, not a real geometry with an interdependent finite and infinite value. The proof is garbage.

[@jamestagge3429]

This is a load of cow pies. 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 you know that the math, if it seems to prove this possible, 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, 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. Claiming that there are other examples of infinite boundaries and finite volumes are pure sophistry. I have been trying to discuss this with a mathematician (I think he is) and he tried to defend himself by claiming that a 1 x 1 box, subdivided 50% each time, infinitely would enclose a finite volume of 1 x 1 and this is just another example of the sophistry of the sciences. 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. You were claiming that an infinite boundary, which is not what is applied to the box mentioned above, could have a finite volume and it can’t. 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, you employ sophistry to win your argument and compare apples to elephants. This is NOT even close to the same scheme as Gabriel’s horn in which the boundary is an infinite extension and your claim is that the volume contained within it is finite WHICH CANNOT be the case. Any thoughts?

[@jamestagge3429]

@@Noah55555 First, thanks for responding. I love these discussions and they are truly very instructive. As for my comment on the math, consider....the claim that the “math proves it” is used all the time. Physicists who study the standard model in particle physics claim that their math proves this or that yet there are those who study sting theory who make the same claim (as long as you materiality has 11 dimensions). Both cannot at once be true. Why does either group then not drop what it is doing and join the other? Additionally, most of those same physicists and most cosmologists have claimed that the universe is likely infinite when the rest of their physics makes clear that it cannot be. These same folks also claim that the universe expanded (via inflation) from about 10 -15 meters (I believe that is the measure), a quantifiable size, to the size of a grapefruit, also a quantifiable size but then to infinity (after inflation ceased). What is it to expand? It is to increase in value, number or measure, progressively, by quantifiable increments. So what would that last measure of expansion be just before the very next one was infinity? As you can see infinity is an abstraction only and cannot exist in any manner within materiality. Their proposition is a gross contradiction. Will you say that it must be so for their math proves it? You argue from authority, a fundamental mistake in the art of debate. I don’t care how smart they all are. When they say something stupid they should be called on it. Continuing my point. I could go on and on with examples of the contradictions proposed by this sort which are easily understood as such, e.g., Hilbert’s hotel, a thought experiment which is sophomoric and also a contradiction but held up as so astonishing and amazing only because the mathematician who authored it is held in such high esteem. He may be a genius and deserving of all the accolades, but he is not for this. He defines his proposition as; there are infinite rooms in a hotel and infinite guests and most importantly he constrains it by saying “and all the rooms are full”. This statement is critical in the definition of his experiment. Then he goes on to employ all sorts of mathematical formulae as to how he could make room for additional guests, even an infinite number. What is wrong with his experiment is that he is the one who made that theoretically impossible. His appeal to a kind of elasticity of the infinite collection of rooms must be made correspondingly to the guests precisely because “all the rooms are full”. It is as if the rooms and guests were like an infinitely tall ladder where on one stanchion the holes represent the rooms and on the other stanchion the holes represent the guests. The rungs which connect the one (each hole) with the other represents the constraint that “all the rooms are full”. By this, logically, if that matters anymore, any consideration of an extension of rooms must also be made with the guests, all the rooms remaining full so there is nowhere to which to shift any guests to make room for new ones. But the math proves it? Back to the horn…in my discussions with others on this matter, examples were brought up such as a 1 x 1 box divided in half and then that half in half, ad infinitum. The claim is that the boundary is infinite but the volume finite, i.e., 1 x 1. But this is sophistry. This is a subdivision of a finite volume and boundary, not a volume or boundary in infinite extension. It is totally different (and there are problems with the example itself not to be discussed here - I resent this kind of thing because it assumes the stupidity and ignorance of those reading or watching). Imagine the boundary defining the horn being removed from the volume it contains. What do we see? The same thing. The boundary cannot be removed because if it were, the edge of the volume would still be there in the exact form and dimension as the original boundary removed. This shows the boundary to be contingent of the volume defined and thus cannot be but as it bounds a volume. It is said that the surface area of the horn is infinite. (remove the horn) and the surface area of the volume (which is the same thing) would then be infinite as well. The volume could only be considered finite (kind a sorta) IF the horn reduced in proportion to the length added in similar manner to the 1 x 1 box’s subdivision which I don’t think is the case. If that is what it does, then is it just another subdivision and NOT what it is purported to be. Perhaps you can clarify that. I have studied this stuff because it find it interesting. I am surprised however, at the lack of spatial perception in so many of those who challenge me on my comments. Often, they will claim that the authors of the propositions I critique “didn’t mean” this or that by the terms they used. Fine…then they should stop using them. Clarify what is meant or proposed or don’t state the propositions at all.

[@SomeoneCommenting]

You don't get something here: *PI IS NOT "FINITE"* just because we write it using a single Greek letter symbol for it. Pi is an infinite value that we can never represent as a "finite" number because its decimal part goes on and on and on forever. So in reality, the volume is *also* infinite. It's physically impossible for one thing to be infinite and the other not. But _mathematically_ we can come up with all this "reasoning". It's the same way in which mathematicians hate when engineers use the impulse function to solve mathematical problems for electric circuits. It's also a trick. Engineers say that "it's finite because the area is 1" yet the mathematicians do not accept that because it goes up to infinity as it's thickness goes to zero. It's _a definition_ , not "a value".

[@SomeoneCommenting]

@Random Name Dude, don't ruin the trolling, I want people to break their heads with this ;-)

[@nosuchthing8]

Haha. So if you pull a coin out of your pocket, a quarter say, and want to paint one side, heads, you can calculate the paint required. Pi r squared. You won't be able to calculate the exact amount of paint because pi is irrational, true.

[@allanreilly5827]

How can Pi be infinite if numbers both larger and smaller then Pi (eg. 3&4) are finite? Pi is irrational, not infinite. Look up what the words mean.

[@nosuchthing8]

I read what you posted. I disagree Consider a cylinder, a cup. The volume for a cylinder is pi r squared times the height. If the radius r is 1, and the height is 1, then the volume is simply pi. Its finite and we can fill it with coffee or paint. The endless digits of pi past the decimal point dont really matter, at some point they are less than the diameter of a molecule and dont affect the volume.

[@nosuchthing8]

@@allanreilly5827 yes they are conflating the infinite sequence of numbers associated with pi and numbers that are infinite