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TheBritishVillain

Calling all math-heads: How did i surpass infinity?

17 hours ago, TheBritishVillain said:

-Only 1cm away from the edge. Ok, move it closer.

-Now only 1mm for the edge. Closer. 

-100 Micrometres? Closer.

Then onto Nanometres and picometres etc. Let's say it's only 1 picometre from the edge. Without changing the unit of choice, we can still get smaller. 0.1 picometre, 0.01, 0.001, 0.0001... You get the idea. 

What you've just discovered is called a "supertask".

 

A super task is essentially a countably infinite sequence of "tasks" that seems like it should work. The commonly cited case of your example is called "Zenos runner" In which Zeno of Elea argues that motion is actually impossible by showing that a runner who moves half the distance to the finish line every "step" will never reach the finish line.

 

Michael from VSauce does a way better job explaining super tasks than I will ever be able to:

 

Edit:: I feel like I should mention that all super tasks are trivially paradoxes, since the definition of a "super task" is "some infinite series of operations that can be completed in finite time". For that statement to work, you would have to be able to complete tasks in "instantaneous time" or, in an equivalent sense, "The function V(t) that describes the velocity of task completion would approach infinity".

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Posted · Original PosterOP

I was looking at my monitor today and was at a long loading screen. Being the weird bored person i was, i tried to move the mouse cursor to as close as to the edge of the screen without actually touching the edge. 

 

This got me thinking. How far could i go? What's the smallest possible denomination i could make without actually touching the edge of the screen? Well, since it's a screen, it would be by 1 pixel.

 

But what if it wasn't a screen?

 

Let's use a ruler and a table as an example.

 

Ruler googled.jpg

 

 

The goal is to put the entire ruler flat on the table and move it upwards vertically (length ways) until any portion of the ruler protrudes past the table and is effectively in the air. Now here is the tricky part. With a monitor, it uses pixels so it has a finite amount of pixels before there is no more. With actual measurements, there is no smallest number thanks to decimals.

 

-Only 1cm away from the edge. Ok, move it closer.

-Now only 1mm for the edge. Closer.

-100 Micrometres? Closer.

Then onto Nanometres and picometres etc. Let's say it's only 1 picometre from the edge. Without changing the unit of choice, we can still get smaller. 0.1 picometre, 0.01, 0.001, 0.0001... You get the idea. 

 

My point is that since infinity is literally supposed to mean 'a number greater than any assignable quantity or countable number' doesn't that mean i should be forever scrolling the ruler upwards aslong as i alter my force? Of course not! But by doing so, we have surpassed infinity.

 

Or have we? Have i simply hit the 'Planck length'? Or does it mean that Force exerted on the ruler (X) surpasses infinity (∞). = X > ∞

 

Another slightly similar way to look at it is the cutting of a cake into 3 perfectly equal pieces. Mathematically it can be done via fractions 1/3, 1/3 and 1/3. However everyone knows that 1/3 as a decimal is only 0.33•. Meaning that you would have to keep infinitely cutting the remainder of the 0.01• cake into 3 equal pieces.

 

Now i'm no math whizz or anything like that. Surely there is a name for this or a paradox or something? Just baffles my mind. We need Thanos to make everything perfectly balanced.


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?? are you confused that three thirds make a whole??


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The border of countries is also infinitely long.

It just depends on how you look at things.

 

But I like the way you think, please continue with that. But also get some sleep to not drive yourself nuts!

 

EDIT: wait, you got me thinking..

You're saying 1 pixel is the least the loading bar could move... But what if anti aliasing.. Computers think in 24 bit color depth, meaning they can display up to 16.8 million colors (rounded up) per pixel.. Meaning this 1 pixel has 16.8 color it could be, so even more loading bar! Plus 8 bit alpha channel (transparency) to add onto that.


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well if you keep cutting the are in half, it will essnetailly be the same concept of a circle (pi, as @Bitter stated) 

 

let's say it is a 1mm distance, 1/2 of 1 = .5, so .5mm (or 1/2mm) and so on, and then 1/4mm, 1/8mm, 1/16mm, 1/32mm, 1/64mm, etc


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You're thinking of some of Zeno's paradoxes - basically his ones regarding motion.

It can also be shown as a geometric series (In this case, approaching 1): Sum from n=1 to infinity of ((9/10)*(1/10)n)

 

Another interesting paradox, similar to the geometric series, is that .999 repeating is equal to 1;

  • Multiply .999 by 10 - you get 9.99999: Multiply 1 by 10 - you get 10
  • Subtract 9.9999 by .999 - you get 9.0; Subtract 10 by 1 - you get 9
  • Therefor .999 equals 1.

 


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5 hours ago, TheBritishVillain said:

The goal is to put the entire ruler flat on the table and move it upwards vertically (length ways) until any portion of the ruler protrudes past the table and is effectively in the air. Now here is the tricky part. With a monitor, it uses pixels so it has a finite amount of pixels before there is no more. With actual measurements, there is no smallest number thanks to decimals.

fun fact: this infinity is bigger in count than  counting whole numbers untill the universe is blue in the face. the infinitely-thin is called an infinitesimal, but there are uncontably infinite infinitesimals

 

 

5 hours ago, TheBritishVillain said:

My point is that since infinity is literally supposed to mean 'a number greater than any assignable quantity or countable number'

strictly speaking, infinity is a concept, not a number. you don't get a valid number if you divide by zero (but outside of mathematics it'll depend on the definition agreed upon in any specific the field of work)

 

5 hours ago, Minibois said:

The border of countries is also infinitely long.

It just depends on how you look at things.

wheeeeeeeee

 

4 hours ago, Imbellis said:

Another interesting paradox, similar to the geometric series, is that .999 repeating is equal to 1;

  • Multiply .999 by 10 - you get 9.99999: Multiply 1 by 10 - you get 10
  • Subtract 9.9999 by .999 - you get 9.0; Subtract 10 by 1 - you get 9
  • Therefor .999 equals 1.

 

there's a much quicker and practical way (although not mathematically rigorous):

  • divide 1 by 1 on paper
  • instead with starting with one, start with 0.9
  • keep dividing.........

you don't get two answers when you divide one number with another, so 0.999999999 has to be the same as 1. otherwise you'll have fundemental problems with division

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I think what you're getting at is infinitesimals, which is the foundation of calculus. Essentially, an infinitesimal is so small as to equal 0 but is does have some sort of "value."

 

The derivative is a ratio of two infinitesimals, rise over run. Since you're looking at a single point with a derivative, both the rise and the run are 0. BUT, their ratio can have a real, nonzero value.

 

Another fun thing about infinity/infinitesimals from calculus is that you can have functions which go out to infinity but have a finite area under the curve. Like e^(-x^2).


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9 hours ago, Imbellis said:

You're thinking of some of Zeno's paradoxes - basically his ones regarding motion.

It can also be shown as a geometric series (In this case, approaching 1): Sum from n=1 to infinity of ((9/10)*(1/10)n)

 

Another interesting paradox, similar to the geometric series, is that .999 repeating is equal to 1;

  • Multiply .999 by 10 - you get 9.99999: Multiply 1 by 10 - you get 10
  • Subtract 9.9999 by .999 - you get 9.0; Subtract 10 by 1 - you get 9
  • Therefor .999 equals 1.

 

This is correct. It is called Zeno’s arrow. The only difference is that Zeno divided by 2 rather than 10.

 

On a related note, I recall that there is an interesting differential equation modeling a bullet penetrating a material that resists the bullet at a rate proportional to the square of the velocity. In such a scenario, the bullet will continue moving forever. It never happens in practice due to the world not perfectly following mathematical models, but according to the equation, the bullet is forever becoming slower while never reaching zero velocity. I should note that there might be some flaw in my recollection of the resistance required to do that.

 

edit: My recollection was wrong. The resistance is linear to the velocity when the bullet never stops if this is correct:

 

https://www.physicsforums.com/threads/differential-equation-for-deceleration-of-a-bullet.725697/

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2 hours ago, Dash Lambda said:

I think what you're getting at is infinitesimals, which is the foundation of calculus. Essentially, an infinitesimal is so small as to equal 0 but is does have some sort of "value."

 

The derivative is a ratio of two infinitesimals, rise over run. Since you're looking at a single point with a derivative, both the rise and the run are 0. BUT, their ratio can have a real, nonzero value.

 

Another fun thing about infinity/infinitesimals from calculus is that you can have functions which go out to infinity but have a finite area under the curve. Like e^(-x^2).

For example, the area bounded by the curve y(x) = 1/x^2 and the curve y = 0 from x = 1 to infinity is 1. Try to make it from x = 0 and it is infinite, but if you get arbritrarily close to x = 0, it is finite. From x = 0.1 to infinity is 10. From x = 0.01 to infinity is 100 and so on. The area in this case is equal to the 1 divided by the lower bound of x.

 

That said, I suggest that people reading this not think that Calculus is scary. It is really very simple once you figure out how to look at it the right way.

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Posted · Best Answer
17 hours ago, TheBritishVillain said:

-Only 1cm away from the edge. Ok, move it closer.

-Now only 1mm for the edge. Closer. 

-100 Micrometres? Closer.

Then onto Nanometres and picometres etc. Let's say it's only 1 picometre from the edge. Without changing the unit of choice, we can still get smaller. 0.1 picometre, 0.01, 0.001, 0.0001... You get the idea. 

What you've just discovered is called a "supertask".

 

A super task is essentially a countably infinite sequence of "tasks" that seems like it should work. The commonly cited case of your example is called "Zenos runner" In which Zeno of Elea argues that motion is actually impossible by showing that a runner who moves half the distance to the finish line every "step" will never reach the finish line.

 

Michael from VSauce does a way better job explaining super tasks than I will ever be able to:

 

Edit:: I feel like I should mention that all super tasks are trivially paradoxes, since the definition of a "super task" is "some infinite series of operations that can be completed in finite time". For that statement to work, you would have to be able to complete tasks in "instantaneous time" or, in an equivalent sense, "The function V(t) that describes the velocity of task completion would approach infinity".


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5 hours ago, ryao said:

That said, I suggest that people reading this not think that Calculus is scary. It is really very simple once you figure out how to look at it the right way.

 

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Another example:

 

Gravity never goes to 0, no matter how far away from a planet (or other celestial body) you are. It decreases with the inverse square of the distance, in other words 1/(r^2). But even though it never stops pulling you back, you can be launched fast enough to never fall back down. That's called escape velocity.


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Posted · Original PosterOP
2 hours ago, straight_stewie said:

What you've just discovered is called a "supertask".

 

A super task is essentially a countably infinite sequence of "tasks" that seems like it should work. The commonly cited case of your example is called "Zenos runner" In which Zeno of Elea argues that motion is actually impossible by showing that a runner who moves half the distance to the finish line every "step" will never reach the finish line.

 

Michael from VSauce does a way better job explaining super tasks than I will ever be able to:

 

Edit:: I feel like I should mention that all super tasks are trivially paradoxes, since the definition of a "super task" is "some infinite series of operations that can be completed in finite time". For that statement to work, you would have to be able to complete tasks in "instantaneous time" or, in an equivalent sense, "The function V(t) that describes the velocity of task completion would approach infinity".

This is the answer i was looking for. I appreciate everyone else's answers too. 

 

Really interesting considering i only did Math at high-school. 

 

If anyone has anymore examples like like the runner or Gravity scenario mentioned above, i would love to hear them!


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Posted · Original PosterOP
2 hours ago, straight_stewie said:

What you've just discovered is called a "supertask".

 

A super task is essentially a countably infinite sequence of "tasks" that seems like it should work. The commonly cited case of your example is called "Zenos runner" In which Zeno of Elea argues that motion is actually impossible by showing that a runner who moves half the distance to the finish line every "step" will never reach the finish line.

 

Michael from VSauce does a way better job explaining super tasks than I will ever be able to:

 

Edit:: I feel like I should mention that all super tasks are trivially paradoxes, since the definition of a "super task" is "some infinite series of operations that can be completed in finite time". For that statement to work, you would have to be able to complete tasks in "instantaneous time" or, in an equivalent sense, "The function V(t) that describes the velocity of task completion would approach infinity".

Wow this vid just blows my mind! 

mind blown.png


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