Proof that pi=4

Quote from: PizzaPlanet on March 18, 2011, 12:04:10 AM

I remind you of our boundaries! What's the middle item of (0;1]? No, it's not 0.5

ems that you are getting confused by the subtle things that it infinite sets can do.

(If this is confusing you might want to read up on open and closed sets.)

No, I'm not. And to answer the question you have entirely ommited, the middle item is 0.5-epsilon/2, where epsilon is a very small number not divisible by 2.

real numbers

Quote from: JoshuaZ on March 18, 2011, 11:16:02 AM

real numbers

There's your problem. Epsilon is not a real number. It is an infinitely small number with the limit of 0.

In order for 0 and 0.000...0001 to be distinct numbers, there has to be a number in between 0 and 0.000...0001. Can you name it?

Quote from: Trekky0623 on March 21, 2011, 08:06:38 AM

In order for 0 and 0.000...0001 to be distinct numbers, there has to be a number in between 0 and 0.000...0001. Can you name it?

First, you need to define what you mean by 0.000...0001.

Quote from: Thork on March 13, 2011, 08:43:50 AM

Squaring the circle huh? Its a very old problem, you can start here
http://en.wikipedia.org/wiki/Squaring_the_circle

No, that isn't what he's doing.

Anyway, the flawed step is the "repeat to infinity" bit. Infinity is not a number, and no matter how many times you repeat this process, you will not perfectly match the circle -- you will only get very, very close to it.

In computer terms, the circle does not exist. In fact, the circle does not truthfully exist as we know it. There is only a very close match. Like with a computer, pixels make up a "circle". It is just tiny squares creating the illusion of a circle.

IMO, if PI has no end? Isnt it just an approximation? And can't really be used in math, or science.. It would seem like more of description than a measurement?

This is obviously false, but does anyone have any disproof, other than proving that pi is approximately 3.14159265358979...?

My initial response was that of fractal dimensions. I don't think you can ascribe a length to this construction in the usual manner, as fractals tend to be 1.x dimensional.

Another, more grounded disproof is through calculus. The derivative as any given point on a circle, in Cartesian coordinates is non-trivial, but solvable or at least defined. However, if the infinite triangles case, the derivative doesn't exist anywhere, as it is not mathematically smooth in the limit sense.

Thoughts?

Look at the sides of the boxes, they are not equal.  You can't reduce them that way and get a perfect circle.  It would be end up being a weird diamond type shape and it's perimeter to area ratio would not be pi, as in a circle.  In other words, they would be different shapes with different dimensions.

Looking at this briefly, I recall the diagonal paradox.

That was more or less what I was saying.  They never become straight lines.

Quote from: crackpipe larry on April 22, 2011, 05:46:03 PM

IMO, if PI has no end? Isnt it just an approximation? And can't really be used in math, or science.. It would seem like more of description than a measurement?

No, Pi is exact. And we can estimate to a degree such that error is negligible.

Quote from: Harutsedo on May 22, 2011, 07:06:37 AM

Quote from: crackpipe larry on April 22, 2011, 05:46:03 PM

IMO, if PI has no end? Isnt it just an approximation? And can't really be used in math, or science.. It would seem like more of description than a measurement?

No, Pi is exact. And we can estimate to a degree such that error is negligible.

Error is negligible? That doesn't sound very exact..

If you Take $10 and divide it three ways, will you end up with an exact number? Or a description of a third? Because 3x $3.33 is not $10..

Exact, I Thought pi had no end?

So If you multiplied it by say 6 would you have an exact number?

Quote from: crackpipe larry on May 28, 2011, 07:42:15 PM

Exact, I Thought pi had no end?

It has an exact value, just not one that we can ever precisely find.

Quote

Maybe, it got lost in the move..

Did you guys figure out the problem with the OP proof yet? I couldn't read the whole thread, but the problem is that you aren't approximating the circle's perimeter by performing the "remove corners" operation.

To get away with approximation unto infinity to find the precise answer for something- you have to make sure that whatever operation you do is actually approaching the thing you are trying to approximate.

This figure does approach the AREA of the circle, but the set of figures with areas equivalent to a particular circle is infinite, and the perimeters of those figures range from the perimeter of the circle to infinity.

You HAVE to show that you are approximating the PERIMETER of the figure. You don't need any "disproof" of the logic beyond showing that it fails this.

Quote from: mightychef on June 02, 2011, 08:33:35 AM

Did you guys figure out the problem with the OP proof yet? I couldn't read the whole thread, but the problem is that you aren't approximating the circle's perimeter by performing the "remove corners" operation.

To get away with approximation unto infinity to find the precise answer for something- you have to make sure that whatever operation you do is actually approaching the thing you are trying to approximate.

This figure does approach the AREA of the circle, but the set of figures with areas equivalent to a particular circle is infinite, and the perimeters of those figures range from the perimeter of the circle to infinity.

You HAVE to show that you are approximating the PERIMETER of the figure. You don't need any "disproof" of the logic beyond showing that it fails this.

 

If someone claims that the perimeter of a circle is 4 we can disprove this experimentally (just measure a circle within at least 1/2 unit of accuracy).

Quote from: three-dimensional-world on June 02, 2011, 09:40:15 AM

Quote from: mightychef on June 02, 2011, 08:33:35 AM

Did you guys figure out the problem with the OP proof yet? I couldn't read the whole thread, but the problem is that you aren't approximating the circle's perimeter by performing the "remove corners" operation.

To get away with approximation unto infinity to find the precise answer for something- you have to make sure that whatever operation you do is actually approaching the thing you are trying to approximate.

This figure does approach the AREA of the circle, but the set of figures with areas equivalent to a particular circle is infinite, and the perimeters of those figures range from the perimeter of the circle to infinity.

You HAVE to show that you are approximating the PERIMETER of the figure. You don't need any "disproof" of the logic beyond showing that it fails this.

 

If someone claims that the perimeter of a circle is 4 we can disprove this experimentally (just measure a circle within at least 1/2 unit of accuracy).

Where exactly do you plan to find a circle?  Grazing out on the Euclidean Plains?

Quote from: three-dimensional-world on June 02, 2011, 09:40:15 AM

Quote from: mightychef on June 02, 2011, 08:33:35 AM

Did you guys figure out the problem with the OP proof yet? I couldn't read the whole thread, but the problem is that you aren't approximating the circle's perimeter by performing the "remove corners" operation.

To get away with approximation unto infinity to find the precise answer for something- you have to make sure that whatever operation you do is actually approaching the thing you are trying to approximate.

This figure does approach the AREA of the circle, but the set of figures with areas equivalent to a particular circle is infinite, and the perimeters of those figures range from the perimeter of the circle to infinity.

You HAVE to show that you are approximating the PERIMETER of the figure. You don't need any "disproof" of the logic beyond showing that it fails this.

 

If someone claims that the perimeter of a circle is 4 we can disprove this experimentally (just measure a circle within at least 1/2 unit of accuracy).

Where exactly do you plan to find a circle?  Grazing out on the Euclidean Plains?

Did you guys figure out the problem with the OP proof yet?

Quote from: John Davis on June 05, 2011, 06:33:45 PM

Quote from: three-dimensional-world on June 02, 2011, 09:40:15 AM

Quote from: mightychef on June 02, 2011, 08:33:35 AM

Did you guys figure out the problem with the OP proof yet? I couldn't read the whole thread, but the problem is that you aren't approximating the circle's perimeter by performing the "remove corners" operation.

To get away with approximation unto infinity to find the precise answer for something- you have to make sure that whatever operation you do is actually approaching the thing you are trying to approximate.

This figure does approach the AREA of the circle, but the set of figures with areas equivalent to a particular circle is infinite, and the perimeters of those figures range from the perimeter of the circle to infinity.

You HAVE to show that you are approximating the PERIMETER of the figure. You don't need any "disproof" of the logic beyond showing that it fails this.

 

If someone claims that the perimeter of a circle is 4 we can disprove this experimentally (just measure a circle within at least 1/2 unit of accuracy).

Where exactly do you plan to find a circle?  Grazing out on the Euclidean Plains?

A rotating black hole.