[Guest]

You can use only paper and pencil. And infinite time. No other material. And you know math as much as an elementary school student. Can you tell me nearly the exact pi number after nearly infinite time?

[Guest]

I knew a guy in high school who was trying to memorize pi to 314 digits. Last I knew he was at about 150, but I wouldn't be surprised if he got it done.

I can't imagine how he managed to fight the ladies off long enough to do that kind of memorization.

[Guest]

I mentioned that method above but I wanted you to get pi without using anything other then a paper and a pencil, both are infinite, and you have a lot of time. Dropping the pencil on the paper is a method but I want you to make only calculation (+-x operations). Monte Carlo method is a magic and an easy method. Few people are aware of it, thus I congratulate you. But I am still asking a method with only calculation.

Another thought, can they use the pencil and paper and by trial and error come up with a good approximation to the sqrt of 2?

Then they can use just one term of a quickly converging series derived by Ramanujan and calculate pi to 14 or so places by computing:

pi ~ 9801/2206/sqrt(2)

[Guest]

Another thought, can they use the pencil and paper and by trial and error come up with a good approximation to the sqrt of 2?

Then they can use just one term of a quickly converging series derived by Ramanujan and calculate pi to 14 or so places by computing:

pi ~ 9801/2206/sqrt(2)

First; it is very hard to get sqrt of 2

second; I ask for an answer that will lead you to -for example- hundred of decimals of pi, if you expense very very much paper, pencil and time.

There is method by using only simple operations but very much time???

[Guest]

I wonder why nobody gave an exactly true answer. Because there is a simple way to calculate pi without using series, complex operations and objects. Here it is:

Consider a square, divide it many small squares as a chess board. Say it: 100 x 100.

And consider a circle in it which touches each border.

Now get into account each intersection point in chessboard.

If that point is inside the circle, tally it(count it) somewhere.

When you're close to corners of the chessboard, it is easy to decide whether that point is inside the circle or not.

When you are not sure, count the small squares from that point to the center of chessboard (and circle of course).

Let x be the count of small squares from our point to center in transverse direction, and y be in vertical direction.

Calculate x*x + y*y if that value is greater than 2500 (50*50) than that point is inside of the circle.

The proportion of counts in the circle and total, gives pi/4.

Actually you don't have to draw such a chessboard, only imagine it.

If you have enough time and divide the chessboard to millions or billions of points you can get closer to pi.

Thus, as I asked, if you have infinite time, you will get nearly exact pi.

Indeed I had tried it by a software 7-8 years ago. In a few hours I had reached 7. decimal of pi.

Of course, today, with a faster computer it would be better.

[Guest]

I wonder why nobody gave an exactly true answer. Because there is a simple way to calculate pi without using series, complex operations and objects. Here it is:

Consider a square, divide it many small squares as a chess board. Say it: 100 x 100.

And consider a circle in it which touches each border.

Now get into account each intersection point in chessboard.

If that point is inside the circle, tally it(count it) somewhere.

When you're close to corners of the chessboard, it is easy to decide whether that point is inside the circle or not.

When you are not sure, count the small squares from that point to the center of chessboard (and circle of course).

Let x be the count of small squares from our point to center in transverse direction, and y be in vertical direction.

Calculate x*x + y*y if that value is greater than 2500 (50*50) than that point is inside of the circle.

The proportion of counts in the circle and total, gives pi/4.

Actually you don't have to draw such a chessboard, only imagine it.

If you have enough time and divide the chessboard to millions or billions of points you can get closer to pi.

Thus, as I asked, if you have infinite time, you will get nearly exact pi.

Indeed I had tried it by a software 7-8 years ago. In a few hours I had reached 7. decimal of pi.

Of course, today, with a faster computer it would be better.

I like your graphical solution and it is the kind of idea I was looking at in my original reply (though I couldn't demonstrate a way of carrying it out) - but it would require a very large circle to be drawn pretty quickly if you were to carry it out practically. However, I also see your point that, for each square, you can *calculate*, with just difference of squares, whether it is in the circle or not (though the number of calculations grows as s^2 compared to the granularity of squares, s)

Essentially, I think you would be approximating a circle (radius r) with an n-sided polygon and estimating the area of that by counting squares. I think the two levels of approximation would make it pretty slow to converge - something borne out by your computer program! However, I wonder how you knew that you were correct to 7 dp? (without knowing what pi actually was, of course!) The calculation of the error and demonstration that you are sufficiently accurate at a particular dp., I think, would be much harder.... but then, I guess, you could do it with a polygon on both sides (inside and outside the circle) to bound the value fairly easily.

[Guest]

I wonder how you knew that you were correct to 7 dp? (without knowing what pi actually was, of course!)

Of course I wasn't able to be sure at which decimal I was correct. I don't know any way of this. But my aim was getting closer to pi, not being sure how much I was correct.

Using poligons, of course, is a faster method but you can't calculate area of a poligon by using simple operations (without sin, sqrt..), by your hand.

I had thought a more easier way, by making rectangles in a quarter circle. But there, I had to use sqrts of numbers. Thus it was impossible to manage it by hand. But by computer, it got much less time than my pisagor method.

[Guest]

Of course I wasn't able to be sure at which decimal I was correct. I don't know any way of this. But my aim was getting closer to pi, not being sure how much I was correct.

Using poligons, of course, is a faster method but you can't calculate area of a poligon by using simple operations (without sin, sqrt..), by your hand.

I had thought a more easier way, by making rectangles in a quarter circle. But there, I had to use sqrts of numbers. Thus it was impossible to manage it by hand. But by computer, it got much less time than my pisagor method.

I think you are approximating using polygons - by placing a grid of squares, you are 'rasterising' the circle with a polygon contained within and then counting squares to approximate this!

I also think you could improve your method by calculating the area of each of each triangle making up your polygon (shown in green below). That would give you the exact area of the polygon, would it not?

[Guest]

I think you are approximating using polygons - by placing a grid of squares, you are 'rasterising' the circle with a polygon contained within and then counting squares to approximate this!

I also think you could improve your method by calculating the area of each of each triangle making up your polygon (shown in green below). That would give you the exact area of the polygon, would it not?

To get pi by poligons, we must exactly calculate the area of poligon by trigonometry. Otherwise it is not more accurate then drawing a circle on a paper, having very small squares on it, and counting the squares. My method also depends on it, but I calculate all border points, and I do it without drawing, by only imaginating.

In your image, borders of poligon are not equal. Medial part is sqrt(10) and others sqrt(8). Anyhow it is very hard to get areas of poligons.

[Guest]

To get pi by poligons, we must exactly calculate the area of poligon by trigonometry. Otherwise it is not more accurate then drawing a circle on a paper, having very small squares on it, and counting the squares. My method also depends on it, but I calculate all border points, and I do it without drawing, by only imaginating.

In your image, borders of poligon are not equal. Medial part is sqrt(10) and others sqrt(8). Anyhow it is very hard to get areas of poligons.

The polygons I propose do not require any trigonometry - it simply adds another step to your calculation. In your method, for a fixed grid resolution, you iterate through every point in the 'grid' and determine whether it is inside or outside the circle and take this as a proportion of the total grid points. I suggest that, with a little extra work, you can take the rectangles around the edge and turn them into triangles which inscribe the circle. The area of these is trivial to calculate and then we can take this as a proportion of the whole area.

In a computer program, you would iterate by each row or column, rather than by individual points.

Archimedes method for the area of the circle used regular n-polygons and sent n to infinity - That indeed requires some trigonometry (sin 360/n or some such). This does not use regular polygons, it just reduces the error at the edges with a set of triangles that approximate the gradient at that row/column.

[Guest]

I wonder why nobody gave an exactly true answer. Because there is a simple way to calculate pi without using series, complex operations and objects. Here it is:

Consider a square, divide it many small squares as a chess board. Say it: 100 x 100.

And consider a circle in it which touches each border.

Now get into account each intersection point in chessboard.

If that point is inside the circle, tally it(count it) somewhere.

When you're close to corners of the chessboard, it is easy to decide whether that point is inside the circle or not.

When you are not sure, count the small squares from that point to the center of chessboard (and circle of course).

Let x be the count of small squares from our point to center in transverse direction, and y be in vertical direction.

Calculate x*x + y*y if that value is greater than 2500 (50*50) than that point is inside of the circle.

The proportion of counts in the circle and total, gives pi/4.

Actually you don't have to draw such a chessboard, only imagine it.

If you have enough time and divide the chessboard to millions or billions of points you can get closer to pi.

Thus, as I asked, if you have infinite time, you will get nearly exact pi.

Indeed I had tried it by a software 7-8 years ago. In a few hours I had reached 7. decimal of pi.

Of course, today, with a faster computer it would be better.

How do you plan to draw both a circle and a regular grid exactly enough, using "only paper and pencil... No other material," to calculate pi to 7 decimal places?

[Prof. Templeton]

How do you plan to draw both a circle and a regular grid exactly enough, using "only paper and pencil... No other material," to calculate pi to 7 decimal places?

Graph paper

(with some very small units).

[Guest]

First; it is very hard to get sqrt of 2

second; I ask for an answer that will lead you to -for example- hundred of decimals of pi, if you expense very very much paper, pencil and time.

There is method by using only simple operations but very much time???

I am not so sure it is hard to get to the sqrt of 2, one just has to compute the series:

1/2 + 3/8 + 15/64 + 35/256 + 315/4096 + 693/16384 + .....

The Ramanujan series for pi can then simply be extended to achieve a greater level of significance, and it is one of the fastest converging series ever developed.

I believe this satisfies the requirements of using only simple operations and given infinite time and paper gets the answer to infinite significance.

[Guest]

Graph paper

(with some very small units).

How would you draw the circle then? Would you buy graph paper with a circle drawn on it as well?

If you're going to do that, why not just buy a piece of paper with however many digits of pi you want printed on it? You'd save a lot of time.

[Prof. Templeton]

How would you draw the circle then? Would you buy graph paper with a circle drawn on it as well?

If you're going to do that, why not just buy a piece of paper with however many digits of pi you want printed on it? You'd save a lot of time.

Maybe you can fashion a compass by wrapping some paper around the end of two pencils...Dude, I don't know. I was trying to add some levity. I'm not the guy looking for 12 decimal places of pi using only a pencil. If I want to know what pi is, I use a calculator which saves a whole lot of time. This topic seems more suited to your area of expertise, I'll just go back to reading this thread, and keep my worthless comments to myself.

[Guest]

Maybe you can fashion a compass by wrapping some paper around the end of two pencils...Dude, I don't know. I was trying to add some levity. I'm not the guy looking for 12 decimal places of pi using only a pencil. If I want to know what pi is, I use a calculator which saves a whole lot of time. This topic seems more suited to your area of expertise, I'll just go back to reading this thread, and keep my worthless comments to myself.

Wow, sorry if my previous comment sounded harsh. I was just amused by the idea of buying a piece of paper with the digits of pi printed on it. No offense intended.

[Prof. Templeton]

Wow, sorry if my previous comment sounded harsh. I was just amused by the idea of buying a piece of paper with the digits of pi printed on it. No offense intended.

I over-reacted. It's hard to interpret intent. I'm also curious as to what the intended solution is. So, back on topic?

[Guest]

Maybe....

I can't remember, but I think it was the division of 2 (two) three-digit numbers (I also think they were prime numbers as well...)

For example (if it seemed confusing ###/###, which (of course) the #'s are numbers which I cannot remember...

But I think this was an approximation to the actual Pi

[Guest]

Simple

pi is also know as 22/7 so you just divide 7 into 22 and keep going

[Guest]

At my post #30 I told my method. You don't have to buy a graph paper, as I told, only imagine such a paper, and a circle on it. Isn't it clear that you don't need to draw a circle on it? I hope it is only my poor english that has leaded you to misunderstand. Give numbers to intersection points on a chessboard of 100x100. Get each number: (1,1):it is in outside the circle, (1,2)(1,3)....(2,1)(2,2)...(3,1)(3,2):all outside. (50,50)(49,50):sure these are inside. But you can not be easily sure that for example (30,20) is inside or out side. Then make a simple operation: (50-30)*(50-30)=400. (50-20)*(50-20)=900. 900+400=1300. Since 1300>2500, this point is inside the circle. Can't you really imagine that and need a graph paper?

To other comment: Sqrt(2) may be calculated, but it is not possible to get pi by only that. You will need too many other sqrts.

Neither 22/7 nor any other division of two or two billion digit numbers gives the exact pi. This is the reason why I want to get pi with a math knowledge of a 13 years old child.

[Guest]

At my post #30 I told my method. You don't have to buy a graph paper, as I told, only imagine such a paper, and a circle on it. Isn't it clear that you don't need to draw a circle on it? I hope it is only my poor english that has leaded you to misunderstand. Give numbers to intersection points on a chessboard of 100x100. Get each number: (1,1):it is in outside the circle, (1,2)(1,3)....(2,1)(2,2)...(3,1)(3,2):all outside. (50,50)(49,50):sure these are inside. But you can not be easily sure that for example (30,20) is inside or out side. Then make a simple operation: (50-30)*(50-30)=400. (50-20)*(50-20)=900. 900+400=1300. Since 1300>2500, this point is inside the circle. Can't you really imagine that and need a graph paper?

To other comment: Sqrt(2) may be calculated, but it is not possible to get pi by only that. You will need too many other sqrts.

Neither 22/7 nor any other division of two or two billion digit numbers gives the exact pi. This is the reason why I want to get pi with a math knowledge of a 13 years old child.

Actually you will not need any other square roots other than the value for 2, this is the beauty of the Ramanjuan series given as:

pi.bmp

pi.bmp

[Guest]

Actually you will not need any other square roots other than the value for 2, this is the beauty of the Ramanjuan series given as:

pi.bmp

You're right then.

But I asked a method with no known series.

[Guest]

You're right then.

But I asked a method with no known series.

Oh, sorry about that I just read the initial post and not the subsequent qualifications

>> You can use only paper and pencil. And infinite time. No other material. And you know math as much as an elementary school student. Can you tell me nearly the exact pi number after nearly infinite time? <<

Perhaps it should have been initially posted as:

I know a way to calculate pi using paper and pencil, infinite time and only knowing as much math as an elementary school student. See if you can guess the method I have in mind.

[Guest]

At my post #30 I told my method. You don't have to buy a graph paper, as I told, only imagine such a paper, and a circle on it. Isn't it clear that you don't need to draw a circle on it? I hope it is only my poor english that has leaded you to misunderstand. Give numbers to intersection points on a chessboard of 100x100. Get each number: (1,1):it is in outside the circle, (1,2)(1,3)....(2,1)(2,2)...(3,1)(3,2):all outside. (50,50)(49,50):sure these are inside. But you can not be easily sure that for example (30,20) is inside or out side. Then make a simple operation: (50-30)*(50-30)=400. (50-20)*(50-20)=900. 900+400=1300. Since 1300>2500, this point is inside the circle. Can't you really imagine that and need a graph paper?

To other comment: Sqrt(2) may be calculated, but it is not possible to get pi by only that. You will need too many other sqrts.

Neither 22/7 nor any other division of two or two billion digit numbers gives the exact pi. This is the reason why I want to get pi with a math knowledge of a 13 years old child.

OK, I understand now. I agree that this is probably the method that requires the least mathematical knowledge.

Would it be more precise for a given number of points to only consider the first quadrant of the circle? In other words, if you are using a 100X100 grid, use a radius of 100 instead of a radius of 50, and place the origin in the bottom left corner of the grid instead of the center.

[Guest]

Would it be more precise for a given number of points to only consider the first quadrant of the circle? In other words, if you are using a 100X100 grid, use a radius of 100 instead of a radius of 50, and place the origin in the bottom left corner of the grid instead of the center.

In fact, you could consider only half of the first quadrant, as each point has a matching reflection: for a grid of 100x100 (with circle radius 100) you could look at the x>=y points.

[Guest]

OK, I understand now. I agree that this is probably the method that requires the least mathematical knowledge.

Would it be more precise for a given number of points to only consider the first quadrant of the circle? In other words, if you are using a 100X100 grid, use a radius of 100 instead of a radius of 50, and place the origin in the bottom left corner of the grid instead of the center.

Of course I didn't make my computer to calculate all 4 quadrants. In fact, I considered x-y analitic plain, and only the points from (0,0) to (+n,+n). But this way may require more math knowledge, thus I wanted to tell my method in square and circle way. It got time but I'm happy to manage to make someone understand my simple method. Thanks to everybody.