Pi

[Guadalupe]

 

PS: As a matter or terminology, the phrase A Pi that is finite doesn’t make sense. Pi is finite: it’s greater than 3 and less than 4. A number being irrational is not the same as it being infinite.

 

 

 

 

Hi! CraigD :eek2:

 

 

It’s true that Pi is not infinite. It is finite, having a value less than 4. It does however have an infinite number of decimal digits. I believe it’s also called “transcendental”.

 

 

 

:QuestionM

[modest]

You are confusing one method of finding pi with the value itself. In terms of geometry, you are confusing the circumfrance of the two polygons with the circumfrance of the circle. The polygons are a bound for pi, and both the inside and the outside polygons bound it.

 

There's a good source explaining here.

 

Archimedies used a 96 sided polygon. The polygon outside the circle measures 223/71 and the inside polygon is 22/7. This means pi cannot be greater than 223/71 nor smaller than 22/7. This does not mean pi itself has 2 different values.

 

The rest of your post extrapolates from this mistake.

 

~modest

 

Hi! modest :)

 

Yes, Archimedes did indeed use geometry by measuring both the inside and outside of a perfect circle and divide it by its diameter.

 

No, he did not. Reread my post and the link therein contained.

 

This would only mean one thing. He found the value for the medium of the line used in drawing the perfect circle and not the true value for Pi.

 

 

 

:cup:

 

You're half right. He did not find the "true" value of [imath]\pi[/imath]. He found an upper and lower bound for its value. That's not to be confused with the circumference of a circle, which is pi of the diameter. The bound does not equal the value, eh?

 

Pi has one value (or is a value) which is revealed in the definition of a circle:

 

A circle is one of the basic shapes of Euclidean geometry. It is the locus of all points in a plane at a constant distance, called the radius, from a fixed point, called the center.

 

-circle

 

Unless you can coheretnly explain how "a constant distance" can have two values, your argument is completely addlepated.

 

~modest

[CraigD]

I stand corrected. I thought we were talking about the thickness of a line used when drawing a perfect circle in finding Pi.

I believe Guadalupe is failing to understand a very critical, fundamental mathematical and geometric concept.

 

Drawing an object geometrically is not the same thing as drawing it physically using pen, pencil, straightedge, compass, etc. Although physical drawing geometric objects is useful as an aid to understanding them, these drawings - I like the term “sketch” to emphasize their limited-precision nature - should not be confused with the “ideal” objects they represent. Unless labeled symbolically - with numeric constants and expressions - sketches are not even adequate descriptions of geometric objects.

 

In physical drawings, “lines” have thickness. Ideally, they don’t. A circle can be defined “perfectly” using numerical mathematical expressions such as [math]x^2 +y^2 = 1[/math]. It can only be hinted at by sketches such as [math]\unitlength 1mm \begin{picture}(87.5,85)(0,0) \circle{10} \end{picture}[/math]

In order for the formula [for pi] to work we need to use the right ingredients which is the circumference divided by diameter or circumference divided by radius square in order to get the true value of Pi and totally leave out geometer all together.

This is how Pi is usually described - as a numeric expression, rather than as analogous to a geometric sketch. As I noted in post #9, expressions such as

[math]4 -\frac4{3} +\frac4{5} -\frac4{7} +\frac4{9} -\frac4{11} \dots[/math]

“…”s aren’t really legitimate math symbols, so a better way to write this is like

[math]\sum_{i=1}^{\infty} \frac4{2i-1} -\frac4{2i+1}[/math].

 

It’s important to understand that, although expressions like these can be used as a guide to write programs that generate approximations of pi, in their pure mathematical form, they are exactly equal to pi. Being irrational (can’t be represented by a fraction of two integers) and transcendental (can’t be represented as the solution to of polynomial equations, such as [math]x^2+x-1=0[/math]), pi can’t be exactly equal to a finite expression using only integers and the operations of addition, subtraction, multiplication, and division, but can be equal to an infinitely long such expression. Because mathematical notation using symbols such as the summation symbol, [math]\sum[/math] allow us to write such expressions with a finite number of symbols, we can write expressions exactly equal to pi and other transcendental numbers.

Yes, Archimedes did indeed use geometry by measuring both the inside and outside of a perfect circle and divide it by its diameter.

 

This would only mean one thing. He found the value for the medium of the line used in drawing the perfect circle and not the true value for Pi.

Archimedes didn’t calculate the circumference of a perfect circle. He calculated the circumference of a 96-sided regular polygon equidistant from a similar polygon inscribed and one inscribing a perfect circle. A 96-sided or other regular polygon is not the same thing as a circle – though it’s difficult to distinguish pen/pencil, straightedge and compass on a ordinary-sized sheet of paper sketches of them with the naked eye.

 

Also, importantly, note that Archimedes didn’t get his results from physically drawing and measuring – he used arithmetic. In terms of the actual calculations done, the main difference between geometric pi estimating techniques and later ones such as Leibniz’s formula appear to me to be the earlier calculations are more complicated to describe using widely known mathematical notation.

 

It’s interesting to me that it’s fairly easy to write pi-estimating algorithms using computer programming languages. For example, this short MUMPS program writes the formulae given by Liu Hui's algorithm, and their approximate decimal values, for 6 to 6144-sided polygons:

s A="1+2**.5*-1+2**.5*6"
f  w A,"=~",@A,! r R s A=(A,"*-")_"+2**.5*-"_(A,"*-",2)_"*2"
1+2**.5*-1+2**.5*6=~3.105828541230249147
1+2**.5+2**.5*-1+2**.5*6*2=~3.132628613281238186
1+2**.5+2**.5+2**.5*-1+2**.5*6*2*2=~3.139350203046867142
1+2**.5+2**.5+2**.5+2**.5*-1+2**.5*6*2*2*2=~3.141031950890509448
1+2**.5+2**.5+2**.5+2**.5+2**.5*-1+2**.5*6*2*2*2*2=~3.141452472285461868
1+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5*-1+2**.5*6*2*2*2*2*2=~3.141557607911856036
1+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5*-1+2**.5*6*2*2*2*2*2*2=~3.14158389214831642
1+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5*-1+2**.5*6*2*2*2*2*2*2*2=~3.141590463227983926
1+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5*-1+2**.5*6*2*2*2*2*2*2*2*2=~3.141592105998927348
1+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5*-1+2**.5*6*2*2*2*2*2*2*2*2*2=~3.141592516690989206
1+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5+2**.5*-1+2**.5*6*2*2*2*2*2*2*2*2*2*2=~3.141592619359959724

[modest]

:confused: :)

 

Fantastic post Craig. I hope Guadalupe appreciates the time you've taken to make so complete an explanation by taking the time to study closely and understand it.

 

~modest

[Guadalupe]

No, he did not. Reread my post and the link therein contained.

 

 

 

You're half right. He did not find the "true" value of [imath]pi[/imath]. He found an upper and lower bound for its value. That's not to be confused with the circumference of a circle, which is pi of the diameter. The bound does not equal the value, eh?

 

Pi has one value (or is a value) which is revealed in the definition of a circle:

 

 

 

Unless you can coheretnly explain how "a constant distance" can have two values, your argument is completely addlepated.

 

~modest

 

 

 

 

Hi! modest :)

 

 

Well, there you go. History does show that Archimedes, Liu Hui, and all the others, only used the method of geometry for measuring a perfect circle which gave us a false value of 3.14159265… as Pi.

 

I believe the equation, c/d = Pi, is approximately 4 thousand years old and was never used to find the true value for Pi.

 

By using this equation, “c/d = Pi”, we need to draw a perfect circle and divided it by its diameter, will gave us a true value for Pi.

 

When finding for Pi, the thickness of the pencil is not important, when drawing a perfect circle; the important thing is to make sure that the outside end of the flat tip lead pencil matched the very end of both sides of its diameter.

 

We should then draw several sizes of perfect circles and precisely measure, with small increments of *fractions, around the circumference and divided them by their diameter which will give us a true value for Pi.

 

Then, submit our findings to the forum so that we may all see the total result.

 

*Recommending using a quarter inch for measuring.

 

 

 

:cup:

[KALSTER]

By using this equation, “c/d = Pi”, we need to draw a perfect circle and divided it by its diameter, will gave us a true value for Pi.

You continually make the same mistake as another guy we had on another forum. You keep thinking of Pi as something that has to be measured. The thing is, even the most precise measurement we can muster can easily be trumped in accuracy by quite basic mathematical methods. Irrespective of what value whoever got way back when, the fact is that we can independently obtain it today to an arbitrary precision. No thickness of the line ever comes into it at all.

[C1ay]

Hi! modest

 

 

Well, there you go. History does show that Archimedes, Liu Hui, and all the others, only used the method of geometry for measuring a perfect circle which gave us a false value of 3.14159265… as Pi.

 

I believe the equation, c/d = Pi, is approximately 4 thousand years old and was never used to find the true value for Pi.

 

By using this equation, “c/d = Pi”, we need to draw a perfect circle and divided it by its diameter, will gave us a true value for Pi.

 

When finding for Pi, the thickness of the pencil is not important, when drawing a perfect circle; the important thing is to make sure that the outside end of the flat tip lead pencil matched the very end of both sides of its diameter.

 

We should then draw several sizes of perfect circles and precisely measure, with small increments of *fractions, around the circumference and divided them by their diameter which will give us a true value for Pi.

 

Then, submit our findings to the forum so that we may all see the total result.

 

*Recommending using a quarter inch for measuring.

 

In the context of the conversation thus far and your contradictory remarks it appears you are being intentionally obtuse. Are you trolling?

[Guadalupe]

You continually make the same mistake as another guy we had on another forum. You keep thinking of Pi as something that has to be measured. The thing is, even the most precise measurement we can muster can easily be trumped in accuracy by quite basic mathematical methods. Irrespective of what value whoever got way back when, the fact is that we can independently obtain it today to an arbitrary precision. No thickness of the line ever comes into it at all.

 

 

Hi! KALSTER :)

 

 

I visited Kenya, Mombasa in the late 1970’s. I enjoyed its beautiful country site and fantastic sunsets. Man, talk about having a Zoo in your very own backyard. Wow!

 

For about 4 thousand year we’ve used geometry for measuring a perfect circle and the diameter never comes into it at all. This method of geometry has gives us an approximation of a perfect circle or false value of Pi. We should ask ourselves why?

 

If we were to use or apply a false value into any type of equation for whatever the reason, wouldn’t it still give us a false value?

 

For about 4 thousand years the equation, c/d = Pi, has existed as the only method in finding the true value of Pi. Not geometry.

 

 

 

;)

[KALSTER]

For about 4 thousand year we’ve used geometry for measuring a perfect circle and the diameter never comes into it at all. This method of geometry has gives us an approximation of a perfect circle or false value of Pi. We should ask ourselves why?

Again with the measuring....

 

There is a clear direct relationship between the radius and the circumference of a circle. As I said, measuring can NEVER be as accurate as is possible with direct and pure geometric derivation. Your formula for deriving Pi requires a measurement of the circumference, so it cannot be as accurate as purely geometric methods.

 

For about 4 thousand years the equation, c/d = Pi, has existed as the only method in finding the true value of Pi. Not geometry.

This is blatantly false. Did you not read CraigD's post?

[Qfwfq]

To put it briefly: [imath]\pi[/imath] is [imath]\pi[/imath] and it has one exact numerical value which can only be defined. We can calculate it to arbitrarily close precision (for any given practical purpose) but we can't and never will be able to write it exactly as a base 10 or base n number; it is not only irrational but it is transcendental too, so we can't even use expressions with roots of rational numbers to write it exactly.

 

The only way to write it exactly is [imath]\pi[/imath] or whatever symbol is chosen to stand for the definition.

[Guadalupe]

Hi! :hyper:

 

I appreciate anybody who takes the time in discussing, explaining, and sharing their ideas and knowledge on any subject that takes place in this forum.

 

Research shows that all the methods used, up to this present date, in finding the true value of Pi have falling short. At best is an approximation of Pi or a false value of 3.14159265… as Pi.

 

It’s not about the symbol Pi that is in question but, the use of a false value of 3.14159265… as Pi. To use this false value with other math formulas is beyond me.

 

My best guess is that, we tired finding the true value of Pi but that’s all we got to work with.

 

 

 

[CraigD]

Research shows that all the methods used, up to this present date, in finding the true value of Pi have falling short.

As numerous previous posts, and nearly any encyclopedia article or introductory math textbook explains, Guadalupe’s conclusion is simply wrong.

 

Guadalupe, my guess is that you are intuitively disturbed by the concept of irrational numbers – numbers that can’t be represented as terminating or repeating decimal numerals. This is a common reaction, that, in my experience, nearly all students experience at some time in their education.

 

However, many numbers, including Pi, have been rigorously proven to be irrational. Such proofs of the irrationality of Pi have been well known for over 200 years.

 

Your insistence that all of the many known mathematical expressions of Pi “fall short” and give a “false value” suggest to me that you need to learn some essential fundamental mathematical principles, not only about number systems, but the principle that rigorous, formal proof is more mathematically credible than intuitive feelings.

[Moontanman]

I don't mean to be out of turn here but can I ask a question? Isn't pi the same as 3 and 1/7? or 22/7

[KALSTER]

I don't mean to be out of turn here but can I ask a question? Isn't pi the same as 3 and 1/7? or 22/7

Those are approximations for ease of use in equations that do not need a high level of accuracy. The true value of Pi, though, can't be represented by a simple fraction.

[Moontanman]

Those are approximations for ease of use in equations that do not need a high level of accuracy. The true value of Pi, though, can't be represented by a simple fraction.

 

Ok, I see what you mean, 22/7 is 3.142857142857142857 and so on.

[Guadalupe]

As numerous previous posts, and nearly any encyclopedia article or introductory math textbook explains, Guadalupe’s conclusion is simply wrong.

 

Guadalupe, my guess is that you are intuitively disturbed by the concept of irrational numbers – numbers that can’t be represented as terminating or repeating decimal numerals. This is a common reaction, that, in my experience, nearly all students experience at some time in their education.

 

However, many numbers, including Pi, have been rigorously proven to be irrational. Such proofs of the irrationality of Pi have been well known for over 200 years.

 

Your insistence that all of the many known mathematical expressions of Pi “fall short” and give a “false value” suggest to me that you need to learn some essential fundamental mathematical principles, not only about number systems, but the principle that rigorous, formal proof is more mathematically credible than intuitive feelings.

 

 

 

 

Hi! CraigD :)

 

The only reason why Pi is irrational and transcendental is because it has a false value of 3.14159265… or an approximation.

 

When I pay for a cup of coffee, I expect to have the exact change and not an approximation.

 

 

 

 

 

:confused:

[Moontanman]

Hi! CraigD :)

 

The only reason why Pi is irrational and transcendental is because it has a false value of 3.14159265… or an approximation.

 

When I pay for a cup of coffee, I expect to have the exact change and not an approximation.

 

 

:confused:

 

Ok, what is the exact value? Unless you know what the exact value is this whole argument is moot. If you don't know it and cannot produce a solution to it then what is your point?