Lots of Pi-related questions

[Shadow]

Hello all,

 

Well, this is my first post on this forum. I decided to find a forum like this since the questions I'm about to ask have been bugging me for some time now. Please note though, that I'm only 15 years old (just entered High School a month ago ) meaning that I'm probably missing something that may seem completely elementary to you University Professor's ;-) So excuse my ignorance

 

Now, we all know that Pi is irrational, meaning that it can never be precise. That's also why we can never measure the exact circumference/area of a circle/sphere/whatever. My imagination is still a little limited, and my head usually hurts, when I try to think in 3D, so I'll stick to plain old "flats".

 

My question begins here: Imagine we took a piece of rope/string or something, measuring exactly x meters, made it into a perfect circle, and then attempted to calculate it's circumference with Pi (let's assume that the diameter of the circle would be a rational number). While the circumference would still be exactly x meters, Pi would be trying to convince us that the circumference was around zero point something-something-something x. Because it's irrational of course. Now, if we did it the other way around. If we had a circle that according to Pi had a circumference of y, while the actual circumference was z (y=zero point something-something-something z). We took the circle apart, forming a straight line from it in the process, and measured it's length. We would then find out that the circumference of the circle was actually z, while Pi was telling us that it was y, or zero point something-something-something z.

The question: Why don't we measure the circumference of circles like this? And also, how is it possible that Pi is irrational? If what I'm saying is not trash, we should be able to compute the circumference of a circle precisely this way, so that means we should be able to compute Pi precisely. This means that I'm wrong somewhere; where? Another thing, is it possible to draw Pi similarly to the way we draw for example [math]\sqrt18[/math] ? (-> We make a right-angle triangle a,b,c, where c is the hypotenuse and a,b=3 [by the way, what is this process called?]. Whatever it's called, how is it possible to make an irrational number (=infinite number of decimal places) into a line that has a beginning and an end? I know I'd never be able to make a precise [math]\sqrt18[/math] line with a ruler and a pencil, but "virtually" speaking...)

 

I know I'm missing something here, I just don't know what. When I asked my math teacher, she didn't have an answer, so she told me I was way to curious for my age. So I've come here

 

Thanks all,

 

Shadow

 

PS.: Brilliant forum you've got here admins!! You should be proud ;-)

[uncool]

As to the first question: The circumference would still be exactly y. If it is y, then it will always be y. The thing about pi is that pi itself cannot be exactly enumerated as a rational number. However, that does not mean that other things that might relate to pi cannot be - for example, pi*(1/pi) = 1 can be. The problem with your example is that you forgot about the diameter - the diameter is what cannot be measured in your example.

 

No, we cannot form pi from constructions. This deals a little in Galois theory, but the only "constructible numbers" are those whose "minimal polynomial" is of a degree that is a power of 2. pi is a transcendental number - that is, it is not a number that solves a rational polynomial a_0 + a_1x + . . . + a_n*x^n, and therefore pi cannot be constructed.

Just remember - a circumference may be exactly calculable in terms of decimals. Just not a circumference and a diameter at the same time.

=Uncool-

[Mr Skeptic]

Pi can be calculated as close to perfectly as you want using various methods. Your measuring method is not one of them, though. Because you cannot measure precisely, your result will be inaccurate (you need to measure both the circle and the radius, and have a perfect circle).

 

One way to approximate pi is by modeling it as a regular polygon with infinite sides. Calculate the circumference of the polygon as the number of sides increase, and you can get as close as you like to pi.

[Shadow]

Wasn't that how Pi was originally discovered ?

 

but anyway, thanks, both of you

 

Another question, is it possible to have a circle with a rational circumference and diameter ?

[insane_alien]

no. since there is a multiplication by an irrational constant(or division if you are coming the other way) the only way that could happen is if pi was rational.

[Mr Skeptic]

Wasn't that how Pi was originally discovered ?

 

but anyway, thanks, both of you

 

Another question, is it possible to have a circle with a rational circumference and diameter ?

 

No. If that were possible, you could just divide the rational circumference by the rational diameter, and you would get pi and pi would be rational. But you can't do that because pi is not rational.

[Shadow]

Yeah, that makes sense....well, thanks all !! :-)

I know where to come now if my math teacher again has no idea what I'm talking about

[Country Boy]

As long as you don't mind if WE have no idea what you are talking about!

[Sisyphus]

 

Another question, is it possible to have a circle with a rational circumference and diameter ?

 

No. Here's a simple, informal proof:

 

Any rational number can be written as A/B, where A and B are integers.

 

Let the diameter equal A/B. The circumference, therefore, is A*pi/B. But since pi is irrational, there is no integer A that you can multiply it by to get an integer A*pi. (If you could, then you could just write pi as a fraction.) Therefore, (A*pi)/B cannot be rational.

 

Similarly if you start with a rational circumference A/B, in which case the diameter would be A/(B*pi). Yatta yatta.

 

EDIT: Oops, apparently already answered.