Uncovering the Irrationality of Pi: Exploring Its Definition and Proof

[Nerditude]

Here's a question. Pi is said to be the ratio of a circle's circumference to its diameter. If this is the case, what does it say about the circumference of a circle that pi is still irrational.

I get that pi is also used in the calculation of a circumference in the first place. Since this is true, it has to be the case that pi has been proven to be irrational independently of its definition.

So, the question is, how did we prove pi is irrational?

 

[Anti-Crackpot]

So, the question is, how did we prove pi is irrational?

Nothing I can tell you that a quick Google search won't tell you. But one thing you might want to be aware of is that pi is also transcendental, meaning that it is the solution to no polynomial equation, or more precisely, "that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients." See: http://en.wikipedia.org/wiki/Transcendental_number

This, in contrast, for instance, to the Golden ratio, which is irrational, but not transcendental.

Incidentally, here is a link to another thread on this forum where the same issue was discussed at some length:

Looking for "Easy" proof of Pi Irrational
https://www.physicsforums.com/showthread.php?t=8193

As for the below question...

Pi is said to be the ratio of a circle's circumference to its diameter. If this is the case, what does it say about the circumference of a circle that pi is still irrational.

It's kind of mind-bending, but if you were to snip a perfect circle and stretch it out into a straight line along an x-axis demarcated into measurement increments as small as you please, then, while that line it would be "this long" and no longer, or "this short" and no shorter, well, good luck being able to locate, or rather, specify, where exactly the endpoint of that line is relative to your 0 point.

 

[Number Nine]

Halls of Ivy mentioned the following theorem in the above linked thread...

Of course, that requires that one know the theorem:

Let c be a positive real number. If there exist a function, f, positive on the open interval from 0 to c, continuous on the closed interval from 0 to c, and such that all its anti-derivatives can be taken to be integer valued at 0 and c (by appropriate choice of the "constant of integration") then c is irrational.

The proof of that is elementary but long.

Does anyone know where I might find a proof? That seems like a wonderful theorem.

 

[SteveL27]

Halls of Ivy mentioned the following theorem in the above linked thread...



Does anyone know where I might find a proof? That seems like a wonderful theorem.

I just Wiki'd this.

http://en.wikipedia.org/wiki/Proof_that_π_is_irrational

 

[blue_raver22]

I would like to clarify that the irrationality of pi is a well-established mathematical fact. It has been proven using various mathematical techniques, including but not limited to, the proof by contradiction and the continued fraction representation method.

The definition of pi as the ratio of a circle's circumference to its diameter does not necessarily imply its irrationality. In fact, there are other irrational numbers that can also be defined as a ratio, such as the golden ratio.

The irrationality of pi is a consequence of its infinite decimal expansion, which means that it cannot be expressed as a finite or repeating decimal. This is what makes it such a fascinating and mysterious number.

The proof of pi's irrationality is a complex and intricate process that involves advanced mathematical concepts and techniques. It is a result of centuries of mathematical research and development, and it is a testament to the power and beauty of mathematics.

In conclusion, the irrationality of pi does not say anything about the circumference of a circle, other than the fact that it is a fundamental constant that is used in its calculation. Its proof of irrationality is a testament to the depth and complexity of mathematics, and it is a reminder that there is still much to uncover and explore in the world of numbers and geometry.