Is the rationality of Pi known

In light of Con's honest exposition of their position, I'm going to take a slightly different tack. For this round, at least, I shall change my direction from trying to win a public debate to persuading Con of the possibility of determining the rationality of pi, in theory. I totally understand your correct position that "it doesn't matter how many trillion digits you determine, you haven't demonstrated irrationality". I offered the idea only as suggestive evidence, not as a serious route to demonstrating rigorous proof. Reading between the lines in what you've written about polygons approximating circles, I conclude that you have deep suspicions about infinite series. I understand this; Infinities and how to handle them have long been hotly contested by professional mathematicians... there is now almost universal consensus, though, and in cases such as the determination of the rationality of pi I doubt that you'll find a single professional mathematician in the World who would admit to even the slightest doubt. I myself, not being a professional mathematician, admit to areas of personal doubt in mathematics (such as Cantor's Diagonal Argument) where I have to deal with the cognitive dissonance of trusting the mathematical consensus whilst harbouring secret reservations. As it happens, I suspect that the handling of infinite series is exactly the nub of your doubt; I've spent a long time soul-searching the questions that arise around this topic and getting comfortable with some infinite series, so perhaps I am in a position of being able to convince you of those things which I have satisfied myself about; I'm gonna try, at least! First up, I'm going to point out something that might seem, at first glance, to throw into question how much we can trust decimal expansions. It is simply this: 0.999... is strictly equal to 1 Now, that's not approximately equal to... that is identical to. Fortunately, since both representations (0.999... and 1.000...) are infinitely recurring decimal expansions, they share the property of being rational... but... what would happen if there were such an equality in which one side was recurring and the other was not? Is such a thing possible? Well, how about we go the other way and consider the possibility of a number that differs from 1.000... by being just one digit larger in the "last" digit? Such a thing, were it possible, would clearly not be a recurring decimal expansion and it would be as close to 1.000... as is 0.999... (literally no difference!) - but, since the focus here is working out what ratio we could use to arrive at our desired number, we can establish that we would need the ratio of (infinity plus one) to (infinity) to produce this number... and since (infinity plus one) is equal to (infinity), we simply get 1... or, you could just say "you can't have infinity plus one" and let the trail go cold. I cannot offer you a rigorous proof that you'll easily grasp at this point, but I hope that you can accept in your heart and mind that constructing an equality with a recurring decimal and a non-recurring decimal is impossible. This is, I contend, one way of looking at the concept that rationality and irrationality are "real" concepts! This is just one way of looking at the limit of the definition of what a recurring decimal *is* (since we can never list all of the digits of any number). The take home idea that I'd like you to consider from this example is loosely expressible as "the infinitieth digit really doesn't matter". Now, I'd like to take a little time to explore how one can convert any recurring decimal expansion into a ratio. I suspect you may have seen the method, but here it is being used to work out what 0.345345345... is as a ratio (I honestly just made that number up off the top of my head): x = 0.345345345... 1000x = 345.345345345... 999x = 345 x = 345/999 x = 115/333 Looking at that third step, we are asked to accept that when we shift the digits three to the left and subtract, *all* of the remaining expansion of the decimal cancels out... even the final three digits, which intuitively don't exist in the 1000x line. I'm hoping that you accept the validity of the logic despite the slightly jarring counter-intuitiveness. Our intuition is sometimes wrong! The take-home idea that I'd like you to consider from this example is loosely expressible as "any arbitrarily large number of digits from the extreme end of a decimal expansion really don't matter"! Now, if the final digit, or indeed the final million digits don't matter, why is that? It doesn't sit well with our intuition! Simply put, though, we can say that differing by an infinitely small amount IS the same thing as being equal. This is not just a "for all practicable purposes" argument... this is a statement of absolute equality. I'd be interested to know where you stand at this point in my exposition! Now to extend into slightly more abstract infinite cancellations... Consider S = 1 + 1/2 + 1/4 + 1/8 ... ad infinitum Then S/2 = 1/2 + 1/4 + 1/8 ... ad infinitum Therefore S - S/2 = 1 Therefore S = 2 Hopefully you are still comfortable that we are on reasonable grounds. There is danger here, to be noted... this is not necessarily going to hold if the terms of our infinite sequence are increasing in magnitude! I'm going to end this round, soon, a little lamely (I'm running out of time this evening and characters to use, and I'd like to know where you stand before carrying on any further). You alluded to polygonal approximations of pi... we can construct a triangle, square, pentagon, hexagon, heptagon, etc, of unit "radius" and get closer and closer approximations of the area of the circle (and, thence, pi) by this method. Since this is an infinite series in which successive terms strictly decrease, we can in theory get a good handle on what the devil pi is! In the light of what you said about multifaceted polygons approximating but never equalling circles, I'd point out that the nature of talking about series that approach the limits of infinity are such that, in fact, an infinitely-sided polygon is identical to a circle... and that's the very nature of calculus in a nutshell... put another way, there is no more than an infinitesimal difference between the two, which should be sufficient. But, even if you are not happy with the accuracy of the physical model in Euclidian space, I hope that I have persuaded you of the truth contained in the infinite algebraic expansions... essentially, I appeal to your reason to accept that a convergent series that approaches the answer we desire at infinity is not just "sufficient for practical purposes" but is, in fact, absolutely, completely, truthfully, mathematically accurately the right answer... and that this level of precision is absolutely sufficient for determining whether a decimal expansion is recurring and whether a given constant is, in fact, irrational. Certainly I have not tried to outline a proof of pi's irrationality here; all I have tried to do is to lift Con's apparent mental block towards the possibility that mathematics has the authority to even talk to the rationality of pi. Depending on Con's response to these musings, I may continue on this path or I may return to trying to win the debate for the next round : )