Is the rationality of Pi known

I do still at this point believe the rationality of Pi (i.e. circumference/diameter) is unknown. and I did say: "having read your arguments I will accept that if 0.9999... equals 1 then Pi has been calculated exactly, and is "known" even if we can't ever write down all of the digits". this is because it makes sense to say Pi is irrational "IF" Pi is a number which can have two different representations, I'm not going to ask for the 2 trillionth digit and declare the rationality is unknown, I wouldn't have a debate where I can't learn anything. When I offered two proofs showing 0.999.. equals 1, it is not because I believe any are correct. In fact I believe they are examples of BAD MATHS (only when dealing with a crazy amount of precision though) Let's look at the second proof that I showed in round 3, with my corrections, see below: S = 0.999.. i.e. 1 - k 10S = 10 - 10k (10 - 10k) - (1 - k) = 9 - 9k S = 1 - k At this point you may be thinking 'hang on I've said 0.00...1 and 0.00...2 etc are the same, so surely I should write 10k as 1k' and you may tell me it should look like the below: S = 1 - k 10S = 10 - k (10 - k) - (1 - k) = 9 9S = 9 S = 1 Though the end result of 10k and k is the same, they are not the same, and therefore should not be treated the same! Doing so is BAD MATHS in my opinion. Infinitely small numbers are much like infinitely large numbers, imagine you can live forever and can press a button to double your money, you will be infinitely rich, pressing a button which can quadruple your money each time will not change how wealthy you can be, however with the second button you can get rich faster. Alternatively imagine if I lost half of my money every second and someone else lost a quarter of their money each time, it is natural to say we will both end up with "nothing" but the pattern actually continues for eternity. Applying this maths someone can have an infinite amount less than me. But it is possible to make it so that a person always has 10 times less than me. Let's move on now.. 0.999... = 0.9 + 0.09 + 0.009 + 0.0009 etc 1 = 1 + 0... So, basically 0.999.. is ALWAYS less than 1, and by an infinite amount i.e. 0.00...1 Onto your comments, why is the internal angle of a circle 179.99...? How to calculate internal angles of polygons - let's take an octagon which as you know has 8 sides, 360/8 = 45 degrees There are 180 degrees in a triangle, so the other angles add up to 180 - 45 = 135 There are 8 sides, so an octagon has a total of (8 x 135) degrees = 1080 degrees If we have a 16 sides shape now 360/16 = 22.5 degrees There are 180 degrees in a triangle, so the other angles add up to 180 - 22.5 = 157.5 It will therefore have total of (16 x 157.5) = 2520 degrees If we have a million sided shape now 360/1 million = 0.00036 There are 180 degrees in a triangle, so the other angles add up to 180 - 0.00036 = 179.99964 It will therefore have (1 million x 179.99964) = 179999640 degrees As you can see the more sides there are in a polygon the closer the angle "OF ONE SIDE" becomes to 180. A circle only has one side! I can place any numbered side polygon inside a circle and the angle will never reach 180, because only a straight line has 180 degrees. The internal angle of a circle is therefore 179.99... degrees Having written that I have come to realize that an infinite sided polygon perfectly represents a one sided circle, and concede to defeat. I would like to thank you for having this debate. You have helped to convince myself that the rationality of Pi is known, and that Pi (circumference/diameter) is without doubt an irrational number, at least in my mind anyway, but I will let voters decide Hope you have found this debate interesting, best regards, Kirk