Is the rationality of Pi known

Thank you for trying to help me understand, I do very much appreciate it. My position: I accept the number calculated using Archemede's method known as Pi is an irrational number, because not seeing any numbers repeating in a sequence in one trillion digits is good enough evidence as any. You have hit the nail on the head about what I disagree on i. e. infinity, and 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. I didn't realize there were so many ways to show 0.999. . equals 1, see below The simplest is: 0.1111. .. = 1/9 multiply both sides by 9 0.9999. .. = 1 Another is: S = 0.999. .. Then 10S = 9.999. .. 10S - S = 9.000. .. S = 1 And then there is your method too, which is equally good. However I believe the decimal system is made more for practical purposes Let's look at the first one, my instinct is not to type 0.999. .. when I multiply 0.111. .. by 9. I'd simply put 1. It just seems like a clever trick to show 0.999 and 1 are the same thing, when really when we say 0.111. .. is 1/9 it is because this is the best way to show what one ninth is in a decimal format. You say there is no number between 0.999. .. and 1, however I believe there is. This number is 0.00000. .. ..1 which is basically an infinitely small number and is the same as 0.0000. .. 2 and 0.0000. ..3 etc. Take a look at the below to see where my idea is from: 0.9 + 0.1 = 1 0.999 + 0.001 = 1 0.999999 + 0.000001 = 1 if I have 1 million 9's after 0, I need 1 million minus 1 zeros before 1. I thus need to make an addition to make a whole number. . You are saying that 0.999999. .. = 1 ,meaning that 0.00. ..1 is equal to EXACTLY ZERO If this is true then 0.9999. .. . + (0.000. ..1 x 0.9999. .. ) should equal 1 Consider another equation 0.9999. .. . + (0.000. ..1 to the power of 0.999. .. .) = _______ The point I'm trying to make is that an infinitely small number is still a number. Mathematicians solve the above by saying that anything multiplied by zero is equal to zero, and you have showed that 0.000. ..1 is equal to zero so you would fill the blank space with either 1 or 0.999. .. I have a deep mistrust of maths, e. g. answer the following, How many degrees are there in a square? How many degrees are there in a triangle? How many degrees are there in a circle? If you have remembered what you are taught you will have said that a square and circle both have 360 degrees when a circle is NOT a square. If we accept that a triangle has 180 degrees and make the infinitely small number be known as K then the internal angle of a circle should be (180 - K) x any reoccurring whole number. I'd also like you to consider the following Time can be continually split in half. Imagine having a sophisticated timer and lamp. The lamp is on, but after exactly 1 minute the lamp is turned off. When 30 seconds pass it is turned on. After 15 seconds the lamp is turned off. It will be turned on after 7.5 seconds, and so on. This demonstrates the existence of 0.000. .. .1 i. e. an infinitely small number. The same can be done with space too, though we like to think space is limited and this is why I think mathematicians say that 0.999. .. is equal to 1. If 0.999. .. equals 1, and thus 0.000. ..1 doesn't matter since it is nothing then you should in my opinion be able to tell me whether the light will be on or off in exactly 2 minutes of the above experiment.