Originally posted by: ndee
so if everything has endless numbers after the comma, it just gets upped to the next bigger number? Like 0.3999999999999... would get 0.4?
Exactly, now you're getting it. Although .388888888..... will not eaqual .39 simply because .3899 is greater then .38888888....... and .3899 is less then .39. But what number is less then .34 but greater then .3999999999999...... I don't think you will find that number and the only case were you cannot find a real number that is less then another real number and greater then a diferent number will of the form were you need to find a number between .34 and .34 which is because .34 = .34
The same holds for 1. If we cannot find a number between 1 and 1 then 1 = 1, now that was hard. I also don't know of a number between 1 and .9999999....... so I guess 1 = .9999999........ strange isn't it. Please prove me wrong, find an x such that 1 < x < .999999......
There was a statement that we don't know what .9999999..... realy is so we can't compare 1 and .99999999....... Yet we can know what the square root of 2 is. Square root of 2 is irrational meaning that it cannot be written in the form of a fraction in lowest terms. And although not an official definition, it means that there will never be an end to the decimal representation of square root of 2 and most importantly there will never be a pattern to the representation of square root of 2. If there is a pattern then theoreticaly if you create a system running a simple infinite loop, then you could output the entire number in an infinite amount of time. And if the loop results in an accurate number then the number much be rational.
Now take a program, print "." then set up an infinite loop to print "9" right after the last character. See it run and after an infinite amount of time you will see .9999........ But if that is true then .9999...... must be rational and can be written as a fraction, but what fraction. I can only think of the fraction of 1/9 + 8/9 = 9/9 as mentioned earlier. But 9/9 = 1
I sometimse like to say that the difference between 1 and .999999..... is .0000000........repeating1 but that in itself is not correct. Since it repeats infinite amount of times and infinity cannot be compared to a number then it would be true, but if we are saying that 1 is not .999999...... and .00000000..........1 is not 0 then we cannot say that infinity = 2 times infinity or even infinty + 1 = infinity. If we say that that 1 after the zeros will ever come then infinity must be treated as an exact term such that only infinity = infinity. Threfore if I take my term .0000000......1 and .99999999......... and add them then I would simply get .9999999999........1 as we have both the 0 and the nine repeating infinetly many times and one after that we add the final 1. So what we need to do is remove one of the 0s from the .0000000..........1 term or basicaly multiply it by ten to get to a .00000000..........*10 1 term. and then you could say if infinity ever stopped then you could provide an exact number to add to .999999........ repeating to equal to 1.
But as you have noticed that this entire argument is irrational and has gaps on to many levels and it also is the only argument that could ever proove that .999999...... is not equal to 1 so you either agree with my ramblings above or aggree that 1 = .9999999........ or you could ingore both options and simply agree that 0 = 1 = 2 = 3 = 4 = 5 = 6 = ..........