Might Tested
Noob
J, how about you just do what people typically do when they don't understand something. Trust the people who have been studying this extensively.
Luckily, in the world of mathematics, no philosophical nonsense needs to come into play. Mathematicians have defined the rules of real numbers. In this instance, to say .9 repeating is not 1 because .9 repeating is a fictitious concept as opposed to 1 being a real number, is on the same level as saying 1/3 is a fictitious concept. A slim margin of mathematicians deny that 1/3 is .3 repeating. A kid could sit and keep dividing 1 by 3 if he lived forever or prove the result by induction (which is an accepted method of mathematical proofs).
.9 repeating has never ending digits, but its value is not never ending. This is the same as saying 1.000000000.... repeating is not a real number because it has never ending digits, but clearly its value is finite. Thus, it should be fairly intuitive to apply that to .9 repeating. You would be correct however, in stating that a value with never ending magnitude is not a real number. This is why infinity is not a real number. You can't explicitly state infinity's magnitude. Pi is a real number, 3.14159265... is pi's irrational representation, just as .9 repeating is 1's irrational representation.
If you want a "real" way of demonstrating that .9 repeating is 1. Imagine you're on a 1 mile racetrack. You know that you have not reached the finish line if there is a quantifiable distance between your current position and the finish line. For example, if you've driven a quarter mile, then there is still the half mile marker in between you and the finish line. Suppose, you are now .75 of the mile. Well, there's still the .8 mark before you hit the 1 mile line. Now, this will continue until you hit say, .99 , at which point some idiot will say "Ha, we're there", to which you would reply "Well, no, there's still the .995 marker we haven't crossed to get to 1." You could keep incrementing forward to say .9999, but there would still be .999995 that you would have to reach, and that is a quantifiable position between you and the 1 mile marker. Now let's say, you are at .9 repeating. Some ignorant rube will at this point say "Oh, you haven't crossed over to the 1 mile marker, because there's .999999999999....99995 in between you and the 1 mile marker!" But .9 repeating does not end. There is no possible number in between a decimal point and infinite 9s and the number 1, thus you can conclude that the two positions are identical. This may seem a bit absurd, but it's actually a common method of identifying limits using the squeeze theorem. You can google the squeeze theorem if you want, it would probably explain this concept better.
That being said, if you want to completely refute this statement, you would pretty much be flying in the face of calculus and Ph.D mathematicians over centuries, but since it is possible that the majority of the learned community is wrong since the advent of mathematics and the scientific method, state your case in a peer reviewed journal, and proceed to collect an absurd amount of money, fame, and titles.
