| MDMAlliance said: Actually, not really. You never "reach" it. That is part of the problem. Even with the theories that say 0.999... is 1, your sums never go to the point where it's like, "now I add this last part to finally get to X." These are all based on theories that people accept by faith. I am arguing more on a semantic level, but also on a mathematical level. For the most part, I am really trying to figure out as much as I can about why people say 0.999... equals 1, but to no avail since no one is really making an argument that details why. All I see are flawed systems to say why 0.999... equals 1. Why not someone argue to me, explicitly why these theories are correct and why they actually make 0.999... equal 1? Really, the hole I see is that if it eventually did hit one, there should also be an objective point where the switch happened. I can also see why 0.999... is looked at "since it goes on forever, and there cannot be a number between 0.999... and 1, 0.999... has to be 1." I understand that. That doesn't change the fact that it isn't a fixed object in my perspective. Infinity would have to be a fixed object, too. |
Yeah, it looks like you're arguing about semantics. When people say it is equal to 1, they don't mean it transfroms from a decimal into 1. It is still expressed as 0.999... It just so happens that 0.999... and 1 are of the same value. It doesn't "hit" 1 in the sense I think you're describing, but it is equal to 1 in terms of its value. The point is there is no difference between 1 & 0.999.... under any circumstances.
The fact that you understand that there's no number between 1 & 0.999.... should be enough reason to believe they are of the same value. Whether or not you feel comfortable calling them the "same number" is really all semantics.
BTW, the sums do get to a point where it equals X. If you have a convergent geometric series, and take the sum of a finite number of figures, then yes you will only approach the limit. But with 0.999... we aren't adding a finite number of figures. We are adding an infinite number of figures (which isn't an objective point as you implied it was), in which case, the the sum will equal the "limit". This is accepted in mathematics.
I feel like I'm starting to repeat myself so I just post a link of this post which I think is more convincing, albeit quite long. Give it a read later if you want.
