Put a simpler way, if they are not the same number, you bby definition should be able to tell me one number that comes between 0.9999~ and 1. If you cannot do that, they are by definition the same number.
Sorry, the number is off by what could be the smallest number next to zero.
| Kytiara said: I'm pretty sure it doesn't work that way txags911. edit: I guess I should extrapolate a little. Take x to equal 0.9999 x = 0.9999 You can then continue adding 9's towards infinity and it never changes. What you are implying is as x nears infinity, the number tends towards 1 which is true, however it never reaches 1, just gets really really close. |
Correct ...
As you can see in my insanely large post, it's not funny how close the number is to 1.
I hope nobody quotes my insanely large post.
| txags911 said: Put a simpler way, if they are not the same number, you bby definition should be able to tell me one number that comes between 0.9999~ and 1. If you cannot do that, they are by definition the same number. |
False ...
Between any two rational numbers there is an uncountably infinite number of irrational numbers between them, your irrational number 0.999... is between 0.99999 and 1.0
Every time I visualise that 1 behind all of those 0s, it keeps getting pushed back by more 0s. My head is exploding with 0s. 0s, 0s everywhere! I can't take it anymore? I can't keep up with that #1. Those 0s keep pushing it back, farther, and farther. I cannot comprehend the infinite amount of 0s. There are more 0s in front of that 1 than the amount of atoms contained in this Universe, times infinity. It's too big. Too big. No limit. I can never find that 1. It is there somewhere. So... Many... 0s!!! There are more 0s in front of that 1 than every single letter that exists on the Internet, times infinity. I'm going blind by the number of 0s! So... many... 0s! Where's that 1? There is only 1, 1, but an infinite amount of 0s. Oh, I think I found the number 1... Nope, more 0s keep pushing it back, farther, and farther behind. It will never stop pushing it farther behind. The amount of 0s pushes the 1 farther beyond existence. How can a 0 mean nothing when there are so many of them? I am going crazy! CRAZY!!!
A quote from my math teach back in high school: "Two numbers are different only if you can find a value that is between them. Therefore .99999.... = 1" This was an actually quote from my math teacher back in highschool.
HappySqurriel said:
False ... Between any two rational numbers there is an uncountably infinite number of irrational numbers between them, your irrational number 0.999... is between 0.99999 and 1.0 |
That is incorrect. The irational numbers are a different number set then the rational numbers. By definition because of this you can not find an irrational number between to rational numbers.
HappySqurriel said:
False ... Between any two rational numbers there is an uncountably infinite number of irrational numbers between them, your irrational number 0.999... is between 0.99999 and 1.0 |
False. 0.9999~ is not an irrational number any more than 0.333~ is. It is 3 * 0.3333~ (or 3 * 1/3) (or 1)
An infinitely repeating number is not necessarily an irrational one. That's the mistake you made.
Incidentally, the true proof is that between any two non-equal REAL numbers, there is an infinite number or other real numbers between them. So if 0.999~ and 1 are not the same, you should be able to name at least 1 number between them.
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