3 ms·
0.999... = 0.999...9 0.999...9 + 0.000...1 = 1 0.999...0 + 0.000..1 = 0.999..1 0.000...1 = 1/∞ 0.999...9 = 1 - 1/∞ 0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/
by brutt 6y ago
0.999... = 0.999...9
0.999...9 + 0.000...1 = 1
0.999...0 + 0.000..1 = 0.999..1
0.000...1 = 1/∞
0.999...9 = 1 - 1/∞
0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞
If x/∞ = 0, then 0.999...x = 1.
If x/∞ ≠ 0, then 0.999...x ≠ 1.
- brlewis 6y agoOk, now you're saying that infinite decimals have final digits.
- brutt 6y agoIf Universe is infinite, then if we compare you to size of Universe, you are infinitely small, so you don't exists at all. Why I should waste my time? If Universe is finite, then finite number of elements can make only finite number of combinations, thus this discussion is repeated infinite number of times again. Why I should waste my time again?
- brlewis 6y agoYes, you're wasting your time if you compare my size to the size of an infinite universe. If you really want to waste your time that way, you don't want to use real numbers. On the real number line my size is exactly zero. You need to go into infinitesimals, which are out of place when you're looking at decimal notation, which is only for real numbers.
- gspr 6y agoNone of these, except 0.999… and 1 are well-known standard objects in this setting. You have to define what you mean.
- brutt 6y agoI defined it: 0.000...1 = 1/10^∞ = 1/∞
- gspr 6y agoThat's not a definition. Neither 10^∞ nor 1/∞ is defined in any standard system, so you'll have to define those too if you want to use them to define 0.000…1.
- brutt 6y agoSee https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number .
- gspr 6y agoYou're working with surreal numbers? This is not what people would expect unless it's explicitly stated. In addition, you're likely going to have a hard time explaining surreal numbers to someone who struggles to grasp that 0.999... = 1 in the ordinary reals.
- drran 6y agoSurreal numbers and infinistemal are simpler to work with when you need to work with infinite series. Here John Conway explains them: https://www.youtube.com/watch?v=1eAmxgINXrE https://www.youtube.com/watch?v=1eAmxgINXrE
- drran 6y agoNice notation. I will steal it.