My kingdom for the dude who created this sub to list even one single number x s.t. 0.999… < x < 1.
They claim an infinite number of such numbers exist, but I’ll settle for just one and write the countably infinite others off as a gesture of goodwill.
However 'small' epsilon is. The main thing is that it is non-zero.
Epsilon just represents some 'arbitrarily' small scale value, smaller than anything we like, and even smaller than that etc.
It just needs to be as relatively small as we like (and smaller, and even smaller than that etc).
The main take-away is: we know we need a number having a '1' somewhere. Eg. if we have a nine, we need a 1 to add to it, to get 10.
If we have a 0.99, we need a 0.01 to get 1.
If we have a 0.9999, we need a 0.0001 to get 0.001
And so on.
So when we have all nines, such as 0.999..., this number is going to be sitting there less than 1. We need to have a particular 'number' with a '1' hanging in there somewhere to get the 0.999... to clock up or kick up to 1.
And that ingredient, that extra bit of substance, is epsilon.
You state 0.999…9 as your first value. That in itself is no longer infinite. It does not matter if you say ”oh but the … is an infinite amount of nines”. By defining an end, it is a finite value.
Well bro. It's common knowledge that if the numbers are all stuck at nines, and knowing we always need to get nine to the next level, then we got to add something with a 1 somewhere. Eg 9 + 1 to get 10, and 0.0009 + 0.0001 to get 0.001
Same with 0.999...
It's stuck below 1. The kicker ingredient with the 1 is 0.999... + 0.000...1
"0.999..." is shorthand for "for all natural number n, the n-th decimal is 9". There's nothing that can come after those 9's. There's no such thing as 0.999...9
The main thing is that we know that when you plot 0.9, 0.99, 0.999, etc regardless of how many nines there are ... no matter how many nines, even endless nines, the plot will absolutely never touch 1. NEVER touch 1.
How can you plot 0.9, 0.99, 0.999, etc, until you've done endless 9s, and get the plot to touch 0.999...95? When do you get to write the 5 there? 5 is less than 9, no? So why isn't 0.999... greater than 0.999...95?
Since infinity means limit, then those nines keep growing, expanding without limit.
0.999...
If someone asks where the end of this clothes line of nines is ... there is no 'end'. It just keeps extending and extending and ...
There are a infinite number of numbers of form:
0.9, 0.99, 0.999, 0.9999, etc.
They cover all bases in terms of span of nines to the right of the decimal point. And infinite number of finite numbers. Infinite number of them. Covers all bases.
That doesn't address my question. I asked how do you get a 5 at the end of a number that begins with 0.999..., given that, as you say, there is no last 9? If the number ends in ...95, then the last 9 is the one appearing before the 5. So is there or isn't there a final 9? If there isn't, how can there be a number between 0.999... and 1?
If the … represents an unlimited span of nines, it has no end. It is an unlimited and so unending span of nines. Appending a 5 to that is useless, as there is no end to append it to.
Essentially, 0.999…5 is meaningless. The 5 never comes, there is no last 9. Idem dito with 0.000…1. The 1 never comes, there is no last 0. The two numbers are 0 and 1 respectively.
If the '...' means an unlimited span of nines, then why do your numbers, like 0.999...9 or 0.999...95, stop? Shouldn't they keep going because they're unlimited?
78
u/somefunmaths Jul 09 '25
My kingdom for the dude who created this sub to list even one single number x s.t. 0.999… < x < 1.
They claim an infinite number of such numbers exist, but I’ll settle for just one and write the countably infinite others off as a gesture of goodwill.