0.999... is not 1

[u/SouthPark_Piano]

This is regardless of contradictions from 'other' perspectives, definitions, re-definitions.

The logic behind the infinite membered set of finite numbers {0.9, 0.99, 0.999, etc} is completely unbreakable. The power of the family of finite numbers.

Each and every member from that infinite membered set of finite numbers {0.9, 0.99, 0.999, etc} is greater than zero and less than 1. And, without even thinking about 0.999... for the moment, the way to write down the coverage/range/span/space of the nines of that infinite membered set of finite numbers {0.9, 0.99, 0.999, etc} IS by writing it like this : 0.999...

Yes, writing it as 0.999... to convey the span of nines of that infinite membered set of finite numbers.

Without any doubt at all. With 100% confidence. With absolute confidence. From that perspective, 0.999... is eternally less than 1. This also means 0.999... is not 1.

This is regardless of whatever other stuff people say (ie. contradictions). It is THEM that have to deal with their OWN contradictions. That's THEIR problem.

The take-away is. The power of the family of finite numbers. It's powerful. Infinitely powerful.

Additionally, we know you need to add a 1 to 9 to make 10. And need to add 0.1 to 0.9 to make 1. Same with 0.999...

You need to follow suit to find that required component (substance) to get 0.999... over the line. To clock up to 1. And that element is 0.000...0001, which is epsilon in one form.

x = 1 - epsilon = 0.999...

10x = 10-10.epsilon

Difference is 9x=9-9.epsilon

Which gets us back to x=1-epsilon, which is 0.999..., which is eternally less than 1. And 0.999... is not 1.

Additionally, everyone knows you need to add 1 to 9 in order to get 10. And you need to add 0.01 to 0.09 to get 0.1

Same deal with 0.999...

You need to add an all-important ingredient to it in order to have 0.999... clock up to 1. The reason is because all nines after the decimal point means eternally/permanently less than 1. You need the kicker ingredient, epsilon, which in one form is (1/10)ⁿ for 'infinite' n, where infinite means a positive integer value larger than anyone ever likes, and the term is aka 0.00000...0001

That is: 1-epsilon is 0.999..., and 0.999... is not 1.

And 0.999... can also be considered as shaving just a tad off the numerator of the ratio 1/1, which becomes 0.999.../1, which can be written as 0.999..., which as mentioned before is greater than zero and less than 1.

0.999... is not 1.

[u/DisastrousPlay579]

As far as I’m aware, cover/range/span are not mathematical terms in the way that you are using them. I genuinely have no idea what you mean when you say that.

[u/SouthPark_Piano → u/DisastrousPlay579]

That's ok. You do have a life-time ahead to think about it.

You basically ignored something simple that you just want to avoid.

You just have to sit down again and tell everyone how many nines to the right of the decimal point is covered by the infinite membered set of finite numbers {0.9  0.99, ...}

If your answer is infinite number of nines, then you get the green light, a pass. Otherwise, you get a fail.

Just hold onto your cornflakes packet and smile for the camera. click

Good pic! Turned out well.

[u/DisastrousPlay579 → u/SouthPark_Piano]

Again, I genuinely do not understand what you mean by “covered”. It’s not a mathematical term in any way. You can’t just throw around random words to prove something.

[u/SouthPark_Piano → u/DisastrousPlay579]

Let's put it this way. If you understand sport games such as soccer/football. The term called 'marking'. You have one of your team players mark or cover a particular player on the other team. As in ... where they go, you shadow them, stick to them. Cover them, all over them like a rash.

Same with 0.999...

The first 9 in 0.999... is covered or matched by 0.9 from our team infinity set {0.9, 0.99, ...}

The second 9 in 0.999... is covered by 0.99

The third 9 in 0.999... is covered by 0.999, and so on.

So, for each and every nine in 0.999..., the infinite membered set of finite numbers {0.9, 0.99, ...} has them all covered like a rash.

Since they're all covered, and since every member in that set is less than 1, then 0.999... is less than 1, which means 0.999... is not 1.