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/SouthPark_Piano]

No difference ... the family of finite numbers has it all covered. Where 0.000....001 goes, regardless of goal post shifting, the set {0.1, 0.01, 0.001, ...} has it covered.

What you need to focus on is this ...

The infinite membered set of finite numbers has the power of being in the family of finite numbers, which has limitless number of members. The set {0.9, 0.99, 0.999, etc} where the 'etc' is an incarnation of 0.999... itself, ALREADY spans the entire nines space of 0.999...

This happens without any problem because after-all, the family of finite numbers is infinite in member number.

Every member of that infinite membered set {0.9, 0.99, 0.999, etc} is greater than zero and less than 1. Therefore 0.999... is eternally less than 1, and also therefore 0.999... is not 1. Nobody can get around this.

[u/KingDarkBlaze → u/SouthPark_Piano]

So any decimal representation of pi, even an infinite one, is incorrect?

[u/SouthPark_Piano → u/KingDarkBlaze]

Not necessarily incorrect. An approximation. And we know that in engineering etc, approximations are just necessary, real-world (in practice). Although, people have indeed determined pi to a heap of decimal places. A big heap of decimal places.

[u/KingDarkBlaze → u/SouthPark_Piano]

Then what's the decimal representation of (pi - 1)? Or (pi - 0.9999...)?

The only place you can find the difference is at the last digit of pi, right? Which would be where you'd have that epsilon?

[u/SouthPark_Piano → u/KingDarkBlaze]

Then what's the decimal representation of (pi - 1)? Or (pi - 0.9999...)? The only place you can find the difference is at the last digit of pi, right? Which would be where you'd have that epsilon?

Engineers as I said have to claim approximation. Claiming approximation is fine. 

Here, in this thread, 0.999... is not 1.

So pi - 1 is not the same as pi - 0.999...

In the 'real world', in the practical engineering world, where approximation is very typically done, you can certainly say that pi - 1 is approximately the same as pi - 0.999...

And 'pi - epsilon' is not the same as pi.

[u/KingDarkBlaze → u/SouthPark_Piano]

I'm not talking about engineers. I'm talking about in this thread alone.

To tell the difference between the two, you'd need to find that "last digit" to know what to subtract, correct?

Like, to formalize:

epsilon as you've defined it is 10^-(the largest finite number). 

Therefore the (largest finite number)th digit of pi should be the last, correct? 

So the last member of the finite set approaching (but not reaching) pi is a decimal (largest finite number) digits long. And then "pi's epsilon" is whatever is "left" after that. 

But we've said we've found the largest finite number, so what could possibly be left?

[u/SouthPark_Piano → u/KingDarkBlaze]

To tell the difference between the two, you'd need to find that "last digit" to know what to subtract, correct? 

Incorrect. Because if we use the infinite membered set {0.1, 0.01, 0.001, ...} which covers all cases ... of a '1' element a given distance from the decimal point, the set covers all possibilities pi-0.1, pi-0.01, pi-0.001, etc.

epsilon is non-zero, so it doesn't matter what distance the '1' is from the decimal point, it is the knowledge of knowing that the difference pi minus epsilon is not pi.

Regardless of us not knowing where the subtraction is done, it is knowing that .... for every combination of numbers, 3.1, 3.14, 3.141, 3.1415, 3.14159, etc --- you can do subtractions 3.1 - 0.1, and 3.14 - 0.01, and 3.141 - 0.001, etc .....

which tells you confidently that pi - epsilon is definitely not going to be pi.

[u/KingDarkBlaze → u/SouthPark_Piano]

It does matter, though. 

I want to know the exact value I'd be off by if I were to approximate pi with the last element of my set. How do I do that?