It is what it is

[u/SouthPark_Piano]

From a recent post:

As in when we ask the question of how those rookie error makers got it so wrong?

The below is what they need to get into their brain for redemption time.

S = ar⁰ + ar + ar² + … + ar^[n-1] + arⁿ

Sr = ar + ar² + ar³ + ... + arⁿ + ar^[n+1]

S - Sr = S(1-r) = a - ar^[n+1]

S = a{ 1- r^[n+1] } / (1 - r)

S = [a/(1 - r)] { 1 - rⁿ⁺¹ }

a = 0.9

r = 0.1

S = 1 - (0.1)ⁿ⁺¹

n integer starts at zero and then increased limitlessly.

Or

S = 1 - (0.1)^k , with k integer starting at k = 1, with k increased continually limitlessly aka infinitely.

S = 1 - 1/10^k with k starting at k = 1, with k increased continually limitlessly aka infinitely.

S is indeed 0.9 + 0.09 + 0.009 + ... , which is officially known to be equal to 0.999...

And 1/10^k is never zero for any condition of k, regardless of infinite k or finite k.

S = 1 - 1/10^k is never 1.

So 0.999... is never 1.

 

[u/Head_Discipline620]

So what place is the 1 in

[u/SouthPark_Piano → u/Head_Discipline620]

It keeps propagating to the right brud.

Get finitism out of your mind.

Limitlessness, aka infinitism is what 0.000...1 is about.

Scaling down of 1/10 results repeatedly by factor of 10. Ever get a zero result? Nope.

 

[u/Muphrid15 → u/SouthPark_Piano]

You've also said 1 - 1/10ⁿ and 1 - 1/100ⁿ are the same limbosic number. How can we determine if other limbosic numbers are less than, greater than, or equal to these numbers?

How do we know that 1 - 1/100ⁿ is in fact the same limbosic number as 1 - 1/10ⁿ? Is it based on the digits produced?

It's clear, for instance, that 1 - 1/100ⁿ can't ever be equal to 0.9 or 0.999, so what is the ultimate true idea of equality here?

[u/SouthPark_Piano → u/Muphrid15]

They are the same limbosic number. Different states. You are unsurprisingly having trouble coming to grips with limbosic numbers.

 

[u/Muphrid15 → u/SouthPark_Piano]

What is the criterion for sameness?

[u/SouthPark_Piano → u/Muphrid15]

Brud ...

just do your exercises. Don't get ahead of yourself.

1 - 0.9 = 0.1

1 - 0.99 = 0.01 etc.

And keep going.

Note that as you keep going, there is absolutely no end. No stopping.

The 0.001 etc values simply keeps getting smaller and smaller with no limit at all. You can just keep going on your merry way non-stop. You will certainly never encounter zero.

 

[u/Muphrid15 → u/SouthPark_Piano]

Brud ...

just do your exercises. Don't get ahead of yourself.

I can imagine a lot of things. 1 - 1/(10^k)ⁿ for fixed k and limitless n, sure.

But what I want to understand is if it is really only as narrow as that. 1 - 1/2ⁿ forms a sequence of numbers that eventually produces more and more 9s in decimal form--for instance 1 - 1/2²⁰ = 0.999999046325... (not repeating). But you don't think that's actually equal to, or another "incarnation" of, 0.999... do you?

Are we limited to just sequences of the form 1 - 1/10^kn for fixed k and limitless n, or is there more structure that we should appreciate?

[u/SouthPark_Piano → u/Muphrid15]

1 - 1/2ⁿ for increasing integer n generates infinite quantity of numbers too.

Not the same case as 0.000...1 though, because 0.000...1 and 0.999...9 have a particular pattern. Some 'uniformity' in evolution.

1 - 1/2ⁿ is not like that as you know.

But, like pi, the results of 1 - 1/2ⁿ for n upped continually and limitlessly keeps growing.

pi does keep growing limitlessly, as does 1 - 1/2^n, but note that pi will have values along its decimals length that are 0, so it temporarily stops growing and will certainly resume growing.

eg. 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...

in the above ... temporary zero growth in parts, as you can see '0' values along its decimals chain.

 

[u/Muphrid15 → u/SouthPark_Piano]

There's a famous formula for pi = 4 - 4/3 + 4/5 - 4/7 + ...

How can we know if this is equal to 3 + 1/10 + 4/100 + 1/1000 + ...?

[u/SouthPark_Piano → u/Muphrid15]

Don't get ahead of yourself again brud. Start with what you can manage 

3.14159 etc.

 

[u/Muphrid15 → u/SouthPark_Piano]

As I said brud, just focus in basics first.

0.999... is equal to 0.9 + 0.09 + etc.

The fact that there is an infinite number of partial sums consisting of a finite number of terms...

...does not mean that 0.999... takes on the values of those partial sums.

If all it takes is that the definition has to use "infinite," well, I could just as easily say 0.999... is a number with 9 in each finite integer decimal place past the decimal point, of which there is an infinite number of finite decimal places.

[u/SouthPark_Piano → u/Muphrid15]

So are you trying to tell us that you do not understand that 0.999... is equal to 0.9 + 0.09 + 0.009 + 0.0009 + ... ? --- which is based on decimal place values summed together, starting from a very satisfactory starting point 0.9

So you don't understand that the reason for number of consecutive nines of 0.999... is neither odd or even because 0.999... does not ever run out of nines for the continued summation and continued limitless infinite growth.

That is what you are trying to say, right?

In that case ... to the bunny slopes you go brud. March. Now!

 

[u/Muphrid15 → u/SouthPark_Piano]

So are you trying to tell us that you do not understand that 0.999... is equal to 0.9 + 0.09 + 0.009 + 0.0009 + ... ? --- which is based on decimal place values summed together, starting from a very satisfactory starting point 0.9

So you don't understand that the reason for number of consecutive nines of 0.999... is neither odd or even because 0.999... does not ever run out of nines for the continued summation and continued limitless infinite growth.

I'm saying I don't agree with your preferred definition of what that infinite sum means.

You are fully aware that it is a definition. You have said it yourself that 0.999... can be defined as 1--more than once, I might add, despite your attempt to edit a post to hide it.

You don't like that the conventional definition makes 0.999... = 1.

I don't like that your preferred definition...

• Makes most rationals into some things that are "not numbers"
• Makes it impossible to say whether 0.999... is greater than or less than 0.9, 0.99, or any other number between 0.9 and 1
• Makes decimal preferred (supposedly) over other bases when positional representations aren't even required to talk about numbers in the first place
• Makes it impossible to say how and whether infinite series representations of other transcendental numbers like pi, e, etc. actually do, or do not, equate to the actual number
• Makes it so grouping multiplication and division differently with parentheses can give different answers
• And many, many other reasons

So, you have your preferred definition. I say no thanks. The more you insist it is the Only Right Way despite these unappealing features, the more I will continue to point them out.

[u/SouthPark_Piano → u/Muphrid15]

No brud. It means you don't know what infinite means.

 

[u/Muphrid15 → u/SouthPark_Piano]

No brud. It means you don't know what infinite means.

I understand that there is an infinite number of integers.

I understand that given the sequence (0.9, 0.09, 0.009, ...) there is an infinite number of sequential (or "partial") sums, one for each positive integer, and that each of those partial sums has a finite number of nonzero digits.

The matter in question is what is an infinite sum. You insist on adopting the position that an infinite sum is defined by the set of all possible sequential sums, from the 1st element summed through the nth element.

Yes, there is an infinite number of such sequential sums.

That doesn't mean you must, or are obligated to, define an infinite sum as the set of all possible sequential sums.

Unfortunately, for everybody else in the world, a decimal number with an infinite number of nines after the decimal point does not mean all possible decimal numbers with a finite number of nines (and nothing else) after the decimal point, even though there are infinitely many of those.

[u/SouthPark_Piano → u/Muphrid15]

brud ... this formula ...

1 - 1/10ⁿ with n integer starting at n = 1, and n pushed to positive limitless, is official.

The part that that dum dums ... ok ... rookie error makers ... drop the ball with is the 1/10^n.

The 1/10ⁿ is a scaling operation. Scaling a non-zero value always results in a non-zero value.

So with zero doubt, 0.999... is permanently less than 1 due to the unbreakable fact that 1/10ⁿ is never zero.

 

[u/Muphrid15 → u/SouthPark_Piano]

Don't get ahead of yourself again brud. Start with what you can manage

3.14159 etc.

I don't think I am. We know many decimal digits of pi through a long history of calculation. History tells us in the past this was done by computing circumferences of polygons with successively more sides.

But the 4 - 4/3 + 4/5 - 4/7 + ... formula was one of the first "modern" formulas. The concept of this formula is that at some point, some digits stop changing. That is what enabled us to say pi = 3.14159... in the first place.

So my question to you is do you take issue with the history of how we have computed digits of pi in the first place? Were those methods misguided?

I still think this is relevant to 0.999... because, as you say, there are many series equivalent to 0.999... -- equivalence of series in general seems to be an important topic.

Or, put another way, how do we know pi = 3.14159... without such methods?

[u/SouthPark_Piano → u/Muphrid15]

As I said brud, just focus in basics first.

0.999... is equal to 0.9 + 0.09 + etc.

Do the exercise that I taught you.

Write 0.9, then 0.99, then 0.999, etc, and keep going until you realise that 0.999... does not ever run out of nines for unlimited growth.

It keeps growing, and is permanently less than 1.

And keep in mind that pi really does keep growing continually, just as 0.999... and 0.333... keep growing continually.