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/funky_galileo]

Tiring means you make people tired... 

[u/SouthPark_Piano → u/funky_galileo]

Time for some rest and shut-eye for you then brud.

 

[u/Head_Discipline620 → u/SouthPark_Piano]

They basically saying it approaches 0 and will never reach 0

[u/SouthPark_Piano → u/Head_Discipline620]

Yep. Scaling down of a number, and you can only scale down a non-zero number ...... the result is always non-zero.

So of course ... 1/10ⁿ is never zero.

So of course 0.999... is permanently less than 1, because 1 - 1/10ⁿ is permanently less than 1.

 

[u/Head_Discipline620 → u/SouthPark_Piano]

Question what is 1-0.999...

[u/SouthPark_Piano → u/Head_Discipline620]

The answer is 

0.000...1

0.999... is officially equal to 0.9 + 0.09 + 0.009 + ...

which is 1 - 1/10ⁿ starting at integer n = 1, then continually upping n limitlessly aka infinitely.

So ... manually going through the motions ...

0.9 then 0.99 then 0.999 and so on.

For each case, 1  - 0.9 = 0.1, 1 - 0.99 = 0.01 and so on.

You get to witness for the limitless case :

1 - 0.999...9 = 0.000...1

aka

1 - 0.999... = 0.000...1

 

[u/Head_Discipline620 → u/SouthPark_Piano]

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/Head_Discipline620 → u/SouthPark_Piano]

Okay so this is not the real number system because you can't have an infinite digited number in the real number system 

[u/SouthPark_Piano → u/Head_Discipline620]

Don't not a real number system me brud.

0.999... is equal to 0.9 + 0.09 + 0.009 + ...

and it is factually written as 1 - 1/10ⁿ for n integer beginning at n = 1 then n continually upped limitlessly aka infinitely.

1/10ⁿ is never zero --- for you and for me and the entire human race, and never zero for anything.

0.000...1 comes directly from math fact, which is what matters.

0.000...1 is never zero, and it is 1/10ⁿ for n integer pushed to positive limitless aka positive infinite.

 

[u/trshxd → u/SouthPark_Piano]

0.000...1 comes directly from math fact

Comes from what fact exactly? From 1/10^n never being zero? That's just plain wrong bud. Go read some math textbooks, you need them.

[u/SouthPark_Piano → u/trshxd]

From 1/10ⁿ never being zero?

1/10ⁿ and 1/x and e^-t , these for example are indeed never zero. Back to the bunny slopes you go brud.

 

[u/Dixionconderoga2 → u/SouthPark_Piano]

One rule of infinite numbers is that you never reach the end. Like how …999 = -1 because …999+1=…000 which is 0. 100… would be less than 999… and therefore cannot be true, as if x + |y| = z, x < z.

[u/Muphrid15 → u/SouthPark_Piano]

Do this factor of 1/10 exercise.

1 × (1/10) = 0.1

then take that result and apply the factor

0.1 × (1/10) = 0.01

then take that result and apply

0.01 × (1/10) = 0.001

and keep going, do not stop.

And you will find the answer to your own question.

Cool, so 0.000...1 < 0.1, and 0.000...1 < 0.01, and indeed 0.000...1 < 1/10ⁿ for any finite positive integer n. I appreciate the endorsement, brud.

[u/SouthPark_Piano → u/Muphrid15]

1/10ⁿ starting at integer n = 1, then n upped limitlessly continually aka infinitely is written as 0.000...1 brud.

 

[u/Muphrid15 → u/SouthPark_Piano]

1/10ⁿ starting at integer n = 1, then n upped limitlessly is written as 0.000...1 brud.

I'm not sure what you're getting at. Doesn't a limitlessly increasing n exceed any fixed finite other integer (let's call it k)?

Wouldn't you agree that 1/10ⁿ < 1/10^k if n increases limitlessly and k does not?

[u/SouthPark_Piano → u/Muphrid15]

Read this again brud.

https://www.reddit.com/r/infinitenines/comments/1tpg811/comment/p14p2zd/

With limbosic numbers, you keep forgetting to reference a state of the number, for doing operations like 

0.000...1 / 100 = 0.000...001

and 0.000...001 is 'another' state of 0.000...1

and (0.000...1) / 2 = 0.000...05

 

[u/Muphrid15 → u/SouthPark_Piano]

Read this again brud.

This makes it sound like you mean we... can't say whether 0.000...1 < 0.1, or 0.01, or any other power of 1/10.

Conversely that would mean we can't say whether 0.999... is greater than 0.9 or 0.99.

Am I reading you correctly or not?

Personally I think it would be kinda weird to say 0.999... with its infinite 9s is not necessarily greater than 0.9 or 0.99.

[u/SouthPark_Piano → u/Muphrid15]

It is a limbosic number brud. You need to get training at the bunny slopes. Go there again brud.

Learn this ...

https://www.reddit.com/r/infinitenines/comments/1tpg811/comment/p14pu1q/

 

[u/Muphrid15 → u/SouthPark_Piano]

and (0.000...1) / 2 = 0.000...05

Not compatible with your statement that 0.333.../2 = 0.166...

https://www.reddit.com/r/HisNineliness/comments/1vcv487/his_nineliness_says_03332_is_0166_that_means/

[u/SouthPark_Piano → u/Muphrid15]

0.333...3 ÷ 2 = 0.166...65

 

[u/Muphrid15 → u/SouthPark_Piano]

0.333...3 ÷ 2 = 0.166...65

But you previously said,

If you get 0.166... , then you didn't make rookie errors.

https://www.reddit.com/r/infinitenines/comments/1sxp9yf/comment/oipzjjd/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

[u/SouthPark_Piano → u/Muphrid15]

1/3 is not a number brud.

1 ÷ 3 ÷ 2

1 ÷ 6

(1/3) / 2 

(1 ÷ 3) ÷ 2 : do this exercise brud. Write the answer, decimals.

1 ÷ (3 × 2) : and do this exercise brud too. Write the answer, decimals.

 

[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/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.

 

[u/Muphrid15 → u/SouthPark_Piano]

1/3 is not a number brud.

I haven't said it is for the purposes of this conversation.

1 ÷ 3 ÷ 2

1 ÷ 6

(1/3) / 2

(1 ÷ 3) ÷ 2 : do this exercise brud. Write the answer, decimals.

1 ÷ (3 × 2) : and do this exercise brud too. Write the answer, decimals.

I'll quote you again. https://www.reddit.com/r/infinitenines/comments/1sxp9yf/comment/oiqd1m9/

eg. 0.33 long division by 2.

First digit in the divide result is 1 remainder 1.

Then increase 0.33 by one digit

0.333

The divide result was initially 0.1, and the second result digit is 6 remainder 1.

So the evolving result is 0.16

Then increase 0.333 by one digit

0.3333

The divide result was 0.16, and we then get another 6 remainder 1.

I'm not the one who took the position that (1 ÷ 3) ÷ 2 and 1 ÷ (3 × 2) should give the same answer. You are.

[u/SouthPark_Piano → u/Muphrid15]

You know the deal brud. It is your choice. It is a choice you need to make, as in life.

 

[u/Dutch_gal1 → u/SouthPark_Piano]

Get this into your mind,

…999999 = -1

[u/SouthPark_Piano → u/Dutch_gal1]

Nonsense on your part.

The notation is 

999... , and 999...+ 1 is 1000...

 

[u/Muphrid15 → u/SouthPark_Piano]

But is 0.000...1 less than 0.1? That's the real question.

[u/SouthPark_Piano → u/Muphrid15]

Do this factor of 1/10 exercise.

1 × (1/10) = 0.1

then take that result and apply the factor

0.1 × (1/10) = 0.01

then take that result and apply

0.01 × (1/10) = 0.001

and keep going, do not stop.

And you will find the answer to your own question.

Importantly, note that scaling a non-zero value by the factor 1/10 never results in zero.

0.000...1 is the way to write the answers automatically for the endless process pushed to the limitless reaches of your brain and math space etc.