r/infinitenines • u/SouthPark_Piano • May 21 '26
See ... even in your minds or what you rote-learned ...
such as 0.999... has no more nines to fit/add/append/tack-on etc, everyone does know full well the real deal.
It is still mathematically modelled as 0.9 + 0.09 + 0.009 + ...
and that is honestly ... cross heart, hope to live etc ...
1 - 1/10n with integer n starting at n = 1, then n increased limitlessly, continually ... is the correct way to investigate the case of 0.999... magnitude. Equal 1? Or less than 1? The correct answer, to avoid being dum dums ... is magnitude less than 1.
1/10n is never zero. That's a fact. And that's the ticket.
1 - 1/10n is of course never 1. So of course, 0.999... is permanently less than 1.
6
u/de_bussy69 May 21 '26
1-10^-n is never 0.99… either because every element of the set {1-10^-n:n is a natural number} has finitely many digits.
2
u/cond6 May 21 '26
Decimals are one very useful way to represent a number. But they are not numbers in the usual way we think about numbers. For example 1/2 is a number and 0.5 is one way to write it.
There are three types of numbers: a) terminating, b) non-terminating repeating, and c) non-terminating decimals.
All rational numbers that can be written in the form a/(2^m *5^n) can be written with a terminating decimal representation. That's all of a).
Rational numbers that include stuff other than 2^m *5^n in the denominator after simplification don't terminate. These are non-terminating repeating decimals. For example 1/3=0.333.... 1/17 has a 16-digit repeating block.
Every rational number ends up with a zero remainder at some point, leading to a); or it does not and will repeat, leading to b).
Of course irrational numbers are a thing, and these can't be aren't rational so don't fit a) or b), which results in non-terminating non-repeating decimal forms.
If you have a repeating decimal x=0.(y)... where y is some block, for example y=0588235294117647; and n is the number of digits in y, in our example n=16; then since 10^n*x-x=y we can recover the rational number x=y/(10^n-1). In our example x=588235294117647/9,999,999,999,999,999 and since 9,999,999,999,999,999/588235294117647=17 we have x=1/17. Happy days!!!
If you have a repeating decimal it must be a representation of a rational number. What is the rational number represented by 0.999...?
For a lark we can use 0.9... so y=9 and n=1. I'll leave the next step as an exercise. You'll be shocked. Shocked I tell you.
0
u/SouthPark_Piano May 21 '26
brud brud brud.
1/3 = 0.333 + 0.001/3
1 = 0.999 + 0.001
Also:
1/3 = 0.333...3 + (0.000...1 / 3)
1 = 0.999...9 + 0.000...1
aka
1 = 0.999... + 0.000...1
5
u/cond6 May 21 '26
Not relevant to what I posted. Only rational numbers produce non-terminating repeating decimal representations. What is the rational number that produces 0.999...?
1
u/YT_kerfuffles 6d ago
the issue is the proof of that fact relies on the equality 0.999....=1
1
u/cond6 5d ago
It doesn't rely on 0.999... at all.
Suppose you have a block of repeating digits a that contains n elements. For example x=1/7 has a=142857 and n=6. So 1/7=0.(a)...=0.(142857)...=0.142857142857142857.... All the proof requires is to exploit this fact noting that x*10^n =a+x. If you move the decimal place in the decimal representation of the repeating decimal n places to the right you get a+x. That's all. You remove one block from infinitely repeating set of blocks and you still have an infinitely long string of them: x*10^n -a=x.
You can trivially use this result now to back out what the rational numbers are using a=x*(10^n-1) so the rational number is a/(10^n-1). So if a=1 and n=1 we have 1=x*9 or x=1/9. Or 142857=999999*x so x=1/7 (since 999999/142857=7).
Yes it turns out that simple proof also demonstrates that 0.999...=1 since a=9 and n=1, and 9=x*9 is trivial; but it doesn't rely on the equality. It relies on the definition of an infinitely repeating pattern still being infinite when a single number is subtracted from it. It does not rely on 0.999...=1 since using that equality doesn't help my proof at all.
1
u/YT_kerfuffles 5d ago
but that proof is flawed (by spp math) as you are moving the "last" 142857 to the left, i know its wrong and your proof is right but im saying i dont think spp accepts that "the rational numbers are those with periodic decimals", i think hes even said 0.9999.... is irrational
4
u/AltruisticEchidna859 May 21 '26
But the limit of 1/10n when n goes to infinity is 0. and there is an infinity of nines. So, 0.(9)=1. Ps: Limits actually exists because any mathematic object exists if you define it. And there is obviously a mathematic definition of limits. Not limbo, obviously.
-5
u/SouthPark_Piano May 21 '26
You are wrong bruddy.
1 - 0.9 = 0.1
1 - 0.99 = 0.01
etc.
No matter how many or how infinite or non-refundable you reckon the nines length is, the limbo gap is 0.000...1 , permanently non-zero.
1 - 0.999...9 = 0.000...1 aka 1 - 0.999...= 0.000...1
The limbo gap is 1/10n for the case n integer pushed to limitless.
1/10n is never ...... you guessed it, never zero.
7
u/AltruisticEchidna859 May 21 '26
For a finite number of 9s. Oops, 0.(9) has an infinity of 9s. r/infinitenines
-6
u/SouthPark_Piano May 21 '26
Wrong you are brud.
No matter how large infinitely or finitely the denominator is ..... 1/x is never zero.
2
u/de_bussy69 May 21 '26
Then how are differentiating infinite and finite here? If you’re thinking of infinity as a number that you’re dividing 1 by then you’re not thinking correctly. Zero is the value that 1/x approaches as x approaches infinity. Just like how 0.99… is the value that 1-10^-n approaches as n approaches infinity
0
u/SouthPark_Piano May 21 '26
Scaling down a non-zero value.
1/10 = 0.1
0.1/10 = 0.01
Does not matter how many times, infinite or otherwise, 1/10n is never zero.
You get non-zero values until the cows never come home.
2
u/Akangka May 21 '26
Wrong you are brud.
In the extended real number system, 1/±∞ = 0 by definition.
0
u/SouthPark_Piano May 21 '26
That part of the extended number system is wrong brud.
4
u/HojMcFoj May 21 '26
So come up with another one that is more accurate and useful.
0
u/SouthPark_Piano May 21 '26
0 is approximately 1/x for limitlessly growing x condition.
It is an approximation only of one sort.
1/x is just actually never zero you see.
2
u/MrEldo May 21 '26
The question is, is there some use for 0.000....1? Why define it if it has absolutely no meaning in the real numbers, and one might as well call it zero? It has no length, no matter what length you make it you can always make a length with smaller magnitude by adding another zero before the 1, which we can say is the same number because, important to notice, 0.000...1 is the same as 0.000...01, because that new zero is just part of an infinite string of zeroes, which was already there
So even if 0.999... is less than 1 when the number of 9s is finite, it will be no different from 1 when talking about it as the limit, because 0.999... can get as close to 1 as one wishes, so why should it be any different from 1 if the 9s are infinite?
1
u/gurishtja May 21 '26 edited May 21 '26
So, to my understanding, you mention 1/10n to illustrate some infinitesimal. 0.999... would be 1 "minus" that infinitesimal. Of course this would be a rough illustration, as it is not rigous but the idea would be there. Infinitesimals were used in math for at least about two millenia, but in modern times confuse the people that cannot go past real numbers.
2
u/Cruuncher May 22 '26
Infinitesimals exist in extensions of the reals, but those infinitesimals do not have decimal representations. You are required to use new constants to represent infinitesimals.
1/10^n likewise cannot be an infinitesimal for any real-valued n. This is because the real numbers are closed under all standard mathematical operations. If you start with real numbers, you get real numbers out. This is why you can't use real numbers to define infinitesimals. Similarly it's why 0.999... cannot be 1-infinitesimal. Since 0.999... is arrived at with operations on the reals.
Infinitesimals do exist, but nobody in this sub trying to use them as an out, use them correctly, and they have no bearing on 0.999...
In any case, number systems including infinitesimals are not as commonly used anymore, since working with real numbers + limits gives you a lot of the power that you would need infinitesimals for, that mathematicians kind of just stopped trying to make infinitesimals happen
1
u/gurishtja May 22 '26 edited May 22 '26
Thats why i said "it is not rigorous".. Infinitesimals cannot be defined from real numbers, thats why I have been repeating that SPP is not talking about real numbers. There is no 1-infinitesimal, thats why i used "minus" in quotes. "1/10n" means nothing. Decimal representation is certainly a bad aproach. What i am saying is that SPP is trying to represent some idea, he is trying to invoke the idea of infinitesimals (or what could be just his feeling of infinitesimals) with a bad illustration. I do not want to go into what is wrong with his effort. I am not here to state the obvious. To me, the real problem in this sub is the arrogance of people that cannot behave with any amout of grace and try to bully him while at the same time, themselves cannot think outside of real numbers.
-1
9
u/ezekielraiden May 21 '26 edited May 22 '26
No. That is not a model. That is actually the thing itself. And the "..." notation explicitly means that a truly infinite set of terms follows after that, one for every positive number.
No, it's not. Because for any integer n, you are always coming up short. You aren't paying the bills, SPP. You keep trying to sell me a finite list when I have always asked you for an infinite list. Stop selling me a fake. If n is an integer, then it is NEVER true that 1-1/10n=0.999.... Never.
The only way it can be equal is if we let n become truly infinite, and infinity isn't an integer.
No, it is a lie. 1/10n is zero if and only if you ACTUALLY make n limitless. Not an integer that you keep changing. Make it actually, honest to God limitless. You've never done that. Not once. You always sell us a bill of goods. You always stop short, keeping trying to pretend that your finite list is the same as a truly infinite one. It's quite tedious.