Is 0.999... Equal to 0: A Discussion on Convergence and Infinite Series?

[Seasons]

I believe that when you say "doing division" you are again referring to the process of getting from the representations of two numbers $x$ and $y$ to the number $z$ such that $x = yz$. This is again a problem of representations and of the methods used to find the representation of a new unknown number, not a problem of division itself, which is very well defined.

The thing is that when it comes to representing numbers there is no "correct" way. There are some ways that are more convenient than others in some circumstances, but then again may be worse in others. For example, representing rational numbers as fractions can be very convenient when you are multiplying and dividing. However, decimal representation can be preferable in other circumstances.

I agree with all of that, except for one glaring problem..

when you multiply 3*3 or any whole number by another whole number (the inverse of division), you never get something like: …999.

Subtraction inverses addition perfectly.

Why is division so weird ?

I think you're just so used to "that's just the way it is", that you won't even admit how bizarre that is. I get that numbers can be represented in a plethora of ways, I get that… but why this totally bizarre way?

Why aren't all decimals non repeating terminating decimals?

I want to answer that question. I think if I can answer that question, I can settle into completely agreeing that something defined as a process that never ends can equal something that's simply there, as in the 1=0.999… issue. I don't doubt that there's no number between the two, with a caveat, there's always another 9 to add, infinitesimally, and is that not making a new number between the two? My take on what I've seen in replies so far, is that 0.999… is it's own number, not a process, and to think of it as a process, is to confuse the issue that it's completed. I think for a lot of people, the declaration that it's completed is really hard to accept without pondering something fishy, and taking the stance that it's an approximation, but not the actual equality. I thought my question was different than the other threads on this topic, but Mark was really good at pointing out my confusion, so now I'm approaching my general concerns and curiosities - maybe there's nowhere to go after this, or maybe you have a great answer. I certainly don't.

 

[Orodruin]

I would argue that the 999... issue arises from decimal representation being particularly ill suited for division. There are other representations where this is not an issue at all. Note that, in base 3, this would be the 222... issue and in base 2 it is the 111... issue.

 

[Mark44]

So, I realized something interesting to me about what you all discussed and seemed to all agree upon.
If you expand a sequence from the inside out, rather than just adding to the end, the end number gets dropped.

I have no idea what you're trying to say here. A sequence is just a list of numbers. What do you mean "expand a sequence" either from the inside or adding to the "end". An infinite sequence has no end. If you insert terms into a sequence, you get a different sequence (list).

And I started thinking: If the end number gets dropped, then there is a new end number, and that gets dropped as well, because everything, again, before it, is infinite. So for example: 0.1, 0.01, 0.001… is not an addition as mark pointed out, it is a subtraction.

No, there is no arithmetic operation being performed. This is a list of numbers, nothing more.

It was made clear by everyone that what this implies, is an infinite amount of zeroes… so that the 1 gets dropped. but actually, the .01, then gets dropped, and then the .001 gets dropped, so it ends up just equalling 0.

No, that's not at all how it works. If you look at the 3rd term in the sequence, it is .001 away from 0. The 4th term is .0001 away from 0. The farther you go out in the sequence, the closer that term is to 0.

BTW, the sequence you wrote is generated by a formula: $\frac 1 {10^n}$, with n being successively 1, 2, 3, 4, and so on.

Now let's do this sequence for 0.4, 0.14, 0.114… since there's always a new end number being added which needs to be dropped (at infinity), this sequence also equals 0!

No. This sequence converges to (gets arbitrarily close to) .1111... = 1/9, which is not the same as 0.

Anytime, you make a sequence y, xy, xxy, xxxy, … it always equals zero. Correct?

No.

I also want to add, to Marks' reply so that Mark can understand what I meant, that division seems baffling compared to addition, subtraction and multiplication. If you missed my last post, I hope you all read that as well.

Why do you think division is baffling? What you seem to be having a hard time with is the ambiguity of lots of numbers with regard to their decimal representation.

The reason this bothers me so much, is that even though the numbers are rational, you can separate these rational numbers into three categories.

Terminating decimals without a repeat
Repeating decimals (though, because they regress, technically, they're not repeating)
and finally
Decimals that are not created through long division, but rather by summing decimals lower than them.

I don't know what you mean by this last category. Rational numbers (the ratios of integers) fall into two categories.
Terminating decimals, such as .5
Repeating patterns of a specific length, such as .383838... (or 38/99)
In both cases there is a repeating pattern. .5 can be written as .5000..., with repeating 0's

Now, all these categories seem extremely odd to me, considering how neat multiplication, addition and subtraction are.

I think people would reply, "Well, they repeat, because they don't go into the number equally"

Ok, but why don't they go into the number equally? why

I don't know what you mean by this, either. In any fraction (a ratio), if the numerator (top) is larger than the denominator (bottom), and the numerator contains a factor equal to the denominator, you get a whole number answer. E.g., 12/6 = 2
If the numerator is smaller than the denominator, division results in a decimal fraction that either terminates or repeats some pattern.
If the numerator is equal to the denominator, the fraction simplifies to 1.

A fraction is a rational number, whose decimal representation either terminates (like .25) or repeats (like .66666 = 2/3). There are numbers that aren't rational (aren't the ratio of two integers), that are called irrational (meaning they aren't ratios), These are vastly more numerous than the rational numbers. An example is .101001000100001... and so on. There's a pattern here, but the pattern has a variable length.
Another very famous irrational number is $\pi$ (pi), which is about 3.14159. Another that is almost as famous is e, the so-called "natural number," and whose name honors the Swiss mathematcian Leonhard Euler.

I actually don't think anyone knows the answer to that. And, I speculate, because nobody knows the answer to that, their other theorems are simply guesses to that extent. We all know why 1 apple and 1 apple is 2 apples. Because there is an identity copy. Addition isn't weird like division. And two of you in this thread seemed totally perplexed why this totally perplexes me.

And if you have 3 apples and give one away, you have 3 - 1 = 2 apples.
If you eat one apple each day, then after a week, you will have eaten 7 * 1 = 7 apples.
If you have a dozen apples, and share them equally with two other friends, you will each have 12/3 = 4 apples.

If you share the 12 apples amongst 8 people, each person will get 12/8 = 1.5 apples.
What's so perplexing about that?

I would point out with only division (and other types of divisory functions), when you operate on whole numbers can you get these bizarre read outs.

You get bizarre results with computers and calculators, even with addition or subtraction. If you add an extremely large number to an extremely small number, you will likely get an answer that is equal to the large number.

Changing the base, just shifts which numbers do and don't give these bizarre read outs, it doesn't make them go away. You accept that it's just a different way numbers are represented, without really explaining why it's so relatively convoluted compared to addition, subtraction and multiplication. When I see something so inexplicably bizarre like this, I can understand why people are confused enough to challenge these notions, particularly when people can't seem to explain why, but then they have all types of rules for this phenomenon nobody understands.

The rules for arithmetic are well understood by anyone who has mastered arithmetic.

I do think to myself when I ponder this stuff, that maybe we aren't doing division correctly. I'm not sure what doing it correctly entails, but the methods we try to explain around, may actually be a form of psychosis.

No. Division has been well understood for hundreds of years.

 

[Seasons]

I would argue that the 999... issue arises from decimal representation being particularly ill suited for division. There are other representations where this is not an issue at all. Note that, in base 3, this would be the 222... issue and in base 2 it is the 111... issue.

Could it be because in some strange way, you are forcing scenarios where you are dividing by zero, which is always the placeholder for base?
Mark stated that the only rule in division is not to divide by zero. Even if that's not it at all (just a flit of thought), you do admit that something is ill suited to division, implying that something might not be ill suited for division perhaps, if so, that's exactly what I said. I also said, I currently have no bright ideas, maybe you don't either.

 

[Orodruin]

Could it be because in some strange way, you are forcing scenarios where you are dividing by zero, which is always the placeholder for base?

No.

you do admit that something is ill suited to division, implying that something might not be ill suited for division perhaps, if so, that's exactly what I said.

You said there was a problem with division. There is no problem with division.

There are no bright ideas needed here, this subject is well understood.

 

[Mark44]

Why aren't all decimals non repeating terminating decimals?

For a terminating decimal, such as .25, when you divide 1 by 4, at one stage of the division you get a difference of 0. So the next digit in the decimal representation is 0, as in .250. Continuing the process, you get 0 at each stage, so what is normally referred to as a terminating decimal actually repeats a pattern of 0's after the .25. So in essence, all rational numbers have a repeating pattern of finite length. With 1/4 = .25, the repeating pattern is '0'. With 1/3 = .3333..., the pattern is '3'.

I want to answer that question. I think if I can answer that question, I can settle into completely agreeing that something defined as a process that never ends can equal something that's simply there, as in the 1=0.999… issue. I don't doubt that there's no number between the two, with a caveat, there's always another 9 to add, infinitesimally, and is that not making a new number between the two?

Something you probably don't realize is that there is a difference between .999 and .999... The three dots (an ellipsis) indicate that the pattern repeats endlessly.

My take on what I've seen in replies so far, is that 0.999… is it's own number, not a process, and to think of it as a process, is to confuse the issue that it's completed.

Correct. .999... is a number (an alias for 1).

I think for a lot of people, the declaration that it's completed is really hard to accept without pondering something fishy, and taking the stance that it's an approximation, but not the actual equality. I thought my question was different than the other threads on this topic, but Mark was really good at pointing out my confusion, so now I'm approaching my general concerns and curiosities - maybe there's nowhere to go after this, or maybe you have a great answer. I certainly don't.

Could it be because in some strange way, you are forcing scenarios where you are dividing by zero, which is always the placeholder for base?

?
No, I'm saying that you can't divide by zero. I listed it as the only rule about division. There aren't any scenarios where you get to divide by zero; such division is undefined.

Mark stated that the only rule in division is not to divide by zero. Even if that's not it at all (just a flit of thought), you do admit that something is ill suited to division, implying that something might not be ill suited for division perhaps, if so, that's exactly what I said. I also said, I currently have no bright ideas, maybe you don't either.

There are lots of numbers other than zero that you can divide by, and there's no problem in doing so.

 

[Seasons]

Mark, I didn't find your explanation of 0.4, 0.14, 0.114, 0.1114… not being equal to zero very compelling considering what I actually wrote. Perhaps, this miscommunication was illustrated when you admitted that you didn't understand what I meant by "expanding from the center". What I meant by this, is that the 4 keeps moving (outwards) as the 1's keep expanding. Everytime you add a new 1, that 1 it's added to the end, just like the 4 was, and it needs to be removed, just like the 4 was, so that it all ends up solving as zero, not as you stated, "an approximation to 1/9".

But that's not what interested me most!

Your revelation that 0.25, always implies 0.25(0…) struck me as a way I could understand this.

So let me get this straight? Is this how you see the number 1?

(…0)1.(0…)

?

 

[Mark44]

Mark, I didn't find your explanation of 0.4, 0.14, 0.114, 0.1114… not being equal to zero very compelling considering what I actually wrote. Perhaps, this miscommunication was illustrated when you admitted that you didn't understand what I meant by "expanding from the center".

I took a guess as to what you meant by "expanding from the center." In any case, you can't "expand a sequence."

The sequence you defined converges, but not to zero.
Here are the first six terms of your sequence:
.4 -- distance from 0: .4 -- dist. from 1/9: approx .289
.14 -- dist. from 0: .14 -- dist. from 1/9: approx .0289
.114 -- dist. from 0: .114 -- dist. from 1/9: approx .00289
.1114 -- dist. from 0: .1114 -- dist. from 1/9: approx .000289
.11114 -- dist. from 0: .11114 -- dist. from 1/9: approx .0000289
.111114 -- dist. from 0: .111114 -- dist. from 1/9: approx .00000289
Your numbers are always more than .11 away from zero, so they aren't approaching zero. But as you can see, they are getting closer and closer to 1/9. The farther out you go in your sequence, the closer a given term will be to 1/9.

What I meant by this, is that the 4 keeps moving (outwards) as the 1's keep expanding. Everytime you add a new 1, that 1 it's added to the end, just like the 4 was, and it needs to be removed, just like the 4 was, so that it all ends up solving as zero, not as you stated, "an approximation to 1/9".

But that's not what interested me most!

Your revelation that 0.25, always implies 0.25(0…) struck me as a way I could understand this.

So let me get this straight? Is this how you see the number 1?

(…0)1.(0…)

?

Yes.
We don't usually write leading zeroes, and if the repeating digit in a decimal fraction is 0, we don't usually write it, either.

 

[Seasons]

That's not the point I was making. You yourself said "0.000…1 is not a number because there are an infinite number of zeroes before the 1"

Yet, you're saying that 0.000…289 is a number in the case of approximation to 1/9th. Directly contradicting the logic you used when you explained how 0.000…1 isn't a possible number. So, what I did, is that I followed your exact steps:

1 cannot be at the end. When you add a zero at the end, that zero can't be at the end either (because there are an infinite amount of zeroes before it); which means none of those zeroes can be there.

I used the same logic for the 0.1111…4. You stated that it cannot be a number, because there's an infinite number of ones before the 4, so I dropped the 4. But just like the last problem, the new end is a 1, with an infinite number of 1's before it, so, I have to drop that 1 as well, because there are an infinite number of 1's before every one, there can be no 1's at all; which makes the result: 0. Perhaps our confusion come from my notation, because the expansion is really this: 0.4, 0.((…1)4); which isn't the same as 0.1…4 - I'm trying to think if it makes a critical difference. All this thinking is really hurting my head. I need a break for a while. I'm sure all of this probably requires almost no mental effort from all of you!

Also in pondering how you explained that zeroes repeat like this, as an implication that we don't generally express (…0)1.(0…), it occurred to me that this is a special exception to expansion and regress, because, one zero cannot be smaller or larger than another one, which makes it seemingly illogical to notate them as implied placeholders for regress and expansion.

 

[Mark44]

That's not the point I was making. You yourself said "0.000…1 is not a number because there are an infinite number of zeroes before the 1"

Yet, you're saying that 0.000…289 is a number in the case of approximation to 1/9th.

That's not at all what I said. I clearly stated exactly what I meant.
First, for each number in the sequence {.4. .14, .114, .1114, ...} I gave a) the distance of each term from 0, and b) the distance of the same term from 1/9. Please don't attribute words to me that I did not write.

For another thing, I would not write 0.000...289, as this is meaningless -- there's no way to tell where 289 is in the decimal expansion.

Directly contradicting the logic you used when you explained how 0.000…1 isn't a possible number.

No. You misunderstood and misrepresented what I wrote.

So, what I did, is that I followed your exact steps:

1 cannot be at the end. When you add a zero at the end, that zero can't be at the end either (because there are an infinite amount of zeroes before it); which means none of those zeroes can be there.

I used the same logic for the 0.1111…4.

I did NOT write 0.1111...4

You stated that it cannot be a number, because there's an infinite number of ones before the 4, so I dropped the 4.

This makes no sense. You cannot have an infinite number of digits in front of a particular digit.

But just like the last problem, the new end is a 1, with an infinite number of 1's before it, so, I have to drop that 1 as well, because there are an infinite number of 1's before every one, there can be no 1's at all; which makes the result: 0. Perhaps our confusion come from my notation, because the expansion is really this: 0.4, 0.((…1)4); which isn't the same as 0.1…4 - I'm trying to think if it makes a critical difference. All this thinking is really hurting my head. I need a break for a while. I'm sure all of this probably requires almost no mental effort from all of you!

Also in pondering how you explained that zeroes repeat like this, as an implication that we don't generally express (…0)1.(0…), it occurred to me that this is a special exception to expansion and regress, because, one zero cannot be smaller or larger than another one, which makes it seemingly illogical to notate them as implied placeholders for regress and expansion.

Your question has been answered (i.e., is .9999 ... = 0?) and it should be obvious that the answer is no. You tagged this thread as level B (basic level math), but understanding what it means for a sequence to converge and for an infinite series to converge are topics usually in the second semester of calculus. Without some background in this level of mathematics on your part, much of our explanations are going over your head. We are more than 40 posts in this thread, and you seem unable to recognize that your sequence {.4, .14, .114, .1114, ...} is always more than .11 away from 0, so could not possibly converge to zero.

Thread closed.