.999... = 1?

[Lance]

So it's basically an infinite set?

What specifically is an infinite set? I think we could have a few possible infinite sets going on here.

Oh. I didn't know that could be more than one thing.

[Мастер]

So it's basically an infinite set?

What specifically is an infinite set? I think we could have a few possible infinite sets going on here.

Oh. I didn't know that could be more than one thing.

I'm just not sure what "it" is.

It is possible to construct an infinite sequence (in fact, many different infinite sequences) of numbers that are all less than one, but strictly increasing.

Here's one:

0.9
0.99
0.999
0.9999
0.99999
0.999999
<and so on>

Every number in the sequence is less than one, each one is greater than the last, and the sequence goes on forever. Pick any number less than one, and eventually, there is a number in this sequence which is greater than the number you have chosen.

This is somewhat related to the notion of "open" and "closed" sets. The set of all real numbers x such that 0<x<1 is an "open" set; the set of all real numbers x such that 0<=x<=1 is a "closed" set. The closed set has a greatest element (and also a least element); the open set does not. There is a least upper bound for the open set 0<x<1 - it is 1. Every number in the set is less than or equal to one, and there is no such upper bound which is smaller than one. But this least upper bound, 1, is not in the set. For the closed set, 0<=x<=1, the least upper bound is in the set. (Technical note - sets do not have to be "open" or "closed" - some are neither.)

[Lance]

Here's one:

0.9
0.99
0.999
0.9999
0.99999
0.999999
<and so on>

Okay, I understand this. What I guess I don't fully get is: what is the difference between "and so on" and "..."?

[Мастер]

Here's one:

0.9
0.99
0.999
0.9999
0.99999
0.999999
<and so on>

Okay, I understand this. What I guess I don't fully get is: what is the difference between "and so on" and "..."?

Ah, I think I see where you are coming from. Let me type up my thoughts while sitting on the balcony at 5am here in the tropics while the monsoon rain comes pouring down.

What I have typed above is an infinite sequence of numbers. Specifically, it is a Cauchy sequence, which (very loosely speaking) just means that as you go further and further into the sequence, the differences between the remaining numbers get smaller and smaller, approaching zero. Cauchy sequences are one of the two standard methods of constructing the real numbers from the rational numbers. Although the numbers in this particular sequence get closer and closer to one, each individual number in the sequence is strictly less than one. The tenth number in the sequence is less than one, the billionth number is less than one, etc. Each number in the sequence (this is not a requirement for Cauchy sequences, but it is a property of this particular sequence) is a terminating decimal, so I assume we have no problem on the meaning of an individual number in the sequence. I.e., the last number I have written explicitly above is 0.999999, which is just the decimal way of writing the fraction 999999/1000000.

In the real number system, every Cauchy sequence has a limit. This is not true in the rational numbers. For example, I can write a sequence,

1
1.4
1.41
1.414
1.4142
1.41421
1.414214
1.4142136
1.41421356
1.414213562
1.4142135624
<and so one>

Each of the numbers in my sequence is a rational number. However, if we are dealing with rational numbers, the limit of this particular sequence (continued according to the same rule that generated the first few elements) does not exist. If you square each number in the sequence, you get a new sequence of numbers which are getting closer and closer to 2. So you could say the limit of the sequence above is the square root of two. But there is no rational number which, when squared, is equal to two, so if we are thinking about rational numbers, we have to say that the limit of the sequence above does not exist. However, there is a real number equal to the square root of two, so if we are thinking about real numbers, the sequence above has a limit - the square root of two.

In the real number system, every Cauchy sequence has a limit - the real numbers are complete. In the rational numbers, some Cauchy sequences have limits that exist (as rational numbers), and some don't. The rational number line has "holes" in it

Going back to my first sequence (the one with the nines), the limit of this sequence is one, which happens to exist in both the real numbers and the rational numbers. (We can discuss limits and their properties if needed.) Every single number in the sequence is less than one; however, the limit of the sequence is one. You can get as close to one as you want by moving far enough into the sequence. If you pick any real number less than one, the elements of the sequence eventually pass the number you have chosen, and if you keep moving farther into the sequence, you get farther away from the number you have chosen, not closer. This rather colourful clown at tommac's forum who used to bill himself as the greatest genius since Archimedes (but who now bills himself as the greatest genius ever) seems to have problems with this - a sequence whose elements are all less than one, but whose limit is equal to one.

So then, what is the meaning of "0.999...."? The only coherent meaning I have ever seen assigned is that it is the number which is the limit of the sequence

0
0.9
0.99
0.999
0.9999
0.99999
0.999999
<and so one>

Every individual number in the sequence is less than one; the limit of the sequence, however, is equal to one. And "0.999...." is normally defined as the limit, not one of the individual elements. The argument which you could have find repeated ad nauseum at tommac's forum (before the greatest genius who ever lived drove most of the members away) was some variation of "but no matter how many 9s there are, it's still less than one!", which is of course true. But the definition of 0.999... is not a decimal point followed by a thousand nines, or a billion billion billion nines, or however many 9s you can write before you develop carpal tunnel syndrome. It's the limit of the sequence, and the limit is equal to one.

That's about the only definition of "0.999..." I've ever seen, at least explained in any kind of coherent fashion. It's the same as the one that tells us 0.333... is equal to 1/3. 0.333<a billion billion billion digits>3 is not equal to 1/3, although it is really close. But the limit of the sequence

0
0.3
0.33
0.333
0.3333
0.33333
<and so one>

is equal to 1/3, even though each individual element in the sequence is less than 1/3. Similarly, if we say 3.14159265... is equal to pi, well - if you stop after a billion billion billion digits, you have a number which is really close to, but not equal to, the number we call "pi". However, the limit of the sequence

3
3.1
3.14
3.141
3.1415
3.14159
3.141592
3.1415926
3.14159265
<and so one>

is equal to pi, even though no individual element in the sequence is equal to pi.

I have seen what I can only describe as vigorous, militant rejections of the idea that 0.999... is equal to one. It certainly is equal to one the way I (and most people I know) define 0.999.... I have therefore asked some of them what definition of 0.999... they are using, but I have yet to see single answer that I can describe as coherent.

Does this address your point?

[Мастер]

I should add that at places where I've seen this issue discussed in depth, nearly all of the confusion seems to come from a failure to distinguish between the abstract things called "numbers", and the symbols we use to write them down. Romans wrote "XII" instead of "12", but their 12 had all the same properties as our 12.

This page, https://en.wikipedia.org/wiki/0.999..., has a section called "Impossibility of unique representation". This is an issue (I won't say problem) with decimal representations of the real numbers. The decimal system provides a way for writing down every real number (many real numbers require an infinite decimal representation, though), but some numbers have two representations. Specifically, any rational number p/q, where p is an integer and q is a non-zero integer with prime factors of "2" and "5" only, has two representations. 1/4 can be written as 0.25, or as 0.24999... If you change to binary notation instead of decimal notation, the set of numbers which have non-unique representation changes.

Another one I have seen comes from people who think the number system was invented by Intel, and numbers are whatever the latest Intel processor can represent.

[Lance]

That was a very interesting read, and I think it help me to better understand my problem with this.

Way back in grade school, I learned that .333... was "as close as you can come to representing 1/3 in decimal" but that you can never represent it exactly.

I find stuff like the below on Facebook all the time. And am often disappointed by how many people get them wrong. (A) is the common wrong answer. I made a comment in a discussion about the order of operations along the lines of:

For sure the rules are arbitrary. But without the rules, different people would get different answers and they'd all be right.

I'd compare it to which side of the street we drive on. It doesn't matter _which_ side, just that we all agree.

This would be so much easier for me if it was just a rule.

[Мастер]

That was a very interesting read, and I think it help me to better understand my problem with this.

Way back in grade school, I learned that .333... was "as close as you can come to representing 1/3 in decimal" but that you can never represent it exactly.

Hmm. Well, I'd say there are two reasonable choices. You could declare that 0.333... is the limit of the sequence

0
0.3
0.33
0.333
0.3333
0.33333
<keep going>

The limit of this sequence is 1/3. If you stop after a billion billion billion 3s, it's not equal to 1/3. But 0.333... doesn't mean what you get after a billion billion billion 3s, it's the limit of the never-ending sequence.

The other approach you could take is to declare simply that infinite decimals are undefined, and 0.333... is meaningless. Personally, I don't see the utility of this approach, but there doesn't seem to be anything illogical or self-contradictory about it. That way, the decimal system can represent 1/2, 1/25, and 1/6125, but it can't represent 1/3, the square root of two, or pi.

The one that I don't quite get is when people say 0.333... is some well-defined quantity that is different than 1/3. I can't figure out a reasonable way to do that.

I find stuff like the below on Facebook all the time. And am often disappointed by how many people get them wrong. (A) is the common wrong answer. I made a comment in a discussion about the order of operations along the lines of:

For sure the rules are arbitrary. But without the rules, different people would get different answers and they'd all be right.

I agree 100% with that. The order of operations we use is not the only one that makes sense. It's just the one we use. Maybe it's better most of the time for the types of problems we tend to solve, I'm not sure.

I'd compare it to which side of the street we drive on. It doesn't matter _which_ side, just that we all agree.

I like that one.

MathProblem.jpg

I would answer B to that question. But I've definitely seen pretty much the same Facebook conversation unfold. More than once.

[Lance]

[Мастер]

[Arneb]

You're giving me loads of reasons to remain off Facebook...

[Enzo]

Is that why you can't have your cake and eat it too?