Lance wrote:Мастер wrote: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?