Hi there @Certainly real. By way of introduction, I’m a veteran of this thread. I participated heavily several years ago, and a little bit last year. I noticed some basic misunderstandings about numbers that I thought I might be able to clarify. My background is in math, and I have enough interest in philosophy to relate to some of the non-mathematical concerns about the mysterious .999… = 1. Let me just say one thing up front. That equation is a valid theorem in standard mathematics. Whether you accept that as the answer; or whether you reject it on philosophical reasons, either way, it is a fact that IF you accept the basic rules of math, THEN .999… = 1 follows as a theorem. For example it’s proved to be true in freshman calculus, as an example of a geometric series.
But I don’t want to talk about that today. I want to simply clarify some basic aspects of numbers and their representations.
- All counting numbers: 1, 2, 3, 4, …, have finite length representations. The rule to generate the counting numbers (also known as: the natural numbers, the positive integers, the positive whole numbers) is that you start from 1 (or sometimes 0); and to get the next number you add 1.
Given that rule, you can see that if I have some number like 88888888, the next one is 88888889. And if I have, say, 99999999, the next one is 100000000. It rolls over like the odometer in your car. The point is that every single number you get by the “add 1” rule has a finite length.
Therefore expressions such as 888888888… or 99999…, where the ‘…’ means that the pattern continues forever, have no meaning in math. So it makes no sense to ask about them. You can’t ask which is bigger, because neither of them has any mathematical meaning. All whole numbers, all counting numbers, have a finite-length representation.
- When you talk about m or km, that adds confusion, because we are not talking about anything in the real world. The question, is .999… = 1 in the real world is meaningless. The fact is that we can’t measure anything that precisely. For one thing, our measuring instruments are imprecise.
For example our most accurate electron microscopes have a resolution of about 1 picometer, that’s (10^{-12}) meters. That’s .000000000001 meters. And even worse, all our theories of physics break down below the Planck length. That is, there is a physical length below which it doesn’t even make sense to talk about anything being smaller. Not that there is or isn’t anything down there; it’s just that our physics doesn’t allow us to sensibly discuss the matter.
So when you talk about what is .999… meters, I have to say that the notion is meaningless. We can not measure or even sensibly discuss anything smaller than about 30 or so 9’s. Smaller than that, and physics itself doesn’t work.
I hope we can agree that questions of physical measurement involving m or km are irrelevant here. All we are concerned about is the purely mathematical meaning of numeric expressions.
To repeat this, because it’s important: We are discussing pure abstract math and NOT physics. Even though .999… = 1 is true in math, I would NOT say it’s true in the physical world. It’s meaningless in the physical world, because we can’t measure anything that finely. Perhaps this is a frequent point of confusion. We’re not talking about the real world here, only pure abstract math.
- Now to the crux of the matter. I think you might be a little unclear about how decimals work. You have stated that an expression such as .1111… is infinite, or infinitesimal. It’s not. In fact .111… is precisely equal to the familiar fraction (\frac{1}{9}).
Before getting into that, though, let’s just review how decimals work. If you have a decimal like 0.1, the digit position to the right of the decimal point stands for the number of tenths. So 0.1 means, literally, (\frac{1}{10}).
The second digit to the right tells you how many hundredths there are. So 0.11 means 1 tenth plus 1 one hundredth, or (\frac{1}{10} + \frac{1}{100}), which is equal to (\frac{11}{100}), or a little bit less than (\frac{1}{9}).
If we add another digit, that’s the number of thousandths. So 0.111 means (\frac{1}{10} + \frac{1}{100} + \frac{1}{1000}), which adds up to (\frac{111}{1000}), which is a little bit smaller than 1/9.
With this understanding, I hope you can see that a number like 0.111111111111111 is even closer to, and still just a little bit less than, 1/9. We don’t even have to believe in infinitely long decimal expressions to see that for any finite number of 1’s, the number 0.1111…1111 is a finite number, larger than 0 and a tiny bit smaller than 1/9, as long as there are only finitely many decimal positions containing ‘1’. If you then allow expressions with infinitely many decimal places, we can show that 0.111… is exactly equal to the familiar fraction 1/9.
Even though there are infinitely many digits, the number that’s represented is just a familiar fraction between 0 and 1, namely 1/9. Like cutting up a pizza into nine slices. Each slice is 1/9. Of course we could never physically measure 0.111111…, but that is no concern of ours. We are in the realm of pure abstract mathematics, and NOT the physical world of solid things around us.
I hope this is clear so far. I want to just leave it at this and not talk about infinitesimals yet, because the concept of decimal notation is important.
Here’s another example. Say we cut our pizza into the more usual 8 pieces. What is that in decimal notation? Well, 1/8 = 125/1000, which turns out to be 0.125. That’s 1 tenth, 2 hundredths, and 5 thousandths. That’s how decimal notation works.
To directly respond to questions you’ve posed: The number 1.11111… is exactly 1 and 1/9, or 10/9. And the number 2.999… is exactly equal to 3. But even if you don’t believe that, you have to at least agree that it’s larger than 2 and certainly no larger than 3. It’s not infinitesimal and it’s not infinite.
Let me know if this is helpful so far.
