I’d like to answer an objection here, one that I hear commonly and that I’m sure I’ll hear again (even despite this post): people say that a number like 1.999…98 is impossible. If you aren’t familiar with that number, it is 1 point 9 repeating followed by a single 8, found by multiplying point 9 repeating by 2. I’ve explained how I came to this conclusion (inductive reasoning) and why it is consistent with what we know about math, but people like to think I’m being inconsistent with the definition of infinity.
You may have noticed the title of this post and thought, “What is reification?” Reification is a logical fallacy in which an abstract idea is treated like a physical object. For example, many people talk about feelings being “bottled up”, and suggest that letting out anger through venting will reduce the anger that you have. Studies have proven this wrong, because anger is not a physical object that can be reduced by “getting rid of it” through expressing it.
The problem with this objection to my method is that mathematicians who think that it’s impossible to “bookend” an infinite series of 9’s with a decimal point and an 8 will bookend other infinities without batting an eye. Consider the number of real numbers between 0 and 1. The obvious answer is “they’re infinite“, despite the existence of a floor and a ceiling to this series of numbers. One might argue that I’m arguing apples and oranges — the series between 0 and 1 is made up of numbers, not digits. So let’s convert that to digits. If you took all the real numbers between 0 and 1 and listed them from least to greatest, then drew a line crossing the first digit of each number, your result would be 01234567891, where every number except that final 1 would be repeated infinitely. Would this new number be consider a “real number”? “Sure!”, say proponents of Cantor’s diagonal argument, which creates a real number through almost identical means.
Do I agree that the number created is a real number? No, but that’s not the point. It’s not important to agree that a real number is made here, but instead to focus on the fact that the digits in this number are indisputably infinite, even though they end abruptly with other infinite series of digits and finally with one finite digit. This is because we’re not talking about a physical reality but a thought experiment. Numbers are just abstract ideas that we use to describe the relationships between mathematical models such as graphed lines, sets of objects, and sometimes just other numbers. While it may offend your sensibilities to try to grasp a concept like .000….01 (point zero repeating ending with a single one), there is no logical reason that such a number can’t exist.
One final note: While I welcome comments, please don’t tell me I’m wrong unless you can explain how I could have been right. If you can’t come up with an alternative explanation, there’s a good chance that there isn’t one.
Why .999... Does Not Equal 1 
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How do you list all the real numbers between 0 and 1 in order from least to greatest? Sure you can take a finite amount and list but you if you tried all of them you’d have nowhere to sink your feet into for the first one. Even if you could you wouldn’t know where to go after that.
Literally list them all? I’m pretty sure you’d need an infinite amount of time for that. No, we’d conceptually list them, and we know that there’s no point at which a number starting with 0.3, for example, will come after a number beginning with 0.4 or before a number beginning with 0.2. It’s rather obvious what digits we would come up with for obvious reasons, and your objection here is pedantic.
Except it’s literally impossible to list the real numbers between 0 and 1. Which is off-topic anyway, but it’s a very important distinction to make, because it shows that the cardinality of the real numbers is larger than that of the rationals.
But like I said on your other post, the problem again lies in the fact that you’re assuming infinity has an end. The instant you put a 1 or an 8 or anything on the end of it, you no longer have an infinite number of 9’s or 0’s or whatever. You’ve got a finite amount, because infinity means “without end”. You can’t add an 8 to the end of something without an end. It’s a meaningless concept.
I think you missed the point, Wani. You’re treating the concept of an infinite number like a physical infinity, which is to commit the logical fallacy of reification. That’s what the whole post is about, and yet you’re still insistent about doing that.
You haven’t shown reification to be a fallacy at all. Yes, there’s infinitely many numbers between 0 and 1, but that isn’t put an end to infinity, and I believe that’s a problem that’s probably solved in a similar way to Zeno’s Paradox (ie, there are infinitely many numbers between 0 and 1 because you can divide 1 by 2 to get a 1/2 and keep dividing like that infinitely and you’ll always get a number between 0 and 1. It’s Zenoish, if you get what I’m saying).
The problem with stating you’ve got 0.000…1 is that what comes after the 1? It’s harder to see here, so I’ll demonstrate it better with your 1.999….998 thing. What comes after the 8? More 9s? If that’s true, suddenly arithmetic has broken and given us a different solution to a problem for no reason whatsoever. 0’s? In which case, there wasn’t infinitely many 9’s, because the 9’s ended suddenly.
I didn’t have to demonstrate that “reification is a fallacy”… a simple google search or check on Wikipedia will do that for you.
Let’s look at it another way. Given a number like .999… , can you start in the middle of the number and find your way to the decimal point? Not in a literal infinity… you can always add 9’s before whatever 9 you started at without changing the number, and you’ll never reach the decimal point. This is a true analogy to trying to find the other “end” of the number. Now you may argue “but I’m at the nth 9, and all I have to do is go n-1 9’s over to get to the decimal point”. But if you’re thinking that way, you aren’t technically starting from the middle of the string of 9’s but rather from the decimal point as usual. You could make the same argument that the “last 9” is only n-1 9’s away to the right, assuming that the 9 you chose is n 9’s away. It’s not a literal infinity — it’s just a conceptual one.
So what comes after the 1 in .000…001? Zero. I argued before that there aren’t zeroes trailing a number like .999…, but I had stopped thinking in terms of induction at that point. There are always zeroes to the end of any number, and there’s no reason that this won’t hold true for an infinite number. To demonstrate this, we simply multiply a number like .999… by N*10, where N is the number of 9’s after the decimal point. You may not like multiplying by an infinite number and again this may distress your sensibilities, but if it can be done for literally any finite N, it should still be possible with an infinite one through induction. So what do you get my multiplying .999… by N*10? An infinite number of 9’s followed by a decimal point and then zeroes.
I know that this doesn’t work like an infinity that you imagine, but the infinity you imagine is immune to math in the way that it typically works… that is, if it is an exception to the math I’m using, it ought to be an exception to the math that you use, as well. It is perfectly rational (given your standards) to look at a problem like 10*.999,.. and say that you don’t know what the product is. It’s a double-standard to argue that this multiplication works but that multiplication like N*10*.999… is impossible.
“I didn’t have to demonstrate that “reification is a fallacy”… a simple google search or check on Wikipedia will do that for you.” Okay, I should rephrase what I said. You haven’t proven that what mathematicians do with 0.999… = 1 is a reification fallacy. If anything, you’re the one making reification fallacies because you’re trying to treat infinitely recurring decimals as something not infinite. You’re trying to make the abstract concept of infinity into something more concrete so that your arguments are correct.
“There are always zeroes to the end of any number, and there’s no reason that this won’t hold true for an infinite number.” Yes there is, a very good reason. Because there is no end to the number. If there is an end, it’s not an infinitely recurring decimal. Like I said, the definition of 0.999… is that, for any decimal place n, there’s a 9 in it. If there’s zeroes to the end, that’s a point, n, where there isn’t a 9 and that’s a contradiction. The general definition of any infinitely long decimal is that for any decimal point, n, there’s a non-zero n after it. You can’t say “Oh there’s infinitely many things before the 0, because that’s an absurd notion and that’s you trying to reify infinity. Like I said somewhere else, infinity isn’t “a really big number, bigger than you can possibly imagine”, it’s an endless number.
“To demonstrate this, we simply multiply a number like .999… by N*10, where N is the number of 9′s after the decimal point. You may not like multiplying by an infinite number and again this may distress your sensibilities, but if it can be done for literally any finite N, it should still be possible with an infinite one through induction. So what do you get my multiplying .999… by N*10? An infinite number of 9′s followed by a decimal point and then zeroes.”
Using induction, let’s do that. I’ve already demonstrated that 10x 0.999… is 9.999 with infinitely many 9’s after it, because if there aren’t infinitely many 9’s after the decimal place, it was never an infinitely recurring decimal. Likewise, if we multiply by 10 again, we get 99.999 with infinitely many 9’s after it, because if there aren’t infinitely many 9’s, it was never infinitely recurring. And so on. Each time we multiply by 10 again, we get infinitely many 9’s after the decimal place and when we go to 10^N (Which is, I assume, what you meant to say, 10 with N many 0’s after the 1), you wind up with infinitely many 9’s before and after the decimal place.
This shouldn’t surprise you either. You’re multiplying a number by an infinitely. You can get any infinitely large number you want doing that. This is why algebra on infinity isn’t allowed.
“I know that this doesn’t work like an infinity that you imagine, but the infinity you imagine is immune to math in the way that it typically works… that is, if it is an exception to the math I’m using, it ought to be an exception to the math that you use, as well. It is perfectly rational (given your standards) to look at a problem like 10*.999,.. and say that you don’t know what the product is.”
But it’s easy to calculate 10*0.999…. Like you’ve said, you just shift the decimal place over and you still wind up with infinitely many 9’s after the decimal place.
“It’s a double-standard to argue that this multiplication works but that multiplication like N*10*.999… is impossible.” That multiplication is possible because N is infinity, and standard algebra doesn’t allow multiplying by infinity. It’s as valid as multiplying by grapefruit.
Also, consider Hilbert’s paradox. Which, if you haven’t heard of it, you’ve got a hotel with infinitely many guests with infinitely many rooms and each room is occupied by a guest. A new guest shows up, but rather than saying your hotel is full, you move every guest to the next room (The guy in room 1 goes to room 2, the guy in room 2 goes to room 3 and so on infinitely), and now you have an empty room for your new guest and nobody winds up losing a room because you can always move the guy from room n to the room n+1. Likewise, if infinitely many new guests show up, you can get everyone to move from the room they’re in, to the room double what they’re in (ie, the guy in room 1 goes to room 2, the guy in room 2 goes to room 4, the guy in room n goes to room 2n) and now you’ve got infinitely many newly opened rooms for your new infinitely many guests. Both of these are very analogous to working with 0.999… except instead of guests in rooms, we’ve got 9’s in decimal places.