Well, technically it is a number with infinitely many nines after the decimal. What most people don't consider is that one first needs to establish what that even means, so in this kind of debate, I generally try to establish is that under any reasonable definition of 0.99999..., 0.99999... is equal to 1. The reason this can be important is that some people will say things like "mathematicians just define it in a way that fits their deluded narrative," which is nonsense, but you still have to deal with it.
Post #1: What? Of course 1 is 1 and 0.99999... is 0.99999..., but also, 1 is 0.99999... and 0.99999... is 1. 0.99999... and 1 are two different ways of writing the same number, as I will demonstrate in this debate.
Post #2: Wait, what? Is that supposed to be an argument? I don't even know what you're trying to say. Also, if you want to defend the position that 0.99999... is not equal to 1, then feel free to accept this debate.
Post #3: 10 * 0.99999... = 9.99999... = 9 + 0.99999... = 9 + 1 = 10. No, that's not an argument that 0.99999... = 1, (I'll save those for the actual debate) I'm just demonstrating that you have not found an inconsistency in my position, as if 0.99999... = 1, 9.99999... = 10, so there isn't a problem.
For reference, for those of you who are not familiar with coding, when you ask a computer if two strings are equal, it checks if they have the same characters, rather than if they have the same value, so the string '1+1' would not be equal to the string '2', but that doesn't mean that 1+1 isn't equal to 2.
I constructed my religious view through my own reasoning alone, and Best.Korea also claims not to have been indoctrinated, so this seems to me to be a truism, or at least a case where definitive evidence is excessively easy to provide.
Typo. In the definition of injective: I meant to write "The idea is that f is one-to-one in that it never sends two different values to the same value."
First, it's pedantic-yet-important correction time: Not all numbers, all real numbers. There is no such thing as all numbers. If you are a bit confused by the distinction, it is important to realize that "real numbers" is a misnomer. They aren't the only ones out there. Also, it isn't that the set of real numbers isn't itself "greater" than the set of natural numbers, it is that there are more real numbers than natural numbers. This is often phrased as "some infinities are larger than others," which I assume is what doesn't make sense to you.
I'm going to assume that you know what a function is, and that you are familiar with the notation f: A --> B. If you aren't, I suggest you read this: https://www.mathsisfun.com/sets/function.html. In the notation f: A --> B, A is the domain, and B is the codomain.
We say f is injective whenever f(x) = f(y) implies x = y. The idea is that f is one-to-one in that it never sends two different values to the same
We say f is surjective whenever for any y in B there is x in A such that f(x) = y. The idea is that no element of B is left out.
We say f is bijective when it is both injective and surjective. The idea is that f provides a one-to-one correspondence between the two sets. For every element of A there is an element of B, and visa versa.
For two sets A and B, it makes sense to say that A and B are the same size if and only if there is a bijection f: A --> B. We represent the "size" of the set A by |A|, and call it the cardinality of A. Numbers such as |A| are called cardinal numbers. Thus for cardinal numbers |A| and |B|, |A| = |B| whenever there is a bijection f: A --> B. We say that |A| < |B| (also written |B| > |A|) whenever there is an injection f: A --> B but no bijection g: A --> B. It is in this way that "greater" is still meaningful.
Cantor proved that |R| > |N|, where N is the set of natural numbers, and R is the set of real numbers. This is what is meant when it is said that there are more real numbers than rational numbers.
You do realize that in the debate I was claiming that 1 - 0.99999... isn't equal to 0.00...1, right?
Response to comment #11 (Savant):
My comments on the video:
1:06: Yes. That's not wrong. Most people assume that the existence of 0.99999... isn't going to be a question, but I did defend it during this debate.
2:05: Yes, actually, I do think that ...99999.0 = -1. If that seems like nonsense, observe: http://math.uchicago.edu/~may/REU2020/REUPapers/Pomerantz.pdf. I know, it's kind of advanced, but math dealing with the p-adic numbers (which is the field in which numbers like ...99999.0 are considered to exist) is inherently advanced. A better example could have been chosen here, like this: 1/0 = 2/(0*2) = 2/0 so 1 = 2. For a layperson though, I suppose its good enough. So long as no one does research that is.
2:13: Oh look, he brings that up. I feel kind of embarrassed now, but I think I'll leave that in, because it's kind of funny. I still think he should have used division by 0 instead though, because then disclaimers like that aren't necessary.
2:40: A good point, and it's why I don't like the "1/3 = 0.33333... so 1 = 0.99999..." argument, but I'm curious to here what he says about the one on the left.
I have things in my life which are more pressing than this right now, so I will finish the video later. It seems to be a pretty good video though.
0.99999... isn't "approaching" anything. It is a fixed number like any other. That misunderstanding is going to make me lose my mind one of these days. Also, it doesn't have to be defined as a limit. It can also be defined as a supremum. Do you agree with the following statements?
1. 0.9, 0.99, 0.999, etc. are all less than 0.99999....
2. 0.99999... is the smallest such number.
If you do, then you agree with the definition of 0.99999... as a supremum, which is that 0.99999... is the unique number satisfying those properties. If you think that there is no such number that satisfies such properties, the completeness axiom assures us that if S is any bounded set of real numbers, then there is x such that:
1. Everything in S is less than x.
2. x is the smallest such number.
Just to clarify, I had misunderstood what RationalMadman was referring to when he said base-9. Based on the context, it had seemed that he was referring to base-10 but calling it the wrong thing, but I have since realized that he was indeed referring to base-9. He hadn't used it within that particular argument, but rather in a previous argument, and that is what threw me.
That's fine! I published my argument before I saw this, but I guessed what you meant, and you can mention it in your next argument if you want. For that matter though, what you are calling base-9 is actually base-10. (https://mathworld.wolfram.com/Base.html#:~:text=The%20word%20%22base%22%20in%20mathematics,in%20which%20logarithms%20are%20defined.)
This reminds me of a conversation I once had with my mom. We were discussing gnelefants (you pronounce the "g"). Gnelefants, are, of course, not a real thing, but we had a good long discussion on their dangers, uses, misuses, health benefits, health concerns, and the importance of knowing your types of gnelefants.
Yes. It means that math exists on its own, independent of human interpretation. Mathematical truths exist before we discover them, and are objective.
I'm so sorry! I forgot about this. You probably would have won anyway though. Thank you for the debate.
It is the limit/supremum, not the "last number."
Well, technically it is a number with infinitely many nines after the decimal. What most people don't consider is that one first needs to establish what that even means, so in this kind of debate, I generally try to establish is that under any reasonable definition of 0.99999..., 0.99999... is equal to 1. The reason this can be important is that some people will say things like "mathematicians just define it in a way that fits their deluded narrative," which is nonsense, but you still have to deal with it.
I think of it as a truism, but there are plenty who disagree. You'd be surprised.
Okay, good! You seemed a little bit too serious!
Post #1: What? Of course 1 is 1 and 0.99999... is 0.99999..., but also, 1 is 0.99999... and 0.99999... is 1. 0.99999... and 1 are two different ways of writing the same number, as I will demonstrate in this debate.
Post #2: Wait, what? Is that supposed to be an argument? I don't even know what you're trying to say. Also, if you want to defend the position that 0.99999... is not equal to 1, then feel free to accept this debate.
Post #3: 10 * 0.99999... = 9.99999... = 9 + 0.99999... = 9 + 1 = 10. No, that's not an argument that 0.99999... = 1, (I'll save those for the actual debate) I'm just demonstrating that you have not found an inconsistency in my position, as if 0.99999... = 1, 9.99999... = 10, so there isn't a problem.
For reference, for those of you who are not familiar with coding, when you ask a computer if two strings are equal, it checks if they have the same characters, rather than if they have the same value, so the string '1+1' would not be equal to the string '2', but that doesn't mean that 1+1 isn't equal to 2.
I hope that's a joke. That would be like saying "My computer says that the string '1+1' does not equal the string '2' so 1+1 is not 2! Proven!"
I constructed my religious view through my own reasoning alone, and Best.Korea also claims not to have been indoctrinated, so this seems to me to be a truism, or at least a case where definitive evidence is excessively easy to provide.
Typo. In the definition of injective: I meant to write "The idea is that f is one-to-one in that it never sends two different values to the same value."
First, it's pedantic-yet-important correction time: Not all numbers, all real numbers. There is no such thing as all numbers. If you are a bit confused by the distinction, it is important to realize that "real numbers" is a misnomer. They aren't the only ones out there. Also, it isn't that the set of real numbers isn't itself "greater" than the set of natural numbers, it is that there are more real numbers than natural numbers. This is often phrased as "some infinities are larger than others," which I assume is what doesn't make sense to you.
I'm going to assume that you know what a function is, and that you are familiar with the notation f: A --> B. If you aren't, I suggest you read this: https://www.mathsisfun.com/sets/function.html. In the notation f: A --> B, A is the domain, and B is the codomain.
We say f is injective whenever f(x) = f(y) implies x = y. The idea is that f is one-to-one in that it never sends two different values to the same
We say f is surjective whenever for any y in B there is x in A such that f(x) = y. The idea is that no element of B is left out.
We say f is bijective when it is both injective and surjective. The idea is that f provides a one-to-one correspondence between the two sets. For every element of A there is an element of B, and visa versa.
For two sets A and B, it makes sense to say that A and B are the same size if and only if there is a bijection f: A --> B. We represent the "size" of the set A by |A|, and call it the cardinality of A. Numbers such as |A| are called cardinal numbers. Thus for cardinal numbers |A| and |B|, |A| = |B| whenever there is a bijection f: A --> B. We say that |A| < |B| (also written |B| > |A|) whenever there is an injection f: A --> B but no bijection g: A --> B. It is in this way that "greater" is still meaningful.
Cantor proved that |R| > |N|, where N is the set of natural numbers, and R is the set of real numbers. This is what is meant when it is said that there are more real numbers than rational numbers.
Response to comment #12 (RationalMadman):
You do realize that in the debate I was claiming that 1 - 0.99999... isn't equal to 0.00...1, right?
Response to comment #11 (Savant):
My comments on the video:
1:06: Yes. That's not wrong. Most people assume that the existence of 0.99999... isn't going to be a question, but I did defend it during this debate.
2:05: Yes, actually, I do think that ...99999.0 = -1. If that seems like nonsense, observe: http://math.uchicago.edu/~may/REU2020/REUPapers/Pomerantz.pdf. I know, it's kind of advanced, but math dealing with the p-adic numbers (which is the field in which numbers like ...99999.0 are considered to exist) is inherently advanced. A better example could have been chosen here, like this: 1/0 = 2/(0*2) = 2/0 so 1 = 2. For a layperson though, I suppose its good enough. So long as no one does research that is.
2:13: Oh look, he brings that up. I feel kind of embarrassed now, but I think I'll leave that in, because it's kind of funny. I still think he should have used division by 0 instead though, because then disclaimers like that aren't necessary.
2:40: A good point, and it's why I don't like the "1/3 = 0.33333... so 1 = 0.99999..." argument, but I'm curious to here what he says about the one on the left.
I have things in my life which are more pressing than this right now, so I will finish the video later. It seems to be a pretty good video though.
0.99999... isn't "approaching" anything. It is a fixed number like any other. That misunderstanding is going to make me lose my mind one of these days. Also, it doesn't have to be defined as a limit. It can also be defined as a supremum. Do you agree with the following statements?
1. 0.9, 0.99, 0.999, etc. are all less than 0.99999....
2. 0.99999... is the smallest such number.
If you do, then you agree with the definition of 0.99999... as a supremum, which is that 0.99999... is the unique number satisfying those properties. If you think that there is no such number that satisfies such properties, the completeness axiom assures us that if S is any bounded set of real numbers, then there is x such that:
1. Everything in S is less than x.
2. x is the smallest such number.
Just to clarify, I had misunderstood what RationalMadman was referring to when he said base-9. Based on the context, it had seemed that he was referring to base-10 but calling it the wrong thing, but I have since realized that he was indeed referring to base-9. He hadn't used it within that particular argument, but rather in a previous argument, and that is what threw me.
Just so you know, you only have 3 hours left.
Could you elaborate on what we would be debating? Also, could we debate this in English?
That's fine! I published my argument before I saw this, but I guessed what you meant, and you can mention it in your next argument if you want. For that matter though, what you are calling base-9 is actually base-10. (https://mathworld.wolfram.com/Base.html#:~:text=The%20word%20%22base%22%20in%20mathematics,in%20which%20logarithms%20are%20defined.)
Friendly reminder: Your time to make your argument is running low!
This reminds me of a conversation I once had with my mom. We were discussing gnelefants (you pronounce the "g"). Gnelefants, are, of course, not a real thing, but we had a good long discussion on their dangers, uses, misuses, health benefits, health concerns, and the importance of knowing your types of gnelefants.