1.999 repeating = 2
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Voting Style:  Open  Point System:  7 Point  
Started:  1/15/2008  Category:  Science  
Updated:  9 years ago  Status:  Voting Period  
Viewed:  8,700 times  Debate No:  1822 
Debate Rounds (5)
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1+1=2
subtract 1 from each side (1+1)1=21 multiply by a, a=1.999 repeating {(1+1)1}A=[21]A 0=0 therefore, 1.999 repeating = 2, because 1.999 feet is pretty close to 2 feet
My main point in this argument is that 1.9999 repeating CANNOT equal 2 because 1.999 repeating is an irrational number, while the number 2 is a rational number. Since a number cannot be rational and irrational at the same time, then 1.999 repeating CANNOT equal 2. The definition of a rational number is a number that can be expressed as a fraction (or ratio) p/q where p and q are integers and q is not equal to zero. The definition of an irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q. The number 2 is obviously a rational number, since, it can be written as 8/4 (since 8 divided by 4 equals 2). However, there are no two integers p and q, that divided (or which ratio) give 1.999 repeating. We should realize that there are other repeating decimals such as 0.333 repeating, 0.666 repeating, or 1.333 repeating that CAN be written as ratios of two integers, and therefore are rational numbers. For the numbers given above this would be: 0.333 repeating = 1/3 0.666 repeating = 2/3 1.333 repeating = 4/3 As we can easily see, by dividing 1 by 3, we get 0.333 repeating. I want to point out that the number of times that the divisor (the number 3 for this case) is contained in the dividend (the number 1 for this case) is zero. Therefore, the first number in performing the division is 0. and then as we keep performing the division we get 0.3, 0.33, 0.333, etc. This shows that 1/3 equals 0.333 repeating. Similar reason can be used with 0.666 repeating and 1.333 repeating. However, if we divide 6 by 3 (for example) we can only obtain 2. It is impossible to obtain 1.999 repeating by dividing 6 by 3 since the number of times that the divisor (the number 3 for this case) is contained in the dividend (the number 6 for this case) is 2. This shows that it is impossible to obtain the number 1 as the first term in the number 1.999 repeating. Since there are no two integer numbers whose ratio gives 1.999 repating then we must conclude this is an irrational number, and as stated above, that 1.999 repeating CANNOT equal the number 2. As a final note, I will like to indicate that operations such as addition, multiplication, substraction, and division are undefined for repeating decimals such as 0.333 repeating or 0.999 repeating. This operations are defined for fractions, but not for repeating decimals. Hence this operations cannot be used to prove that 0.999 repeating equals 2. Thanks. 

All rhetoric (j/k everything on here is all rhetoric). There is no reason why a rational number cannot be equal to an irrational number. Your definitions are great, but they don't necessarily preclude any possibility that the two could be equal to each other, only that the two cannot be IDENTICAL to each other. Tarzan's proof is rock solid and your semantical arguments are attempts at struggling to find any way you can not to believe something that is hard to accept. A board that is 1.999 repeating feet long is just as long as a board that is 2 feet long, because you can never identify the difference in length. How much less than 2 is 1.999 repeating? If they are in fact different values, than you should be able to extrapolate the difference between them. Also, a full third is 0.333 repeating. But if you simply look at the number, 0.333 repeating SEEMS like it is *just* a little less than a third, because the numbers don't do it justice, in the same way that 1.999 repeating doesn't quite do justice to represent 2. If you multiply .333 repeating by 3 you get .999 repeating, which is equal to 1 because three thirds are a whole.
A=1.999 repeating multiply by 10 10A=19.999... subtract A from both sides 9A=18 < this is the part most people are uneasy about. but 19.999... minus 1.999... is equal to 18. How can you refute this? divide by 9 A=2 elementary, my dear watson How can you not add a repeating decimal??? Does this mean I can't add three thirds to get a whole? Aren't all numbers repeating decimals, with zeros on each end?
First of all, I want to indicate that this is not a philosophical debate. This is a mathematical debate, and anyone that has taken a course involving mathematical proof knows that definitions in mathematics are the basis or building blocks of any proof. You indicated that: "There is no reason why a rational number cannot be equal to an irrational number". By mathematical definition (not philosophical), an irrational number cannot be equal to a rational number. You can check this definition in any mathematics book. You indicated that: "Your definitions are great, but they don't necessarily preclude any possibility that the two could be equal to each other, only that the two cannot be IDENTICAL to each other". Well, first of all, there are not "my" definitions of rational and irrational numbers. This are the definitions that have been established for a long time, you can find them in any math book, and all mathematicians agree with them. And the definition clearly indicates that an irrational number cannot be equal to a rational number. Again, this is a mathematical debate, not a philosophical one were people can debate whether an apple can equal to an orange or whatever. You indicated that: "Tarzan's proof is rock solid and your semantical arguments are attempts at struggling to find any way you can not to believe something that is hard to accept". I don't see how your subjective opinion is PROOF that 1.999 repeating equals 2. You indicated that: "A board that is 1.999 repeating feet long is just as long as a board that is 2 feet long, because you can never identify the difference in length. How much less than 2 is 1.999 repeating? If they are in fact different values, than you should be able to extrapolate the difference between them" The fact that I cannot identify the difference between a 1.999 repeating feet board and a 2 feet board does not mean these two are equal. Using your logic, then 1.88 is equal to 2 as well, since I am pretty sure I wouldn't be able to identify the difference between a board that is 1.88 feet long and a board that is 2 feet long. You indicated that: "Also, a full third is 0.333 repeating. But if you simply look at the number, 0.333 repeating SEEMS like it is *just* a little less than a third, because the numbers don't do it justice, in the same way that 1.999 repeating doesn't quite do justice to represent 2" I indicated above that 0.333 repeating comes from the ratio of two integers, namely 1/3. If you calculate this ratio, and divide 1 by 3, you get 0.333 repeating. Therefore this shows clearly that 1/3 equals 0.333 repeating, and that 0.333 repeating is a rational number. You indicated that: "If you multiply .333 repeating by 3 you get .999 repeating, which is equal to 1 because three thirds are a whole" This is the origin of the problem, repeating numbers cannot be multiplied by integers since this operation is undefined in a similar way that 1/0 (one divided by zero, not the limit, just the number one divided by zero) is undefined. What I mean by "repeating numbers cannot be multiplied by integers" is that there is no mathematical definition or property that indicates that 3*0.333 repeating = 0.999 repeating, and you cannot get from one point to the other by using mathematical properties or definitions. However, you could show that 2*0.333 repeating equals 0.666 repeating by using the following mathematical properties: 2*0.333 repeating = 2*(1/3) This comes from the mathematical definition of dividing 1 by 3 = (2*1)/3 Associative property of multiplication = 2/3 Identity property of multiplication = 0.666 repeating This comes from the mathematical definition of dividing 2 by 3 The problem that arises with 0.999 repeating is that you cannot obtain it by dividing any two integers. Therefore you CANNOT show that 3*0.333 repeating = 0.999 repeating. You indicated that: "How can you not add a repeating decimal??? Does this mean I can't add three thirds to get a whole? Aren't all numbers repeating decimals, with zeros on each end?" As I indicated, you cannot add, multiply, substract, or divided, repeating numbers directly using math properties. There are properties that allow you to do this operations on integers or fractions, but not on repeating (or infinite numbers). And yes, all numbers can be written as repeating numbers, but the difference is that if you can write a repeating number as an integer or a fraction then you can perform addition, multiplication, substraction, or division on these forms of the number (NOTE: all numbers can be written as repeating numbers, however not all of them can be written with zeros at the end). In Tarzan's proof the flaw is in the fact that he multiplies 10 by A, then substracts A, and gets 18. What you should really get if you perform 10*A minus A, you will get 18.0000...(infinite zeros)...0001 As you can see, there will always be a one at the end of this infinite number (I know that for the human brain this is hard to grasp, but that is infinity). 

Your best argument is an inability to do the addition/subtraction to get the result in the proof. But this makes no sense. Addition cannot be undefined in this way, because we work with these numbers all the time. A third plus a third is two thirds. The fact that a third equals a repeating number does not automatically mean that a third is undefined and you can't add it. You try and make a distinction BETWEEN a third and 1.999 rep., because you can't divide two numbers to get it, but that leads us back to the same old argument: why can't a rational number not be equal to an irrational number?
I still believe that your arguments are just not as convincing as Tarzan's proof is. You say that 19.999 repeating minus 1.999 repeating is equal to 18.000 repeating plus one. Maybe I should instruct you on what infinity is, so that you don't botch it in our debate. Infinity goes on forever, so you would never have the chance to put that one in there, and besides, I don't see any reason AT ALL to even think that there MIGHT be a one at the end anyway. your definition did not include that a rational number cannot be equal to an irrational number. You say that it can't but us debaters on here are probably not just going to take your word for it. Saying that I could check it in a math book is not enough to show your case. Give us some exact wording that can be cross checked so that we can indeed see that a rational number can never be equal to an irrational number. your definitions are simply not as convincing as actually doing the math, kennard. The math works, and your definitions seem somewhat indirect and possibly misinterpreted. If my opinion of tarzan's proof is subjective, then so is your interpretation and application of your definitions. I fail to see why your position is one of objectivity and mine is one of subjectivity. "I am pretty sure I wouldn't be able to identify the difference between a board that is 1.88 feet long and a board that is 2 feet long." Why did you write this? The difference is .12 feet. If you meant to say 1.888 repeating, then the difference is .111 repeating feet. This is such a bad argument that I am hoping you just mistyped something in there, otherwise the fact that you can't see something that is over an inch long is your error, not tarzan's. 1/3 + 1/3 + 1/3 =1, just like 0.333 rep. + 0.333 rep. + 0.333 rep. =1 . This leads to the conclusion that 0.333 rep. must be equal to that number that is *just* a little higher (~0.3334) that would more satisfyingly represent a full third, in the same way that 1.999 rep. represents that number that is *just* a little higher that is 2.
You wrote the following two statements: "why can't a rational number not be equal to an irrational number?" and "your definition did not include that a rational number cannot be equal to an irrational number. You say that it can't but us debaters on here are probably not just going to take your word for it. Saying that I could check it in a math book is not enough to show your case. Give us some exact wording that can be cross checked so that we can indeed see that a rational number can never be equal to an irrational number." The definition of a rational number is a number that CAN be expressed as a ratio of two integers (where the denominator cannot equal zero). The definition of an irrational number is a number that CANNOT be expressed as a ratio of any two integers. Therefore a rational number CANNOT equal an irrational number (I honestly don't know how to make this point any clearer). By the way, here are some references: http://en.wikipedia.org... http://en.wikipedia.org... First of all I want to clarify that by saying repeating numbers I specifically mean the repeating representation of the number, such as 0.333 repeating. The number 1/3 is a fraction, it is composed by only two integers, while the number 0.333 repeating is composed of infinitely many digits. These two numbers can be proven to be equal by using simply dividing 1 by 3. The math property of dividing 1 by 3 is a well defined property in mathemetics because it involves two finite numbers. Once you convert 0.333 repeating to 1/3 then you can use all the math properties (associativity, commutativity, addition, etc.) that you want on this representation of the number. However, you cannot add, multiply, substract, or divided, on the repeating representation of the number directly using math properties because repeating numbers go on infinitely. The reason for not been able to use addition, substraction, multiplication or division on infinite numbers is because this properties break down when dealing with infinity. For example: infinity + 1 = infinity substracting infinity from both sides gives: 1 = 0 This was a very simple example, but it shows that basic operations break down when dealing with infinity because these operations are undefined for infinity. Therefore, to FORMALLY do an operation on a repeating number, this number will have to be rewritten as a fraction and then these basic operations can be performed on the fraction representation of the number. The main problem with 1.999 repating is that it cannot be written as a ratio of any two numbers. I will briefly write the proof of this again (PLEASE CHECK THIS ARGUMENT CAREFULLY TO UNDERSTAND WHY 1.999 repeating CANNOT equal 2): If we divide 6 by 3 (for example) we can only obtain 2. However, it is impossible to obtain 1.999 repeating by dividing 6 by 3 since the number of times that the divisor (the number 3 for this case) is contained in the dividend (the number 6 for this case) is 2. THIS SHOWS THAT IT IS IMPOSSIBLE TO OBTAIN THE NUMBER 1 AS THE FIRST TERM OF THE NUMBER 1.999 repeating. Since there are no two integer numbers whose ratio gives 1.999 repeating then we must conclude this is an irrational number, and as stated above, that 1.999 repeating CANNOT equal the number 2. You wrote: "You say that 19.999 repeating minus 1.999 repeating is equal to 18.000 repeating plus one. Maybe I should instruct you on what infinity is, so that you don't botch it in our debate. Infinity goes on forever, so you would never have the chance to put that one in there" I am really glad you wrote this statement down. What makes you think that I would never have the chance to put that one in there, but you would have the chance of substracting all the terms in the repeating representations of 19.999 repeating and 1.999 repeating? You wrote: "your definitions are simply not as convincing as actually doing the math, kennard. The math works, and your definitions seem somewhat indirect and possibly misinterpreted" Have you seen how many post are out there indicating that 1=2? These statements are posted by people who think that "the math works". However, people who really know about math realize that mathematical definitions must be used to PROVE things. That is why anyone who knows math well looks at the statement 1=2 and immediately realize that it involves the old trick of dividing by zero, which is (as you know) UNDEFINED. You wrote: "Why did you write this? The difference is .12 feet. If you meant to say 1.888 repeating, then the difference is .111 repeating feet" I apologize for not been clear on that line. The point that I was trying to prove there was that the fact that I cannot distinguish (visibly) the difference between a 1.9998888777 feet board and a 2 feet board those not make the two numbers equal. I really hope that now that I rewrote the definition of rational and irrational numbers, and how they clearly can't be equal to each other, that you proceed to prove that 1.999 repeating is a rational number. 

Your arguments do make some sense to me, as a nonmathematician, but when I look back at the proof I am still not convinced that you can't subtract 1.999 rep. from 19.999 rep. to make 18. Your wiki references did not say "a rational number can never equal an irrational number", unless you can direct me to the spot they do. Also, please give me some evidence that specifically says that you absolutely cannot subtract a rep. decimal, especially when it is simply to cancel out a similar rep. decimal. With all the tricks mathematicians use to find answers, I have a hard time believeing that this one is not in their arsenal.
I think the following example is good evidence that addition, substraction, multiplication, and divison break down when used on the repeating form of a number. For example, people who claim that you can add repeating numbers do the following: 000.333 repeating + 0.666 repeating  000.999 repeating (Sorry for writing the three extra zeros in front of the decimal point, but I wanted to align the three repeating numbers) So as you can see, they add the numbers as if they were finite numbers, and then write repeating at the end. But let's see what happens when we add the following two repeating forms of the following numbers: 000.999 repeating + 0.999 repeating  001.998 repeating (Again, sorry for writing the extra zeros in front of the decimal point, but I wanted to align the three repeating numbers) As we can clearly see, addition breaks down in this case (unless now you want to debate that 1.998 repeating equals 2). This is because, as I stated earlier, this forms of writing a number go on infinitely, and we can never finish completing the full addition of all the terms. Similar logic can prove that substraction, multiplication, and division cannot be performed on repeating forms of a number. By the way, if you click on the second wiki reference that I gave, the very first line says: "In mathematics, an irrational number is any real number that is not a rational number". You say that you are a nonmathematician, and that is fine, but it clearly says that "an irrational number is any real number that is not a rational number", therefore an irrational number can never equal a rational number. If you read the definition of the two numbers, it indicates that rational numbers can be written as the ratio of two integers, but irrational numbers cannot. In other words, rational numbers can be obtained by dividing two integers, but irrational numbers cannot. By the way, I don't want to make this personal, so I apologize if there has been some tension between our comments. Actually I am glad we are having this debate because it has made me think a lot, haha. 

Well there is tension in a lot of debates. The ones about religion can get really bad... That is actually the reason I took this debate, because since I am not a mathematician I could do this without caring TOO much about it, and joke around a little in the process. As far as your argument goes, I would say that I have run out of good arguments about 4 rounds ago and I am just holding on for the sake of it! I would really like to ask an older, more educated math professor about this, but that is the thing: the most respected math teacher I know at my university did Tarzan's exact proof on the board for us in class! I ended up failing out of calc 2 and ditching math afterwards... Honestly, my opinion is of no consequence and I only argued this case for the sake of arguing, and because people in the comments section seemed to be egging it on some. Otherwise, good debate, you DID make some good points...
Well, for this last Round I'll just respond to some of the comments that have been posted down. Lazarus Long indicated that two integers whose ratio give 1.9999... repeating are 4/2. The proof that he gave is the following. "Since 1.999... = 2, then 4/2 = 1.99999.... Simple, no?" The only simple thing I see in this proof is realizing how flawed it is. I'll just change one number so that people can realize how foolish this "proof" is: Since 500 = 2, then 4/2 = 500 Simple, no? Then he claims the following: "how else would you explain: 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3 = 2 which is the same as 0.333... + 0.333... + 0.333... + 0.333... + 0.333... + 0.333... = 2, which, if we simply do the addition, IS 0.999... + 0.999... = 2 which, of course, requires that 0.999... = 1 (actually the same as the question at hand here), so Q.E.D. " Well, the way I already explained why the "proof" written above is wrong is because addition, substraction, multiplication, and division cannot be performed directly on repeating representations of numbers. I showed this on the Round 4 of the debate. Therefore, 0.333... + 0.333... + 0.333... DOES NOT EQUAL 0.999... Another comment written down was made by beem0r with respect to the points I made in Round 4 are the following. Beem0r wrote: "I think we should realize that it isn't the 8 that's repeating, it's an 18 that's repeating. The 1 in the tens place will always change the previous 8 to a 9. You can do addition and subtraction with repeating numbers, you just have to know what's repeating. .999... + .999... is 1.999..." The fact that he has to "realize" what is repeating proves my point that addition and all the other operations cannot be performed directly with repeating representations of numbers. I do understand what he is saying, so I'll rephrase it here. What he is saying is that the 8 at the end (end?) will constantly be changing to a 9 because of the 1 that is carried from the sum of the nines on the column that is directly to its right. So the number that we will have is 0.999...(infinite nines)...9998 with the eight constantly and forever changing to a 9. This is why addition and the other operations cannot be performed on infinite representations of numbers, because this operations are only DEFINED on finite representations of numbers. Therefore the only way to make algebraic operations on inifinite representations of numbers is to write them in a finite form. This could be done either by writing them as a fraction or representing them by a variable (or symbol, as is the case for Pi). Since 1.999 cannot be obtaind by dividing any two integers (which means it cannot be written as a fraction) then the only way to do algebra on it is by representing it with a variable. The proof that 1.999 repeating cannot be written as a ratio of any two integers is written in Round 1 and rewritten in Round 3 of the CON side. Finally, I want to indicate that many people (including many mathematicians) don't really understand the concept of limits and infinity. For example, the limit of 1/x as x goes to inifinity equals zero. What this means is that as x goes to infinity the function will get closer and closer to zero, but it will NEVER equal zero. Many mathematicians make the wrong claim that, "at infinity the function will equal to zero". This statement is wrong because it assumes that there is a point where the function will actually equal zero. But then I ask, after it reaches zero then what? Does it remain being zero, does it become negative? The point that I am trying to make here is that infinity is not a point that can be reached, infinity just means that it will keep on going forever. Therefore 1/x will get closer and closer to zero forever, but it will never equal zero. The best we can do when dealing with infinity is to use limits to indicate what value some function will get closer to, but that is as good as it gets. This means that 1.999 repeating is by definition the number that is closest to 2 from the numbers that are smaller than 2. 
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You've admitted that .333... is equal to the fraction 1/3.
However, the same logic that has made you conclude that .999... contains some nonzero infinitesimal could be applied to .333... and 1/3.
However, the notation in which 0.999... is written is completely valid in the reals. There is no separate component. It's just a number.
Also, .999... is also 1 in the hyperreals. Please check whatever source you think has told you otherwise, as they are miserably wrong.
So to conclude:
1. 0.999... is a valid decimal representation in the reals.
2. Just using simple thought, the only possible difference between 0.999... and 1 would be an infinitesimal.
3. There are no nonzero infinitesimals in the reals.
4. There is no mathematical difference between 0.999... and 1 in the reals.
The extent to which they are different is the same as that of 1/3 and 2/6. Same number, two representations.
Now that I have read some more on hyperreal numbers, I think that 0.999 repeating, and numbers that have repeating nines after the decimal are not real numbers at all, they are hyperreal numbers. First of all it should be realized that hyperreal numbers are an extension of the real line in the same way that complex numbers are an extension of the real line. In other words, all the real numbers are a subset of the hyperreal numbers and of the complex numbers. However, as we know, not all complex numbers are real numbers. In the same way, not all hyperreal numbers are real numbers. In the hyperreal number line 2 and 1.999 repeating are two different numbers, this means that 2 cannot equal to 1.999 repeating in the real line because 1.999 repeating does not exist in the real line.
This is analogous with complex numbers. For example, 1 and 1+i, are different numbers in the complex plane. However, we cannot say that in the real line, 1 and 1+i are equal just by claiming that the imaginary part in 1+i does not exist in the real line. The complex number 1+i is not part of the real numbers at all because it has an imaginary part. Similarly, 1.999 repeating and all the repeating nines numbers are not part of the real numbers at all because they have a nonzero infinitesimal component with them, and nonzero infinitesimals do not exist in the real line.
>>The last thing I have to say about the argument is that two REAL numbers are only equal if their difference equals ZERO. However the difference between 2 and 1.999 repeating is not zero, it is an infinitesimal.<<
In the reals, an infinitesimal is zero. Ergo, the difference is zero. I thought I was clear about this previously, my apologies if not. I can often be too longwinded or nonsequential in my arguments.
http://en.wikipedia.org......
http://en.wikipedia.org...
A lot of skeptics like yourself there, using arguments quite similar to yours, and it seems they are unable to hold up their side with people with perhaps more knowledge and interest than myself. There's a whole page dedicated to people making arguments against .999... = 1, and I believe all their objections have all been answered quite well.
Well, I can't say I didn't enjoy our battle, so thanks. Be seeing you.
" Also, con is not backing up his "you can't add/subtract/mult/div repeating decimals."
Pro asked why you can't, con said "That's just how it is, you can't do it. "
The last thing I have to say about the argument is that two REAL numbers are only equal if their difference equals ZERO. However the difference between 2 and 1.999 repeating is not zero, it is an infinitesimal. So 1.999 repeating is a representation of the theoretical "closest" number to 2 in the same way that infinity is a representation of the theoretical "largest" number. It would seem to me that accepting 2 = 1.999 repeating is the same as accepting that their exist a real number which is larger than any other real number.
Anyways, I must admit that you seem like a smart guy, and even though I honestly think that the arguments that I have made are stronger I am pretty sure by now that I won't be able to convince you regardless of what I say.
If a page's contents are debated, like the .999... page I linked, there are usually an abundance of references used. Pages with much less disagreement, such as kennard's division by zero page, don't tend to use quite as many, since no one demands sources on widelyknown information.
Wikipedia _can_ be changed, but any user can look at the changelog to see what was changed when. If some troll keeps changing the page, it will be obvious.
Wikipedia is probably the best onestop source for information anywhere on the internet. It's a great source, but sometimes you have to chack the references and/or history if something seems off kilter.
You say:
infinity = 1/infinitesimal
We are talking about _REAL_ numbers. Not surreal numbers, not hyperreal numbers, not extended real numbers. We're talking about reals here.
Neither infinity or infinitesimals are reals.
Infinitesimals are treated as 0, infinities are treated as undefined. That is how the real numbers treat these two concepts.This is why you have never gotten either of these as an answer to a problem, nor can they be input into equations. As you yourself said,
"it is well known that since infinity is not a real number, then algebraic operations break down with it. Similarly, since an infinitesimal is not a real number then algebraic operations break down with it as well."
Get that? NOT a real number. It is 0 if we're using reals.
The difference between the two is 0.
Also, 1.9... is not an infinitesimal. It is a real number, so there is no reason one cannot use arithmetic on it. Therefore, the proof given by Pro in round 2 is completely valid.
I really thought that would be enough, but at least now you can't claim that I should 'wait until I graduate college,' since it's held by mathematicians that .999...=1.
Zero is a real number, all mathematical operations can be performed with it (except dividing by zero).
Some people might think that 1/0 equals infinity but this is wrong, 1/0 is undefined.
(Reference: http://en.wikipedia.org...).
Infinity is not a real number, it is just a symbol used to indicate that the real numbers grow unbounded. In other words, infinity is a symbol that indicates that the real numbers will keep on going forever.
Most people don't have much trouble accepting this, but they have trouble accepting the fact that there are infinitely many numbers between 0 and 1. So how do we represent this type of infinity? We represent it with infinitesimals. The symbol for infinity and the symbol for infinitesimal are related in the following way:
1/infinity = infinitesimal
In other words, in the same way that infinity represents how real numbers get larger and larger forever, infinitesimals represent how two real numbers get closer and closer forever. Let me rephrase this statement because it could be misinterpreted: In the same way that infinity represents how you can always get a number that is larger, an infinitesimal represents how you can always get a number that is closer.
Also, it is well known that since infinity is not a real number, then algebraic operations break down with it. Similarly, since an infinitesimal is not a real number then algebraic operations break down with it as well.
Since
infinity = 1/infinitesimal
then we can conclude that an infinitesimal can never equal zero, because then we will end up with the equation that is undefined, namely:
infinity = 1/0
So 2  1.999... = infinitesimal
Hence, 1.999 repeating represents how you can always get a number that is closer to 2, but it itself is not equal to two.