First of all, I would like to say that 0.99… equaling 1 is pretty much a fact. Search any serious mathematics sources and ask any mathematician and the overwhelming majority will agree that this is true. Of course this does not prove that it is true. But I’m just bringing this up because some people appear to be under the illusion that 0.99… obviously isn’t 1 and anyone who thinks it is obviously is uneducated. I would just like to say that that is not the case and most educated people, people good at math do agree with this thread. So those who disagree with this thread should at least know that they are in the minority.
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It seems that many people are not understanding what 0.999… really is. Some people are saying it’s getting close to 1 but never reaches it. This is false based on the fact that it’s not getting close to anything. It’s not getting anywhere; no more than 9 is “getting” to 10. 0.99… does have a value and that value doesn’t hover from close to 1 to even closer to 1. That doesn’t make sense. It has one static value which does not magically change on its own.
0.999… can be written as the sum of an infinite series, which is one static number. It is the sum of the infinite sequence (9/10) + (9/10)(1/10)^2 + (9/10)(1/10)^3….As I have already stated, this is equal to 9/10 divided by 1-9/10, which is 1 (read about the sum of convergent geometric series). I think the thing most people have trouble understanding is the fact this number is the result after you add an infinite amount of the numbers in the sequence. People are saying things like “you can never add enough numbers to get to 1, because you can always add another 9.” But that isn’t what 0.999…is. There are already an infinite number of 9s so adding more would not make sense. I am not talking about 0.9999 or 0.9999999 or even 0.99999999999999, I’m talking about infinite 9s and what its value would be.
Some people may be upset because they may be thinking “you can’t add an infinite number of values.” Sure you can; people do it all the time. Think about the sequence 1+1+1+1+1…. What would be the sum of an infinite amount of 1s? I am not talking about the sum of a thousand 1s or a million 1s. I am talking about infinite 1s. What would their sum be? Most people have no problem with this and would kindly say the answer is infinity. Sure, infinite 1s cannot be written down on paper or expressed visually, but with mathematics we can be sure that an infinite amount of 1s does equal infinity. Not a billion and not a trillion; neither of these would be infinite 1s. Infinite 1s would equal infinity.
Going back to 0.99…. No we cannot write an infinite amount of nines. We can’t visualize an infinite amount of nines. We cannot physically construct an infinite amount of nines. But like adding an infinite amount of nines, we can use mathematics to add an infinite amount of nines and we can be sure of its value. And we can represent that number with the appropriate symbols. Yeah, if you physically add nines, we will only get closer to 1 and will never reach 1. But we can’t physically reach infinity. That doesn’t mean we don’t know what will happen if we did theoretically add an infinite amount of nines. It would be equal to that number that you approach (which is 1), when you add a finite amount of nines. But when you add an infinite amount of nines, you REACH that number.
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To all the people saying 0.99… is less than 1, I have one question: Give me a number higher than 0.99… and less than 1. As I have already stated, if two numbers are different, then they will have not just one, but an infinite amount of other numbers between them. I’m only asking you to give me one number. One number higher than 0.99… and one number lower than 1. It shouldn’t be too difficult if they are in fact different numbers. And no, you can’t say 0.99… is the next number lower than 1, because there is no “next” number after any other number.
And before you say “infinite isn’t a number” so we can’t tell. That shouldn’t matter. Just because a number expressed (in decimal notation) has an infinite amount of digits doesn’t mean we can’t find numbers higher than it or lower than it. And it certainly doesn’t mean we can’t see if it’s the same as another number. Take pi for instance. Let’s say someone said pi is equal to 3.141. You could easily see that this is false by simply finding more precise value of pi. We could figure out that pi goes to 3.1415, so it’s between 3.141 and 3.142. We can get more precise and see that pi goes to 3.141592, meaning it’s between 3.141592 and 3.141593. We could play this game forever and there would never be a number that’s equal to pi, because pi would always be higher or lower. Same with 0.333 or 0.258714285714…. or any other infinite decimal unless the number you compare it to is equal.
You can even do that with 0.999… Say someone said was equal to 0.99999 was the number that was higher than 0.999…. and lower than 1. We could easily disprove this by getting more precise with 0.999… and seeing that it equals 0.99999999, which is higher than 0.9999. We could play this game forever and we could NEVER find a value higher than 0.9999 AND lower than 1. This would never happen. The fact that there is no number higher than 0.999…. and lower than 1 is proof that they are of the same value. Again, if there is no number higher than 0.99... and lower than 1, then they are equal. If they weren't equal, there would be an infinite number of values between them. And just because 0.9999 has an infinite amount of digits, that shouldn't harm anything.
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The main reason 0.999… equals 1 is because it works. It doesn’t break any mathematic rules at all. You can manipulate it all you want and it would never break anything. In fact, 0.999…equaling 1 would make things wonderful. It would make 0.333…. (or 1/3) perfectly one third of 1 in decimal notation. It makes 0.111111 (or 1/9) perfectly one ninth of 1 in decimal notation. We can multiply it, divide it, add it, subtract, do whatever we want to it, and 0.999…=1 would not have ANY harmful implications.
