The Instigator
Con (against)
Anonymous
does 0.9999999.......... equal 1
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Voting Style:  Open  Point System:  7 Point  
Started:  7/14/2018  Category:  Philosophy  
Updated:  3 years ago  Status:  Debating Period  
Viewed:  772 times  Debate No:  116576 
Debate Rounds (4)
Comments (20)
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Con
Argument 1: The argument by mathematical proof: Let x=0.999... (equation 1) Therefore: 10x=9.999... Therefore: 9x=9.999...x Therefore: 9x=9.999...0.999... (substituting in equation 1) Therefore: 9x=9 Therefore: x=1 (equation 2) Substituting for x using equations 1 and 2 we arrive at: 1=0.999... 

Con Every mathematician across the entire globe has at some point pondered the question of whether 0.999.... equals one. The standard argument that it doesn't equal one is that if it is 0.9 it is shy of one, if it is 0.99 it is shy of one if it is 0.999999 it is still shy of one. No matter where you stop the nine sequence, it will never be one. This seems to show that the finite human experience of this is shy of one, but maybe the infinite, beyond human experience of this equals one. How do we know? My opponent has presented an equation in attempt to prove that 0.9999... does indeed equal one. The equation follows: x=0.99999....... We multiply both sides by ten and get 10x=9.99999...... We then subtract x from both sides 9x=9 This is because we subtract the original x value which is 0.999.... from both sides and the result is x=1 Great, sounds really convincing but unfortunately there is just a bit more to it than that. Suppose we used the same math in the same equation except we have infinitely many nines going off to the left. This time the equation will go as follows: ..........9999999999999=x we multiply both sides by 10 and we get ...........9999999999990=10x We subtract the original x value from both sides and get 9=9x We divide both sides by 9 x=1 So this equation is telling me that .......999999999 equals 1. Do you believe that? Most people think that the first equation is fine, but then they look at the second equation and think what I did is ridiculous. But it is exactly the same mathematics as so you can't just pick and choose when it is right and when it is wrong. If you believe 1=0.9999... then you also must believe that ......999999=1. But let's make matters worse. Let do an equation with infinitely many 9's going of to the right and left. The equation looks as follows: ........9999999.99999999......=x We multiply both sides by 10 ........99999999.9999999......=10x We subtract the original x value from both sides and get 0=10x....9999.999999..... ......99999.99999.... is the original x value the equation is now 0=10xx We simplify and get 0=9x We divide both sides by 9 and get x=0 So this is telling me that ....9999.99999.... equals 0. Honestly, do you believe that? The same mathematics is being applied to different circumstances and because it cannot be false in some and true and others, it just shows to prove one thing. It shows that if you believe this, this, and this, you simply must believe this, this, and this. But however you can choose the more believable option like me. You can believe that these equations don't have a meaningful answer in the first place.
You screwed up your maths dude! What you've written doesn't actually make any sense. There's no way multiplying a positive number greater than 1 by ten and then subtracting itself can produce a negative (and no, this isn't because my equation was wrong, you've just fallaciously extended a bastardised form of it's reasoning to the conceptual case of infinity). I think the problem is you have a fundamental misunderstanding of the nature of infinity. If you have an infinite number of nines in the positive direction that is just infinity. So your equation should read as follows: "(infinity)=x Multiply both sides by ten: 10(infinity)=10x We subtract the original x value from both sides and get: 0=10x10(infinity) x=(infinity)" So this is saying x=(infinity) which is exactly what the first statement said, and your mathematical argument has been one big circle. The reason your mathematical reasoning fell apart is infinity doesn't always obey the same rules that standard numbers do. One can consider infinity to be more of a concept than a number. A lot of people have issues mathematically working with infinity because it is simply so incomprehensibly large that people fall into a number of reasoning traps, which if you go on to do higher level mathematics you might begin to understand the basis of. I can provide you with some other proofs if you'd like which show that 0.999...=1. I don't really want to have to type them out, so here's a link to some. (Please note it also includes the one I have already demonstrated). http://www.purplemath.com... I'd like to now spend a bit of time looking at some of the common reasoning flaws people fall into when they fail to realise 1=0.999... 1. It is common for people to assume that 0.999=/=1 because if you take 10.999... you will get 0.000...1. This is mathematically unsound as it demonstrates a misunderstanding of the nature of infinity. It is not possible to have an infinitely repeating decimal series followed by different digit because you will never reach that digit. There is an infinite number of zeros in the example 0.000...1, just as many zeros as there are in 0.000... As such it can be considered that you will never "reach" the 1 at the end of this infinitely many zeros. (I put reach in quotation marks because it is mathematically incorrect way of describing this scenario but it might make it easier to conceptualise for you). 2. Another common misconception that people make is they describe 0.999... as being similar to having an "asymptote" at 1, or getting infinitely close to, but never quite reaching 1. Again this is mathematically incorrect, and it displays a misunderstanding of both limit theorem, and the difference between a graph and a number. Numbers cannot express asymptotes. They do not "move" like a graph moves. They are at a fixed position on a numberline. If that position is anything less than 1, then there would theoretically be a number that was both greater than 0.999..., and less than 1. Such a number obviously does not exist. As such, there is no position besides the position of 1 on the numberline, that could be occupied by 0.999... without being mathematically incorrect. 3. A third misconception that is commonly made is people say 10.999... = 1/(infinity). This is produced again as a lack of mathematical understanding of the nature of infinity. "1/(infinity)=0" (by applying limit theory). In fact any noninfinite number when divided by infinity will be equal to zero. (Again I have taken a mathematical shortcut here, and infinity is more of a concept than a number, so we can't actually perform a simple operation like the one I just did, and instead you would need to apply limit theory to produce an answer, but based on your argument so far I don't think you would have covered mathematical limit theory yet, so I've used this shortcut to make it simpler for you. Although it's really interesting and I recommend you take a look at it on Khan Academy or something if you're not planning on studying it at university.). Here's the link to Khan Academy's series on limits and continuity if you'd like to take a look: https://www.khanacademy.org... I hope this has cleared things up for you. 

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Hope this clears things up for you.
You can't just "add an infinite string of nines to the left" it's not mathematically valid. Any such number with an infinite string of nonzero digits "to the left" is just represented as infinity. Your maths doesn't hold up because you fail to realise that infinity is a concept, not a number, so the algebraic techniques you're attempting to apply to it don't work. I'll wager the person who made that video actually doesn't have a particularly good understanding of mathematics because what he's doing is wrong.
Both examples are exactly the same as both are just equal to infinity. In neither case is your algebra consistent because in neither case are you correctly applying algebraic principles to actual 'numbers', you can consider it like trying to multiply "limit" by 10, this is obviously nonsensical because you cannot multiply a concept by 10.
Next I'd like to address what Masterful had to say:
0.999... isn't infinity? It does not behave like infinity and is a real number. I can say 0.999... is less than 2 and more than 0.9. I can use it in equations, I can apply a variety of completely valid operations to it precisely because it is a real number. I'm not entirely sure what your argument is and it seems largely nonsensical to me. First of all "this truth" does not create horrors for those who believe 0.999...=1, quite conversely it is essential for proving it, and I'd suggest entirely contradictory to your statement, the vast majority of people who recognise 0.999...=1 will do so on the basis of a firm understanding of mathematics, including this concept of infinity. I think you misunderstand what a real number is? Real numbers include basically any number that isn't the square root of a negative (the imaginary numbers). So the irrational numbers (like pi), integers etc. are all considered real numbers.
Hope this has cleared things up, but I think the best thing for you guys is to at limi
x=........99999999.9999999..........
Multiply both sides by ten we get
10x=........99999999.9999999......
We get exactly the same number as before because infinity goes on forever in both ways. This proves that if you multiply infinity by 10 it does nothing
10x=x
9x=0
x=0
The funny thing is it is actually consistent with the other equations because the algebra is telling me .......99999999 is 1 and 0.999999..... is 1 and .......999999999.9999999....... is 0 so 1+1=0.
x=..........9999999999
Multiply both sides by 10
10x=........99999999990
But look at ....9999990. It is exactly the same as the original x except minus the final 9.
10x=x9
9x=9
x=1
It is exactly the same thing you did with your equation.