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The Contender
Pro (for)

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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pro starts


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...
Debate Round No. 1


Thank you pro for accepting this debate. Let's jump right into it.

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:

We multiply both sides by ten and get
We then subtract x from both sides
This is because we subtract the original x value which is 0.999.... from both sides and the result is

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:

we multiply both sides by 10 and we get
We subtract the original x value from both sides and get
We divide both sides by 9
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:

We multiply both sides by 10
We subtract the original x value from both sides and get
......99999.99999.... is the original x value the equation is now
We simplify and get
We divide both sides by 9 and get
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:

Multiply both sides by ten:
We subtract the original x value from both sides and get:

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).

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 1-0.999... 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 number-line. 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 number-line, that could be occupied by 0.999... without being mathematically incorrect.

3. A third misconception that is commonly made is people say 1-0.999... = 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 non-infinite 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:

I hope this has cleared things up for you.
Debate Round No. 2
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Debate Round No. 4
20 comments have been posted on this debate. Showing 1 through 10 records.
Posted by A1DS 3 years ago
No it's not because 0.999... isn't infinity, it's a real number (see previous comment). The key distinction is the infinite repeating digits are after the decimal place instead of before it, and therefore 0.999... doesn't fit the definition of infinity as I can list a number (e.g. 2) which is larger than 0.999... so it's an entirely different scenario. In the case of infinity, it is impossible to list a number of greater value, so clearly 0.999... represents a different scenario.
Posted by DeletedUser 3 years ago
But the thing is your equation was doing the exact same thing
Posted by A1DS 3 years ago
Which, by definition makes it infinity, so you cannot manipulate it with the algebra you have been trying to use. You'll never see a proper, valid mathematical proof using ...X.Y because it's not essential, any such "number" (and I use that term loosely because infinity is a more of a concept than a number) can be represented as infinity. Your algebra isn't actually mathematically valid.
Posted by DeletedUser 3 years ago
I didn't mean "add an infinite string of 9's to it". I meant right out the rest of the number which happens in this case to be an infinite string of 9's.
Posted by A1DS 3 years ago
It may help you to look into what constitutes, real, rational and irrational numbers. Rational numbers is the set of numbers that can be represented by an integer fraction. 0.999... falls into this category as it can be represented by 9/9 (which is equal to 1 funnily enough). Irrational numbers cannot be represented this way, and include examples such as pi and the square root of 2. Both of these can include examples of numbers with an infinite number of decimals places. Such as 0.333... and 0.999... from the rational numbers and both of the aforementioned examples for the irrationals. Together, the rational and irrational numbers form a set that we refer to as the real numbers. You can google the mathematical axioms on which the reals are defined if you'd like to learn more, but they get quite complicated quite quickly.
Posted by A1DS 3 years ago
Reflectively, I think where your problem is actually coming from Masterful is you don't know what a real number is. Something with an infinite number of decimals is still a real number. Pi is a real number, the square root of 2 is a real number, just because decimals go on infinitely doesn't mean these numbers behave like infinity. So while you've correctly identified that infinity is a concept rather than a number, you're incorrectly applying that same assumption to real numbers.

Hope this clears things up for you.
Posted by A1DS 3 years ago
Limit theory* sorry I ran out of characters.
Posted by A1DS 3 years ago
Okay so first I'm going to address what jackgilbert had to say:

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 non-zero 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
Posted by DeletedUser 3 years ago
What about this one.
Multiply both sides by ten we get
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
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.
Posted by DeletedUser 3 years ago
the original x is infinitely many 9's going off to the left. Multiplying it by ten takes the last 9 away and replaces it with 0. Infinity times 10 is still infinity. Let me right out an equation for you

Multiply both sides by 10
But look at ....9999990. It is exactly the same as the original x except minus the final 9.
It is exactly the same thing you did with your equation.
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