The Instigator
Con (against)
Anonymous
The Contender
Topaet
Pro (for)

does 0.9999999.......... equal 1

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Voting Style: Open Point System: 7 Point
Started: 7/13/2018 Category: Philosophy
Updated: 3 years ago Status: Debating Period
Viewed: 704 times Debate No: 116545
Debate Rounds (4)
Comments (1)
Votes (0)

 

Con

for the first round please just accept. Arguments start next round.
Topaet

Pro

I accept
Debate Round No. 1

Con

The primary argument people use to prove that 0.9999... does not equal one is very simple. If you take 0.9 it is shy of one, if you take 0.99 it is still shy of one, if you take 0.999 it is still shy of one and so on. No matter how far you go down the line 0.999999.... will still be shy of one. It goes on for infinity but 0.9999999.... will never be one. It would be like this. If you walked 10 feet away from a wall and go half the distance to the wall each time you will never reach it. First you get within 5 feet of the wall, then 2.5 feet of the wall, then 1.25 feet of the wall, then 0.625 feet of the wall, and so on. Same thing with 9's. No matter how many 9's you go down the whole number 0 will never be one.
Topaet

Pro

My opponent seems to have misunderstood the concept of actual infinity and mistaken it for potential infinity. It is correct, that if there was a "0." followed by a limited amount of 9s, it would never equal 1. However, if there is an unlimited amount of 9s following the "0.", then it does equal 1. To prove this I will present a few mathematical proofs that 0.999... equals 1:

1. Reductio ad absurdum:
If 1 and 0.999... are not equal, then we should be able to find a distinct number in between them (their average). There is, however, no number in between 0.999... and 1. Therefore, this is a reductio ad absurdum argument that proves that 0.999... can not not equal 1 and thus 0.999... equals 1.
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2. The decimal expansion for 1/3:
1/3 = 0.333...
0.333... * 3= 1/3 * 3 = 0.999... = 3/3 = 1

Therefore 0.999... equals 1.
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3. Subtracting off the infinite sequence:
x = 0.999... |*10
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x | -x
9x = 9 |"9
x = 1

Therefore 0.999... equals 1.
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4. Arithmetic proof:
1-0.999... = 0.000...
Since any number subtracted from an equal value is zero, 0.999... equals 1.
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In conclusion, there are more proofs that 0.999... equals 1 but the four that I have presented should be sufficiently convincing and unless Con manages to refute them all I have affirmed the resolution that 0.999... does equal 1.

Sources of arguments:
Norton, Anderson & Baldwin, Michael. (2018). Does 0.999" Really Equal 1?.

Math Wiki. (2018). Proof:The Decimal 0.999... is Equivalent to 1. [online] Available at: http://math.wikia.com... [Accessed 15 Jul. 2018].
Debate Round No. 2

Con

my opponent has presented several of his proofs to try to show that 0.999... equals one. I will refute them in order.

1. You said that If 1 and 0.999... are not equal, then we should be able to find a distinct number in between them. There is a number between them but it is not a finite number. It is 0.9999.....1. We call it infinitesimally small.

2. Your second argument is also flawed. If you press 0.33333333 x 3 into googles calculator, you get 1 and not 0.999..
This is because it rounds up to 1 because 0.99999.... goes to infinity

3. Your 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
But look at that. ......99999990 is the original x value minus the final 9
x-9=10x
-9=9x
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 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=10x-x
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, you must choose to either believe all cases have a meaningful answer, like -1, 1, or 0, or there isn't a meaningful answer because we have the concept of infinity in each equation.

4. It is the same as 1.

Another thing, let me show you a fundamental flaw in your equations. You assume that infinity minus infinity equals 0. This is not true and I will use a simple equation to prove it. First let's assume you are correct. Infinity minus infinity equals 0.

Infinity-infinity=0
Lets add 25 to both sides. Infinity plus 25 equals infinity
infinity-infinity=25
0=25 obviously 0 does not equal 25, so the equation is incorrect.
This shows that infinity minus infinity does not equal 0.
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Debate Round No. 3
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Debate Round No. 4
1 comment has been posted on this debate.
Posted by MrMolter 3 years ago
MrMolter
Pro is absolutely right. Plus another mathematical solution :
1/9 = 0.111111
2/9 = 0.222222
3/9 = 0.333333
......
7/9 = 0.77777
8/9 = 0.88888
Therefore 9/9 = 0.99999 Means 1=0.9999999.
Also you said something about adding half and half and half, infact it has been proven mathematically that if you have a glass bottle filled halfway, if you keep filling the half of the empty volume you will get to a place where the bottle will be 100% full. Having a number between 0.99999999 and 1 is logical but is just an illusion.
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