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

.999...(repeating) =1

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Voting Style: Open Point System: 7 Point
Started: 8/16/2017 Category: Science
Updated: 3 years ago Status: Debating Period
Viewed: 1,652 times Debate No: 103562
Debate Rounds (4)
Comments (32)
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This is a debate on whether or not .999...(repeating)=1.

First round is acceptance.

I'm looking forward/curious to see any opposing arguments on how .999...cannot equal 1.

No semantics, no trolling.



no, it never reaches 1
Debate Round No. 1



In order to prove that 0.999...=1, I will be showing this via 3 different mathematical methods. While the numerical portion of this argument is the main argument, I would briefly try to explain using words how exactly .999...=1. 0.999 repeating is a number that can only exist within the realm of mathematics. 0.999 repeating is a number with infinite accuracy, so it's not something that can possibly exist in real life. However, using mathematics, a number that has an infinite number of "9s" after the decimal place would evaluate to 1. This is simply because the limit of 0.999 repeating written as an infinite geometric series would evaulate to 1, something I will show later in this argument. 0.99... may not seem like it ever reaches 1, but if we evaluate the limit of a geometric series that represents 0.999 with an infinite number of 9s, it would indeed be calculated as 1.

C.1 Fractions

This is the simplest argument to show that 0.999...=1.

Adding 1/3=Adding 0.333...
Adding 1/3=Adding 0.333...

And as it turns out, 3/3 which simplifies to 1, does indeed equal 0.999....

C.2 Algebra

10x=9.999....(Multiply by 10 to both sides)
10x=9+0.999....(We broke up 9+0.9999)
10x=9+x ( equal to x, as seen in step 1)
9x=9 (Subtract both sides by x)
x=1 (Divide both sides by 9)

C.3 Limits and geometric series

Let's try writing a fraction, and see what we get. Since this number isn't irrational, it can be written as a fraction. In order to do that, we would have to rewrite this number as an infinite geometric series, and then find the sum of that series. The limit of that series as the number of terms tend to infinity, would be 1. While the limit isn't telling you the value of the function at that number, it is telling you what that function is approaching as you get closer and closer to some number (in this case, it's infinity). Since a geometric series has an infinite number of terms, the limit would tell us what the value if 0.999...had an infinite number of "9s". Which it does.

Sum of finite geometric series:



Sn=Sum of n terms
a1=First number in series
n=Number of terms
r=common ratio

If con requests, I can derive this formula.

However, if we assume that r is greater than 0 and less than 1, we can take the limit of the function as n tends to infinity. Therefore, the sum of an infinite number of terms would be...

Sinf.= a1/(1-r).

(r^n=0, as n tends to infinity. Therefore, we're left with a1/(1-r))

If con requests, I can do a more rigourous proof, though this is the formula for the sum of an infinite geometric series.

So, we can now rewrite 0.9999... as...


Which is equal to


In this series,


Since |r|<1, we can use the formula of Sinf.


And that's it. Using the definition of the sum of an infinite geometric series, we figured out that a fraction does indeed evaluate to be 1. Since we can rewrite 0.999 as an infinite geometric series, we can evalaute it's sum in order to figure out it's fractional form. And that form turns out to be 1/1.

This debate doesn't really need sources, since it's just math. However, if Con likes, I can show sources of the formula for the sum of a finite and infinite geometric series.


but it dosnt equal true, if thats what you are saying, it never gets to equal as something true like 1 soda, what is a 0,9999 soda?

dividing it into 3 parts and saying those 3 which neither equates to 1 is 1 all togeather, as if you have changed or added something... then you are simply wrong

its like all you are arguing is that 3/3 is a FULL fraction.. that dosnt make it valid, you cant just change around the value of numbers, they have actual usage.. what does, 0,3 point to in real life? a stone? which is one?


your faker 3(0,33..+0,33..+0,33..)=not 3

i assume the rest of it is non sense as well
Debate Round No. 2



"it never gets to equal as something true like 1 soda, what is a 0,9999 soda?"

Con's example fails here since Con is trying to use a real world example/object to describe a concept such as infinity. It literally cannot be done since we can never have infinite accuracy and everything in our universe is bounded by an finite number of "stuff". If you could theoretically measure 0.999... of a soda, then you would have 1 full soda. The reasoning behind this is something I have already posted, something that Con fails to adress.

"its like all you are arguing is that 3/3 is a FULL fraction.. that dosnt make it valid, you cant just change around the value of numbers, they have actual usage"

Again, this seems to be an argument from incredulity. Just because 0.999... doesn't look like 1, doesn't mean it's not 1. 0.999... and 1 have the same exact same value, just like how 0.5 and 1/2 have the exact same value. 3/3 and 0.999... are equivalent statements.

It's also worth mentioning to point out that a decimal number, it's a real number. A number that gets infinitesimally close to 1, but not 1, is also a number, but it's not 0.999... A number that would somehow get infinitesimally close to 1, would not be real number, whereas

Con seems to be getting confused between this statement

lim(x->1+) f(x)=x

(where a number get's infinitesimally close to 1)

and this statement

lim(n->inf.) 9/10((1-(1/10)^n))/(1-9/10)

(where a repeating decimal number IS 1)

"your faker 3(0,33..+0,33..+0,33..)=not 3"

Again, Con states that this cannot be 3, yet provides no mathematical reasoning or evidence.

"i assume the rest of it is non sense as well"



0,0000000000.. infnitiy

you are adding things unnecessarily

a soda is 1, not 0,999

0,999 is not 0,999? ...
Debate Round No. 3
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Debate Round No. 4
32 comments have been posted on this debate. Showing 31 through 32 records.
Posted by Anonymous 3 years ago
It can be better explained like this,

If you have 100 and you're trying to reach 200, but every time you add to that 100 you reduce the amount you add by one tenth it will look like this.

We need 900% more of what we've just added to reach 200.

We need roughly 8999% more of what we've just added to reach 200.

111+0.1= 111.1
We need roughly 89998% more of what we've just added to reach 200.

The point is, as you reduce the amount added by a factor of ten, you increase the percentile amount needed by a factor of ten, thus you can never truly reach the number 200. My maths might be off slightly, but the point remains.
Posted by Anonymous 3 years ago
All you're doing is adding a 10th smaller each time.

Imagine it like this using 10's just to make it easier,


If you keep going you can only add more 1's, but you can never reach a 2 in any of these decimals, the pattern so far is consistent, what reason do we have to ever assume we will see a 2?

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