The Instigator
The_Tom
Pro (for)
Winning
10 Points
The Contender
Ryvn
Con (against)
Losing
0 Points

0.999..(repeating) is equal to 1

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Post Voting Period
The voting period for this debate has ended.
after 2 votes the winner is...
The_Tom
Voting Style: Open Point System: 7 Point
Started: 12/12/2013 Category: Science
Updated: 3 years ago Status: Post Voting Period
Viewed: 799 times Debate No: 42200
Debate Rounds (3)
Comments (1)
Votes (2)

 

The_Tom

Pro

I am arguing that 0.9999... (going on forever) is equal to 1. My challenger is arguing that this is false. He may use the first round to make his argument.
Ryvn

Con

.9999999999999999999999999999999999999999 is not equal to .9999999999999999999999999999999999999998, by reason that two values, structures, objects, concepts, or any measurably comparable entity, however similar in structure they may be, must retain ALL characteristics of one another to be 2 of the same entity.

Therefore, 0.9999(repeating) is NOT EQUAL to 1.
Debate Round No. 1
The_Tom

Pro

0.9999... = 1
(Note ... means repeating)
This is a mathematically proven fact which most people find hard to believe. I will try an convince the readers that this is indeed a fact.

My opponent claims "two values, structures, objects, concepts, or any measurably comparable entity, however similar in structure they may be, must retain ALL characteristics of one another to be 2 of the same entity."

There are many ways to write the number 1: (4/2), (5 - 4), (0.25 x 4), these don't have all the same characteristics as each other, but they are all ways of writing the number 1

Here I will attempt to prove that 0.9... is another way of writing 1

Logical Proof:
If there is no difference between two numbers then they are logically the same number
So, 1 - 0.999999.... = 0.000000...
Hence, they are the same number. Please note that because there are an infinite number of 9's, that there will be an infinite number of 0's

Another way of saying this is that if you cannot fit a number between two other numbers, there is no difference between them.

Proof using arithmetic:
1/3 = 0.333333...
0.333333... x 3 = 0.99999...
And (1/3) x 3 = 1
Therefore 0.99999... = 1

Algebraic proof:
Z = 0.999999...
(multiply by 10)
10Z = 9.99999...
(subtract Z from both sides)
10Z - 1Z = 9.99999... - 0.99999..
(reduce)
9Z = 9
(divide by 9)
Z = 1
Here Z has started as 0.999 repeating, and has ended up as 1
Therefore 0.99999... = 1

I am sure that my opponent will argue that there will always be an infinitely small difference between .999.. and 1. I will refute that argument here:
The rules of infinity:
It is an accepted mathematical fact that an infinitely small number is equal to zero (1). Another way of writing this is
(1/infinity = 0)
Therefore, THERE IS NO GAP BETWEEN 0.999... AND 1

This seems counter-intuitive, but it has been proven true (2)(3). For other mathematical proofs and in depth explanations why, follow my references, or do a quick google search for "0.9 repeating = 1".

References:
1. http://www.wolframalpha.com...(x)%3D+1/x+as+x+approaches+positive+infinity
2. https://www.khanacademy.org...
3. http://polymathematics.typepad.com...
Ryvn

Con

Ryvn forfeited this round.
Debate Round No. 2
The_Tom

Pro

Well, since my opponent didn't refute me, I will refute myself.
Dear the_tom, in the previous round you had stated that 4/2 is another way of writing 1. Well let me just say sir, this is blatantly incorrect! Not only this, but the number one looks bigger than 0.999.. therefore your proof is null!

Seriously though, arguing with yourself is not fun, so I leave the floor to my opponent.
Ryvn

Con

Ryvn forfeited this round.
Debate Round No. 3
1 comment has been posted on this debate.
Posted by idontexist 3 years ago
idontexist
Wikipedia has very detailed information about this very topic. I doubt you can do better with just five rounds of debate.

Reference: http://en.wikipedia.org......
2 votes have been placed for this debate. Showing 1 through 2 records.
Vote Placed by Ore_Ele 3 years ago
Ore_Ele
The_TomRyvnTied
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Total points awarded:40 
Reasons for voting decision: Conduct for the forfeit. Pro provided a number of arguments (the only true proof being the algebraic) of which Con did not respond.
Vote Placed by ndedo 3 years ago
ndedo
The_TomRyvnTied
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Total points awarded:60 
Reasons for voting decision: Con ff and Pro made great arguments. Interesting debate.