Zero point nine repeating is equal to one
Resolution: Zero point nine repeating (0.9r) is equal to one
Burden of Proof: Burden of proof is on Pro. Pro must prove that 0.9r = 1 and Con must disprove Pro's arguments.
Voting: This is a maths debate and the resolution is true or false; the "Select Winner" point system will be used on this debate.
Additional notes: The real number system will be used in this debate. The first round is for acceptance.
A note on the real number system
The real numbers are an ordered field which satisfies the completeness axiom (the least upper bound property). This means that the real numbers are complete, algebraic operations like addition and multiplication are defined and behave normally, and for any real numbers X and Y exactly one of the following holds: X > Y , X = Y , or X < Y. And if X > Y, then X + Z > Y + Z. 
Hence, if 0.9r ≠ 1, then |1 - 0.9r| > 0. I will argue that there are no non-zero infinitesimals in the real number system and therefor 1 - 0.9r must equal 0 which implies that 0.9r = 1.
The difference between 1 and 0.9r is 1/∞
Consider the following summation:
The real numbers do not contain non-zero infinitesimals; therefor 1/∞ must be 0
Recall that the real numbers must satisfy the least upper bound property. This means that for any set of the real numbers [X, Y] there is a smallest number which is greater than or equal to Y called the supremum. For example, consider the set of real numbers between and including 0 and 3: [0, 3]. There are many numbers which are upper bounds to this set (which are greater than or equal to the largest number in the set); 4, 5, and 300 are all upper bounds. The supremum to this set (the number which is greater than or equal to all numbers in the set and less than or equal to all upper bounds to this set) is 3. 
Let us assert that the real numbers contain the set of completed rationals and the set of infinitesimals. An infinitesimal is defined as a number δ such that |δ| < 1/m for all natural numbers m = 1, 2, 3, ... , n. The set of completed rationals are the set rational numbers combined with the set of irrational numbers. 
If the real numbers contain non-zero infinitesimals, then there must be a supremum for the set of infinitesimals [-δ, δ], and since some infinitesimals are positive the least upper bound to the infinitesimals must be positive. Either the supremum of the infinitesimals is a member of the positive infinitesimals, or it is a member of the completed rationals.
r is a positive member of the completed rationals, hence r is an upper bound to the set of infinitesimals. If r is the supremum to the set of infinitesimals, then it must be less than or equal to all other upper bounds. However, r/2 is also a positive member of the completed rationals so r/2 is also an upper bound to the infinitesimals. r is not greater or equal to than r/2, therefor there is no member of the completed rationals which is the least upper bound to the set of infinitesimals.
δ is a member of the positive infinitesimals. If δ is an upper bound to the set of infinitesimals, then all other infinitesimals must be less than or equal to δ. Since δ is less than all numbers 1, 1/2, 1/3, …, 2δ is also a positive infinitesimal. 2δ is not less than or equal to δ, thus there is no member of the infinitesimals which is an upper bound to the set of infinitesimals.
Consequently, the inclusion of non-zero infinitesimals in the real numbers violates the completeness axiom. The completeness of the real numbers is crucial to mathematical analysis, enabling us to take limits and do calculus. The real numbers cannot contain non-zero infinitesimals; therefor 1/∞ must be 0.
Hence, the difference between 1 and 0.9r is 0; 0.9r is equal to 1
If 1 > 0.9r, then 1 + Z > 0.9r + Z. If Z = -0.9r, then 1 - 0.9r must be greater than 0. As established above, this is false; 1 - 0.9r = 0, therefor 1 = 0.9r. The resolution is affirmed.
My opponent must prove either that the value of the difference 1 - 0.9r is a real number greater than 0, or that the real numbers contain non-zero infinitesimals.
Thanks to my opponent for inviting me to debate this topic.
Essentially my opponent has conceded the debate with the following statement “ will argue that there are no non-zero infinitesimals in the real number system and therefor 1 - 0.9r must equal 0 which implies that 0.9r = 1.” If the real number system does not contain any infinite numbers then the resolution is not upheld. To elaborate, if 0.9r is not an infinite number then the number 0.9r cannot be used in any of the reasons my opponent has given in their opening round argument.
My opponent rightly asserts that the real numbers must satisfy the least upper bound property, however 0.9r is an infinite number. To clarify and infinite number is defined as follows.
Infinite number: Is a number that displays infinity.
Infinity: is an unbound quantitiy that is greater than any real number.(1,2)
So this means this number is unbound and hence not a real number. This means it does not have a “least upper bound”, and so my opponents arguments from round 2 are invalid.
This means that 1-0.9r is not equal to 0.
Now to give some more information about infinite numbers, I do this to make the debate easier to follow as the infinite concept leads to conclusions that sometimes define reality in the same way that conclusions from quanumn mechanics defy logic.(3) I think the easiest way to do this is to follow some basic mathematicalm formulas.
1 + 1 = 2 and
1 + 0.9r = 1.9r, however if we assert that 1 = 0.9r then it holds that
1 + 0.9r = 2. This is clearly illogical!
1 - 1 = 0 and
1 - 0.9r = ?, where ? Is some number that is infinite in nature which shows that infinite numbers defy logical thought. However if we assert that 1 = 0.9r then it holds that
1 - 0.9r = 0, which defys logic in itself.
1 X 1 = 1 and
1 X 0.9r = 0.9r however if we assert that 1 = 0.9r then it holds that
1 X 0.9r = 1. This is clearly illogical!
1 / 1 = 1 and
1 / 0.9r = ?, where ? Is some number that is infinite in nature which shows that infinite numbers defy logical thought. However if we asert that 1 = 0.9r then it holds that
1 / 0.9r = 1, which defys logic in itself.
Non-terminating decimals are real numbers
My opponent claims that 0.9r is an infinite number, referencing two webpages on infinity and one webpage on Schrödinger’s cat. None of these webpages make any comment on non-terminating decimals nor support my opponent’s claim that 0.9r is unbounded or infinite, so I’m unsure why they were cited. My opponent’s source, Mathworld, does have a relevant page on repeating decimals; however it does not support my opponent’s claim that non-terminating decimals such as 0.9r are not real numbers.
Mathworld states that “All rational numbers have either finite decimal expansions (e.g., 1/2 = 0.5) or repeating decimals (e.g., 1/11 = 0.0(9)r).” Mathworld further notes that“Numbers such as 0.5 are sometimes regarded as repeating decimals since 0.5 = 0.5(0)r = 0.4(9)r.” Non-terminating, repeating decimals are rational numbers. 
The mathematics used to construct the real numbers by completing the rational numbers with the irrational numbers works because non-terminating decimals are real numbers. If non-terminating or infinite decimals weren’t real numbers as my opponent claims, then no irrational numbers could be real numbers since all irrational numbers are non-terminating, non-repeating decimals. This would be problematic, because then the real numbers would be incomplete. Hence, non-terminating decimals must be real numbers.
0.9r satisfies the least upper bound property
The same mathematics used to construct the real numbers works because non-terminating decimals satisfy the least upper bound principle. What is the least upper bound of the set of all numbers less than one? Simple: it’s 0.9r or 1.0, both of which are two different decimal representations of the same number. “The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.” 
My opponent has confused infinity with non-terminating decimals. Infinity does not have an upper bound, but it also is not a real number.
Summary of arguments
My argument is simple:
P1: The difference between 1 and 0.9r is an infinitesimal.
My opponent has contested P1, arguing that 0.9r is not a real number because it is a non-terminating decimal; therefor the difference between 1 and 0.9r does not exist or is not a real number. This is false; non-terminating decimals are real numbers – my argument stands. My opponent must prove either that the value of the difference 1 - 0.9r is a real number greater than 0, or that the real numbers contain non-zero infinitesimals. Simply asserting that basic arithmetic operations are “clearly” illogical or “defy logic in itself” does not do this.
The resolution is affirmed, vote Pro.
Firstly, I just want to say thanks to Enji for a fun debate. Now onto my final rebuttals.
To clarify the sources I cited in round 2 (1,2) had to do with infinity and not with real numbers. I cited them to show the illogical nature of infinity. The citation (3) dealt with Schrödingers cat was to show that like in quantum mechanics when we deal with something abstract (like infinity in math) we can reach bizarre conclusions.
My opponent has said the following in summary in their round 3 argument,
P1: The difference between 1 and 0.9r is an infinitesimal.
This argument effectively concedes the debate again. Let me clarify how.
If we accept P1. Then we know that the difference between 1 and 0.9r is infinite. This means it is not 0 which is what my opponent is asserting in this debate.
So while my opponent is correct to say “There are no non-zero infinitesimals in the Real number system in P2” this does not mean we can make an infinite number (which was what P1 says) a real number to win a debate.
Thus I have show both P1 and P2 to be blatantly false for the proposition of the debate. As such the conclusion that “0.9r is equal to 1”cannot hold.
Additionally, my opponent has said this debate does not have to do with infinity. Unfortunately, it does have to do with infinity. A recurring number is number that repeats forever.(4) Forever is another term used for infinity. As such, a recurring number can be said to be repeating for infinity. This means my conclusions from round 2 hold.
Using both my arguments and proving my opponents arguments faulty, I believe I have successfully shown that 0.9(r) is not equal to 1.
Remember if 0.9(r) is equal to 1 then the following MUST hold
1 + 0.9(r) = 2.
I now hand the debate over to the voters.
|Who won the debate:||-|
|Who won the debate:||-|
|Who won the debate:||-|