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# Zero point nine repeating is equal to one

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iamanatheistandthisiswhy
 Voting Style: Open Point System: Select Winner Started: 5/15/2014 Category: Science Updated: 3 years ago Status: Post Voting Period Viewed: 1,999 times Debate No: 54794
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25 comments have been posted on this debate. Showing 1 through 10 records.
Posted by FuzzyCatPotato 3 years ago
I'm with Enji on this one.
Posted by Enji 3 years ago
But there's no problem with recurring decimals; they're real numbers and we can prove it!
Posted by iamanatheistandthisiswhy 3 years ago
@Enji: Yes I am arguing exactly what you are saying. I would disagree that my reason is arbitrary however. I am pointing out a problem with recurring numbers that is very real and very relevant to mathematics and how people use numbers without understanding what the implications are.

Its like people like saying the cat is not dead or undead, not realizing that then defies the uncertainty principle. There are implications to actions and they need to follow logic and reason.
Posted by Enji 3 years ago
Any number which cannot be represented as a fraction with a denominator which is a product of 1's, 2's, and 5's necessarily has a non-terminating decimal representation. n/3, n/6, n/7, n/9/, n/11, n/12, n/13, n/14, n/15, n/17 etc. all have infinite decimal representations. Any terminating decimal can be represented as a fraction which is a multiple of 1's, 2's, and 5's. This is a simple feature of base 10 mathematics, and it's easy to see why -- 1, 5, and 2 are the prime factors of 10.

You're arguing that fundamental axioms of the real number system simply don't hold for a vast portion of the real numbers without any reason (or for the arbitrary reason that you don't like numbers with non-terminating decimal representations).

Non-standard analysis and the hyperreals make use of non-zero infinitesimals, but if you're not a career mathematician you're unlikely to ever encounter them much less make use of them. This is why I framed the debate to pertain to the real number system which is regularly used in science, engineering, school, and day to day life. The real numbers don't contain non-zero infinitesimals because their inclusion contradicts the assumptions made to construct the reals mathematically, and these assumptions are what make the reals useful.
Posted by iamanatheistandthisiswhy 3 years ago
*such not suck

ROFL
Posted by iamanatheistandthisiswhy 3 years ago
@ Enji: What I am arguing is that if it is an infinite number and as suck all the reasoning/proofs fail.

If we accept however that it all the claims are true then it works, but then we are working on an assumption. "Although the existence of such numbers makes no sense in the real number system, many worthwhile results can be obtained by overlooking this obstacle."

Its an assumption. Thats why it does not work and cannot work.

Until we accept the assumption then it cannot work. I reject the assumption. As the assumption cannot be proven.
Posted by Enji 3 years ago
You're acknowledging the fact that there can't be an infinitesimal difference between 1 and 0.9r in the real numbers, but still arguing that they're unequal?
Posted by iamanatheistandthisiswhy 3 years ago
I think this explains it perfectly.

It comes form Mathwords.com

Infinitesimal: A hypothetical number that is larger than zero but smaller than any positive real number. Although the existence of such numbers makes no sense in the real number system, many worthwhile results can be obtained by overlooking this obstacle.

Note: Sometimes numbers that aren't really infinitesimals are called infinitesimals anyway. The word infinitesimal is occasionally used for tiny positive real numbers that are nearly equal to zero.
Posted by iamanatheistandthisiswhy 3 years ago
@imperfectPefection:

Proof 1: 0.1r = 1/9
Then 0.1r*9 = 1 (if we multiply both sides by 9) which is the assumption you are working with, that 0.9r = 1. This is circular reasoning.

Proof 2. Fails for the same reason. Its circular reasoning.

All my argument are that its an infinite number. So it is purely that it is an illogical argument unless you use circular reasoning.
Posted by ImperfectPerfection 3 years ago
Proof 1:
If we agree that 0.1r = 1/9
Then 0.1r*9 = 0.9r
1/9*9 = 1
(I have applied the same multiplication to both sides of the equation, balancing it)
Therefore, 0.9r = 1

Proof 2:
Let x = 0.9r
10x = 9.9r
10x - x = 9.9r - 0.9r
(Because x = 0.9r)
9x = 9
Therefore x = 1
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