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# Is 1/3 = 1

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lannan13
 Voting Style: Open Point System: 7 Point Started: 4/24/2014 Category: Education Updated: 7 years ago Status: Post Voting Period Viewed: 1,121 times Debate No: 53256
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 Suppose this was not the case, i.e. 0.9999... != 1. Then 0.9999... < 1 (I hope we agree on that). But between two distinct real numbers, there's always another one (say x) in between, hence 0.9999... < x < 1. The decimal representation of x must have a digit somewhere that is not 9 (otherwise x == 0.9999...). But that means it's actually smaller " x < 0.9999..., contradicting the definition of x. Thus, the assumption that there's a number between 0.9999... and 1 is false, hence they're equal.Report this Argument Contention 1: 1/3<1Let's do some comparisons. 1/3 as a fraction, is equal to .333 repeating. Assuming that we are only using real number and not the imgainary number i. 1 is indeed =1. Let's apply this to real life. I have 1 pizza and I have to split it between my two friends and I. I would only get 1/3 of the pizza and not the entire pizza. My friends would 2/3 of the pizza which is .667. The two would =1. The reason for this is that we have to consider the appropriate number of significant figures. (http://www.chem.sc.edu...) You see at the end of the allotted significant figures one must round up the last digit. so indeed .9999 repeating would =1 due to the sig fig rule, but the resolution is weather or not .333=1. I have shown through the significant figures rule that indeed it would not. as .333 would not be able to round up to even .334, as 3<5 so .333 would remain as .333 and as a fraction .333 would be 1/3 of the whole number of 1. Report this Argument chilli0919 forfeited this round. All points extended.Report this Argument chilli0919 forfeited this round. All points extended. Please vote Con.Report this Argument 10 comments have been posted on this debate. Showing 1 through 10 records.
Posted by PeacefulChaos 7 years ago
Because while it is plausible to form an argument for the claim that 0.999 ... = 1 because there is no number "x" between the two, you can't use the same argument for the fact that 0.333 = 1, since there are numerous numbers "x" between the two.

E.g. 0.4, 0.45, 0.5, etc.
Posted by PeacefulChaos 7 years ago
I'm confused

I can understand both sides of the argument of how 0.999 ... = 1, but Pro is arguing for 1/3 = 1, or that 0.333 ... = 1.

There are an infinite amount of numbers between 1/3 and 1.
Posted by creedhunt 7 years ago
That isn't the topic of this debate though, do it's irrelevant
Posted by creedhunt 7 years ago
The difference is currently debated amongst mathematicians, but they are both potentially valid. It all depends on infinitesimal theories. https://www.google.ca...
Posted by The_Scapegoat_bleats 7 years ago
Not true.
1/9 = 0.111...

0.111... x 9 = 0.999...

1/9 X 9 = 1

Hence, 0.999... = 1

No number inbetween. Identical numbers.
Posted by creedhunt 7 years ago
Actually, the number in between 0.999... and 1 is just infinitely small. That means there is no digit for it, as it is the opposite of an irrational. 0.000...1.
Posted by The_Scapegoat_bleats 7 years ago
http://en.wikipedia.org......

0.999...=1
It's a mathematical fact, and you've proved it with your final statement. What are you debating?
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