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# 0.9999 (repeating) = 1

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 Voting Style: Open Point System: 7 Point Started: 10/23/2014 Category: Science Updated: 2 years ago Status: Post Voting Period Viewed: 1,569 times Debate No: 63811
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23 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Jzyehoshua 2 years ago
Again, the issue is that what we are debating is the hypothetical not real numbers, the number 0.999... does not exist in reality, you cannot write it out, so there is no way to write out a difference between it and something else. What is being debated is a concept, not an actual number. The hypothetical number is as Freedom just acknowledged, ".9999 repeating always approaches 1 but never reaches it." If it never reaches it then by definition it is not the same thing as 1. The infinitely repeating numbers are theoretical in nature because the infinite is by definition something we have difficulty measuring or addressing in reality. Thus one must get into hyperreal numbers.
Posted by FreedomBeforeEquality 2 years ago
... forgive me i mispoke ... In my third sentence i said it is a real "number". I meant to say it is real ... but it is, as i said further down, a function when he puts on the term "repeating".
Posted by FreedomBeforeEquality 2 years ago
I see where youre going with the literal side of things. Its not necessary though. It is mathematically impossible for those to be equal also. His rendition of .999 out to infinity is wrong in decimal form. The number he is describing with ".9999 repeating" is a real number. Its a fraction. You have to use the mechanics of fractions coupled with the function for infinite decimal places to disprove what he is saying. .9999 repeating always approaches 1 but never reaches it. .9999 repeating is a function ... not a number at all. It certainly can never equal a single number because then it ceases to be repeating!

Please challenge me on this. Ill start drawing up a graph for you and everything ...

It truly scares me that such a lie could go on for so long. Every argument like this that ive looked up on here went to the Pro that 1 = .999 repeating. How can people be so blind? This trend really is making a statement.
Posted by TheTom 2 years ago
An interesting point nomagic. 0.999... is actually a real number. Just because a number goes on forever doesn't mean it is not real. Pi is an infinite sequence, however it is a real number. 0.3.... or 0.8... or the square root of 2 (which is a never ending sequence) are all real numbers. The idea is numbers that break math cannot be considered a real number. An example of imaginary or non real numbers would be the square root of -1.

Hence, you can subtract 0.999... from 1 because they are both real workable numbers.

My opponents contention was that there is an infinitely small number between 0.999... and 1. The problem here is that infinitely small numbers break the laws of regular math. For instance, what is 5 divided by an infinitely small number? is it infinity? What is infinity multiplied by an infinitely small number? is it 1? In other words, infinitely small numbers (infinitesimals) aren't real numbers. However infinite sequences (such as 0.999 repeating) are real numbers.
Posted by NoMagic 2 years ago
I've done very little reading or thinking about this subject. However, I have a question regarding the debate. Pro asserts that 1 is the same as .99999... Pro asserts that .9999... is equal to 1. .9999... is 1. Would it be fair to say that .9999... doesn't actually exist if it is actually 1? If .9999... isn't a real number, can Pro then claim 1 subtracted by a non real number equals zero? Can you subtract a non real number from a real number and get a real solution? Should Pro have argued that .9999... isn't a real number instead of arguing it is one?
Posted by TheTom 2 years ago
Thank you Josh, I would be interested in debating you again sometime.

The reason infinitely small numbers cannot be used in regular mathematics is because it literally breaks ordinary math. 3 / (infinitely small number) = infinity? Infinity + 3 = ?
However 0.999... doesn't have that property. 0.999... + 3 = 3.999... = 4
You can input 0.999... into an equation without employing advanced calculus and pseudo-real math.

You were brave to take on this debate by the way, it is not an easy subject to argue.
Posted by Jzyehoshua 2 years ago
By the way, great debate topic choice TheTom! It is very rare I see a debate that makes me think this much, and challenges my thought processes so much. This is one of the most interesting and fun debates I've had in a while. You really made me think through the concepts involved. I still agree with my original position but you definitely made me consider everything carefully.

Actually, I don't think 0.333 can perfectly equal 1/3 either. It's actually impossible to perfectly convert from fractions to the decimal system to whole numbers all the time, but for purposes of our imperfect mathematical systems we simply go with that structure. It's kind of a consequence of having multiple different formats like fractions and decimals that don't always perfectly mesh.
Posted by Jzyehoshua 2 years ago
Well, largest percentage short of 100% of course. It seems like there is new impetus to declare this a rule of mathematics per Common Core from what I am seeing, and like many other aspects of Common Core it defies the laws of logic and reason.
Posted by Jzyehoshua 2 years ago
I think 0.999... is an infinitely large number if using the percentage system, because it essentially equates to 99.99%. It represents the largest possible percentage imaginable and thus violates the Archimedean Principle and cannot be treated with standard arithmetic.
Posted by Jzyehoshua 2 years ago
I just don't see how you can say an infinitely small decimal can't exist (e.g. 0.000...1) without likewise denying that infinitely large decimals (0.999...) can't exist as well. If one is impossible than so is the other. Infinite repeating exists only in theory, not physical reality. If infinitely small numbers cannot exist, then infinitely large ones can't as well, per the Archimedean Property. The entire discussion is only so confusing because a hypothetical, infinitely large number is being introduced in the first place.
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