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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: 1,347 times Debate No: 63811
Debate Rounds (3)

 Pro Normally debates are about opinions. This is about fact. I maintain that 0.9999 (going on forever) is the same thing mathematically, and realistically as 1. I expect my opponent to have an understanding of math, such as basic algebra, and rules of infinity. First round is for acceptance AND posting arguments. For the sake of simplicity, the term 0.999... will mean 0.999 repeating. FAIR WARNING: This is a mathematical fact. The evidence is on my side. If you choose to accept this argument, you will have to argue against something that is a fact not an opinion. In the sake of fairness, I will post my argument first so as not to blindside someone who did not realize that this has already been proved. My arguments: The philosophical argument: If there is no difference between two things, they are the same thing. This goes for numbers, objects, colors, or anything else in existence. That being said, if you subtract (finding the difference) 0.999... from 1 you will get 0.000...(repeating for infinity) In other words, there is no difference. Therefore 1 and 0.999.. are the same thing.. (1-0.999...) = 0.00000000... You will never reach the point 1 ( 0.000....0001) at the end of the sequence, because the sequence has no end. IN ORDER FOR TWO NUMBERS TO BE DIFFERENT, THERE HAS TO BE A NUMBER BETWEEN THEM. If 0.999... and 1 where different numbers then there would exist a number between them. No such number exists. This is because you cannot put a digit at the end of an infinite sequence. The fractional argument: If 1/3 is equal to 0.333... (repeating) than 1 = 0.999... Proof: (1/3) + (1/3) + (1/3) = 1 (0.333...) + (0.333...) + (0.333..). = 0.999... Therefore 0.999... = 1 The Algebraic argument: assume that x = 0.999... (multiply both sides by 10) then 10x = 9.999... (subtract x from both sides) 10x - x = 9.999... - 0.999... 9x = 9 x=1 Notice how we started out with x = 0.999 and ended with x = 1, this is only possible if 0.999... = 1 http://www.mathsisfun.com... http://polymathematics.typepad.com... http://www.newton.dep.anl.gov... http://homepages.warwick.ac.uk...Report this Argument Con I'll accept this debate and argue, not so much from a mathematical standpoint as from a logical standpoint, because I disagree strongly with your claim that two clearly different numbers are "the same thing mathematically and realistically." For all intents and purposes they might be in practicality, but ultimately it comes down to the fact that they are different, regardless of the amount by which they differ. 0.9999, regardless of the 9s, involved, will never be 1, they are separate numbers. We may round up to 1, but our practices do not define reality. We may do this in speech even for far lesser distinctions simply for the purpose of simplicity, but that does not mean they are the same. They are not. OPPONENT ARGUMENTS"That being said, if you subtract (finding the difference) 0.999... from 1 you will get 0.000...(repeating for infinity). In other words, there is no difference. Therefore 1 and 0.999.. are the same thing."Even if infinite zeroes are involved, there must also be a 1 involved after the infinite zeroes, or 0.9999 with infinite 9s would equate to 1. They are not the same number, therefore there must be a 1 involved as well as infinite zeroes. Under such a model of what in essence is "infinity minus something" you imply that there is something past infinity, a number following the infinite number. Furthermore, all those infinite 9s will still never result in a 1.0, however "infinitely small" the difference may be. And there will still be a difference however infinitely small.Furthermore, even if we may not be able to measure the difference now, does not mean it does not exist. For example, there has been progression in thought over time as to what the smallest particle is. As newer more exacting systems of measurement are produced we can better determine smaller and smaller discrepancies. If for example we discover a new, smaller particle to exist, does that mean it did not exist in the past because we weren't aware of it? No, it existed, we simply lacked the ability to measure it at the time. Likewise, our difficulty in measuring a minute fraction does not mean said fraction does not exist. "IN ORDER FOR TWO NUMBERS TO BE DIFFERENT, THERE HAS TO BE A NUMBER BETWEEN THEM. If 0.999... and 1 where different numbers then there would exist a number between them. No such number exists. This is because you cannot put a digit at the end of an infinite sequence."There is a number between them, and that number is 1 minus 0.9999 with the nines infinitely repeating. The number by which they differ is only difficult to define because the number in question (0.999 with the nines infinitely repeating) is difficult to define. Ultimately you cannot argue a difference does not exist without arguing that an infinite number does not exist, for the logic is the same. And if an infinite number cannot exist, then there is no question to be discussed, is there? If an infinite number of 0.9999 can exist, then an infinitely small difference can exist as well. Report this Argument Pro This is an inherently counter-intuitive subject. When first confronted with the statement 0.999... = 1 our brain immediately rejects this statement. Almost everyone who first see's this will fall back on the argument that there must be an infinitely small difference between the two numbers. We do this, because our minds have a hard time perceiving infinity. My opponent has attacked the logical side of this argument., however the mathematical evidence remains uncontested. A sharp move on his part, because the math is in my favor. I could easily reference 10 more sites where mathematicians demonstrate various proofs, however I will move on to the logical side of this argument. If two numbers are different, there must be an average between the two numbers that is not the same. For instance 3 and 5 have an average of 4. So tell me, what is the average between 0.999... and 1? On to refuting my opponent: When I stated that 1 - 0.999... = 0.000... my opponent said: -"Even if infinite zeroes are involved, there must also be a 1 involved after the infinite zeroes, or 0.9999 with infinite 9s would equate to 1. They are not the same number, therefore there must be a 1 involved as well as infinite zeroes." So you are saying that because they cannot possibly equal each other that there must be a 1 after the infinite series? First of all, you cannot put a digit after an infinite series. That violates the very definitions of infinity. If you don't believe me, maybe you'll believe a professor of mathematics (http://mathforum.org...) That would be like saying infinity plus one. There is simply no such thing. There is no "after" infinity. -"Furthermore, even if we may not be able to measure the difference now, does not mean it does not exist." " As newer more exacting systems of measurement are produced we can better determine smaller and smaller discrepancies." You are thinking in finite terms here. We are not inventing microscopes to find smaller and smaller numbers. While this logic may apply to physics it has nothing to do with math. -"If an infinite number of 0.9999 can exist, then an infinitely small difference can exist as well." An excellent point. My opponent brings up infinitely small numbers. However this brakes whats called The Archimedean property. This principal states that, for any positive real number, r, we can choose a natural number, N, large enough so that their product is greater than 1. In other words, in order for a number to be real, it can be multiplied by a large enough number that the product of those two numbers is greater than 1. Unfortunately, 0.000000... would not fit into this category. Therefore the difference between 1 and 0.999... cannot be defined as real. (http://tme.coe.uga.edu...) In conclusion: If you cannot find a difference between two numbers, and you cannot find an average between them, they are the same number.Report this Argument Con Ultimately 0.999... is what could be termed a "hyperreal number." The difference between it and 1 will be an infinitesimal, or 0.000...1. Hyperreal numbers are defined with infinite values (such as a 9 repeating) while infinitesimals are their contrasting opposites, between them and a whole number, and extremely small. They are paired as equivalently as antonyms and synonyms. Just as you must use the concept of infinity when expressing the number 0.999 with the nine infinitely repeating, so you must use the concept of infinity when expressing the difference between it and 1 of 0.000...1 with the zeroes infinitely repeating before a 1 at the end.http://en.wikipedia.org...;https://www.princeton.edu...http://www.math.harvard.edu...; Encyclopaedia Britannica defines an infinitesimal as "in mathematics, a quantity less than any finite quantity yet not zero... no such quantity can exist in the real number system. http://dictionary.reference.com...;Calculus in essence is the application of non-standard analysis to address these infinitesimals and hyperreal numbers by representing them with epsilon-delta procedures, or functions that represent the infinitely small amount.http://en.wikipedia.org...;Pro argues that infinitesimals break the Archimedean property, but this is only because the number they themselves are arguing, 0.999..., is itself a non-Archimedean number or hyperreal number. http://en.wikipedia.org...;In essence, Pro is producing a hyperreal number to be subtracted from a real number, and saying it should produce a real number. But the inverse of a hyperreal number will be an infinitesimal, the corresponding opposite of an infinitely large number is an infinitely small one. You cannot in essence produce an infinitely large hyperreal number, and argue the infinitely small number between it and a real number does not exist. If "infinitely large" can exist, then so too can "infinitely small." As noted by professor Thomas J. Crow of Blue Ridge Community College, "Sequences can converge to numbers other than zero. For example, the hyperreal number x=(n/n+1)=(1/2, 2/3, 3/4, 4/5...) is sequence that converges to 1. Consequently x=(n/n+1) is a finite hyperreal number that's infinitely close to 1."http://academic.brcc.edu...;Pro is trying to apply standard analysis, in essence, to non-standard numbers. They want to produce a hypothetical number that is infinitely close to 1 without being 1, and then claim the infinitely small difference that would exist between it and 1 cannot exist, and that the hypothetical number would be the same as 1. Yet the entire field of calculus exists to address and quantify such problems. Report this Argument Pro For my final piece of evidence I direct you to wolframalpha, a website designed to do any type of math. This site allows you to enter a mathematical statement to see if it is true or not { http://www.wolframalpha.com...} If you enter the statement 0.9... = 1, it confirms my point, it says the statement is true. The flaw in my opponents argument: My opponent has introduced the idea of hyperreal numbers in order to justify the existence of infinitely small numbers. Con has failed to make an important distinction though; he has has mixed up the difference between infinite numbers, and infinite SEQUENCES. Infinite numbers break the rules of regular mathematics (infinity + 1 = infinity?) so mathematicians invented a new form of math (hyperreal number systems) to deal with infinitely large and infinitely small numbers. HOWEVER, AN INFINITE SEQUENCE (such as 0.999...) is not an infinite number, and can be treated with the same laws of regular mathematics (1). You don't need a PhD in mathematics to know that 1/3 = 0.333... Ergo, my previous simple mathematical proofs still hold. In fact, there are many theoretical mathematical models that exist only to explore impossible area's of math, such as "imaginary numbers" For the sake of fairness however, there are more advanced methods of proving that 0.999... = 1, these methods employ the use of geometric sequencing and take into account infinities. (2) Heck, this subject has it's own Wikipedia page.(3) There are many was to write the number 1 (3/3, 0.5 *2, Cos 0 degrees) I maintain that 0.999... is just another way of doing just that. I reassert my original point, if there is no difference between two numbers, they are in fact the same number. 0.9999999 with 50 pages of 9's to follow does not equal one. At any time, if the sequence of .9's is stopped, the number is no longer equal to 1. However, if the 9's go on FOREVER there can be no difference. You cannot fit a number between 5 and 5, you cannot fit a number between 2 and 2, and you cannot fit a number between 0.999.... and 1. If you subtract 0.999... from 1, you will get 0.000.. There is never any 0.0000.....0001 at the end, because THERE IS NO END. In conclusion, if two numbers x and y have no difference, can be changed into one and other using algebra, and can be shown to be equal using simple arithmetic, they are indeed the same number. That is the case for 1 and 0.999... and that is why I assert they are equal, mathematically and literally. For those of you who remain unconvinced, I implore you, spend 2 minutes of your time, do a quick Google search, and find out for yourself. This is truly a fascinating topic, and I'm glad to be debating it. It's a case where simple math disagree's with intuition. Thank you Con for giving me a good fight. References: (1) - http://www.xamuel.com... (2) - http://www.jstor.org... (3) - http://en.wikipedia.org......Report this Argument Con Ultimately the decimal system itself shows that 0.999 does not equal 1, or quite simply it would have a 1 in front of the decimal. It does not, therefore 0.999 will always be less than 1. If 0.999 were the same as 1, it would have a 1 in front of the decimal, and because it does not it will always be less than 1. In essence 1.0 has the zero infinitely repeating, and 0.999 has the 9 infinitely repeating, but the difference is that one will always have a 9 and the other a 0 repeating, so one will always be less than the other. The only reason you cannot fit a number between 0.999... and 1, is that you can't write 0.999... in the first place. An infinitely repeating number is an imaginary, hypothetical concept, thus 0.999 is a hyperreal number rather than a real number, and its inverse is an infinitesimal. Regardless of the WolframAlpha site, all that would show is that there are conflicting beliefs on both sides of the debate. Because one source believes it's equal to 1 doesn't negate the fact that there are other experts who disagree. That there is room for debate consistent with the arguments I have laid out is seen from an article in the Mathematics Educator by Anderson Norton and Michael Baldwin, "Does 0.999... Really Equal 1?" As Norton and Baldwin point out:"Students may be justified in rejecting the equality if they decide to work in another system-namely the non-standard analysis of hyperreal numbers-but then they need to understand the consequences of that decision... Starting from 0, the point gets nine-tenths of the way to 1, then another nine-tenths of the remaining distance, and so on, but there is always some distance remaining (cf. Zeno’s paradox). This conception aligns with Aristotle’s idea of potential infinity and his rejection of an actual infinity: 0.999… is a process that never ends, producing a decimal expansion that is only potentially infinite and not actually an infinite string of 9’s..."http://tme.coe.uga.edu...Norton and Baldwin also point out that 0.999... only "approximates" 1, you must in essence apply rounding to an infinitely repeating number to make it equal to 1. But a rounded number will never be the same as the number it is rounded to, even if representing it that way might help mathematicians sleep easier at night. They conclude that:"Nonstandard analysis provides a sound basis for treating infinitesimals like real numbers and for rejecting the equality of 0.999… and 1 (Katz & Katz, 2010). However, we will see that it also contradicts accepted concepts, such as the Archimedean property. "Again, per the source by Thomas J. Crow, hyperreal numbers include those that infinitely converge towards 1 as 0.999 most certainly does. http://academic.brcc.edu... An intriguing topic, and I likewise thank my opponent for a thought-provoking original discussion. Report this Argument
23 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Jzyehoshua 1 year 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 1 year 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 1 year 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 1 year 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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