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The student's acceptance or rejection of 0.999...=1 
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#1
Dec612, 01:32 PM

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I was searching the web for some perspective on this nonintuitive equation, 0.999...=1. The consensus is that if a student rejects it as a "parlor trick" there are a handful of reasons to explain the student's "confusion." One reason given is that the student cannot help but see the number as a very large number of 9's after the decimal place, when in fact there are infinitely many 9's.
I argue that the student is not confused. The parlor trickery is the invocation of infinity to explain the equation. Infinity is an operationally useful concept but does not actually exist. I'm sure there are counter arguments and I'd like to hear them. 


#2
Dec612, 01:46 PM

P: 3

Clarification: I am not implying that the equation is wrong. Rather, I am reinterpreting the common student response. The equation is logically consistent in a mathematical system that happens to rely upon a conceptinfinitythat is rather imaginary. The student's rejection is a result of their encountering the utilitarian, imaginary side of mathematics, whereas the concept of 1 at least seems "realistic."



#3
Dec612, 01:51 PM

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If you refuse to accept the existence of infinite sets (which is mathematically acceptable), then the notation 0.999999.... would also not exist. So in that case, there is no equality to accept or reject, since 0.999... would not have any meaning or existence.



#4
Dec612, 02:49 PM

P: 3

The student's acceptance or rejection of 0.999...=1
Well they certainly exist in our imagination, which is not to say anything of their import. However, examples of infinite sets are not represented in the material aspect of existence to my knowledge. Examples of 1 are represented materially. So, when we equate a materially grounded concept to a purely imaginary concept, without explaining that we are doing so, the student is naturally challenged by the omitted information.



#5
Dec612, 04:42 PM

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[tex]1=1.0000000...[/tex] The righthand side can also only be represented if you use infinite sets. If you don't accept the existence of an infinite set, then the righthand side also doesn't exist. So the righthand side is also a "purely imaginary concept". Somehow students don't really have troubles with that equality however. 


#6
Dec612, 05:44 PM

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My understanding is that when Riemann invented his nonEuclidean geometry, it did not, as far as he knew, correspond to any physical reality, it was just something that he found intellectually interesting. Later, Einstein found that it does correspond to physical reality but that is irrelevant to Riemann's interest. Math does not have to represent reality. Students will find LOTS of things far more weird than 1 = .999999... if/when they study cosmology and/or quantum mechanics. Things that are true to SEEM to be impossible. 


#7
Dec612, 07:05 PM

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Except when students are introduced to irrational numbers they are taught "the decimals never repeat". So they do in fact understand infinite decimals. Not well of course, but well enough.



#8
Dec612, 09:40 PM

P: 16

The original equality is as easy to understand and accept as this one:
1/3 = 0.3333333.... Multiply both side by 3 and there ya go. 


#9
Dec612, 09:47 PM

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#10
Dec712, 09:28 AM

P: 295

To be honest, all students who have trouble grasping this have an intuition that tells them there must be a last 9, because humans can't see something infinite just as they can't see the fourth dimension. A mathematician can tell you that a fourdimensional cube, tesseract, has 16 vertices. The thing with this is the mathematician can't see four dimensions as well, but he can understand it.
I usually go to the talk page of Wikipedia about this topic and I even read some arguments like "What does 0.9999.......988888........ equal?" This shows that people have trouble that "an infinite number of nines" actually means "there can't be anything other than another nine after the nines, because there is a nine after every nine". The usual argument used to show that 1=0.999.... is as follows: x=0.999... so 10x=9.999... and 10xx=9x=9, hence x=1=0.999... Of course, the reason we can just multiply by 10 and subtract is because a sequence multiplied by a scalar (or a constant) converges to the limit of the original sequence times that scalar. Oops! What is convergence? The logic here is actually clear but convergence can't be formally explained at the level that this is taught. Normally, the first argument is enough to convince most; but I have seen people who ask "Isn't there one nine less in 10x?", which is a pretty natural question for a student who can't grasp infinity to ask. It would be reasonable to answer "Because infinity minus one is still infinity", but not having defined "infinity" properly, and without limits, our argument is in vain. I do think this is the most intuitive explanation, and this small lack of justification relies on the reader's intuition, which is to be expected as this is an intuitive proof. After this, I'd like to reply to the OP: Mathematical concepts do not have to exist in reality, just like a 5sphere doesn't exist (or does it? String theory might indicate it actually does.), infinity does not have to exist either. Mathematics seeks to create a logically consistent set of theorems and axioms. The science that actually tries to model the real world is physics, which has laws that are based on experimentation; and it uses mathematical spaces where these laws hold true to model the nature. Mathematical axioms are not laws and a mathematician can easily work in another axiomatic system as long as it is consistent. 


#11
Dec712, 10:44 AM

P: 350

As others have suggested, I think the main block for students is that they don't really know much about convergence. Somehow the "infinite" nature of taking a limit is difficult to grasp until you have studied convergence and limits on their own. Another example that was, for me at least, difficult to grasp at the time was the proof that the area of a circle is equal to pi*r^2. It involves inscribing a regular polygon inside the circle and then cutting and rearranging the pieces into a parallelogram (you all probably remember it from high school geometry).



#12
Dec712, 02:42 PM

P: 295

I prefer using the fact that the area of a region is given by the double integral taken over it for the proof of the area of a disk, [itex]\displaystyle \iint_{D}dy\,dx=\iint_{C}r\,dr\,d\theta[/itex], which immediately reduces to [itex]\displaystyle \int_{0}^{2\pi}\int_{0}^{a}r\,dr\,d\theta=\int_{0}^{2\pi}\frac{a^2}{2}d \theta=\pi a^2[/itex] for a disk of radius a.
To me, this sort of system where a concept is taught before its proof can be fully grasped is confusing and creates what the Fundamental Theorem of Algebra does: It is a theorem of algebra, but it can't be proven resorting only to algebraic methods. This leaves us with the important question that whether we should teach it in an algebra class or leave it until we get to an analysis class. 


#13
Dec712, 03:34 PM

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#14
Dec712, 04:47 PM

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#15
Dec712, 05:06 PM

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#16
Dec712, 05:31 PM

P: 615

...Then again, my philosophy does not necessarily account for the fact that most algebra students simply don't care enough to thoroughly understand the basic material for algebra, let alone analysis. My personal opinion on 0.999... = 1 is that if they don't believe it, teach them more parameters of the number system they are working with. If we are talking about the set of all real numbers, reveal that infinitesimals are not truly part of the set of real numbers. Thus, there is no nonzero number [itex]x[/itex] in the domain such that [itex]10.999...= x[/itex]. If you are working with a number system that does include infinitesimals...they aren't actually wrong if they say that 0.999... ≠ 1 . 


#17
Dec712, 08:39 PM

P: 350

Only presenting concepts that can be rigorously proved? From what age? Are you suggesting that students should remain ignorant of the great achievements of science and math because they can't understand the rigorous proof?
Personally I think the proof using inscribed/circumscribing regular polygons is a bit more illuminating since it clearly shows the link between pi as a ratio of lengths and pi as an area. Using the Fundamental Theorem of Calculus just seems a bit overpowered for such a fundamental fact. But htat's just a matter of taste. 


#18
Dec712, 09:05 PM

P: 350

Ok, just to modulate the tone of my previous message, I think it is a good idea to learn about stuff like the Heisenberg uncertainty principle, special relativity, orbitals (in chemistry), and other stuff that, mathematically, is way over the heads of most science students. Rigor is good, but facts are important too. I'm sorry if I came off as overreactive and misinterpreted your comment.



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