The student's acceptance or rejection of 0.999 =1

[thoughtweevil]

I was searching the web for some perspective on this non-intuitive 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.

 

[thoughtweevil]

Clarification: I am not implying that the equation is wrong. Rather, I am re-interpreting the common student response. The equation is logically consistent in a mathematical system that happens to rely upon a concept--infinity--that 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."

 

[micromass]

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.

 

[thoughtweevil]

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.

 

[micromass]

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.

Clearly this is not so, since we can also write

$$1=1.0000000...$$

The right-hand side can also only be represented if you use infinite sets. If you don't accept the existence of an infinite set, then the right-hand side also doesn't exist. So the right-hand side is also a "purely imaginary concept". Somehow students don't really have troubles with that equality however.

 

[phinds]

... examples of infinite sets are not represented in the material aspect of existence to my knowledge ...

If your interest is to only use math that EXACTLY corresponds to physical existence (2 apples plus 3 apples = 5 apples) then you are doing experimental physics, not math. Math things are not required to exist in the real world, they are only required to be internally consistent.

My understanding is that when Riemann invented his non-Euclidean 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.

 

[pwsnafu]

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.

 

[EebamXela]

The original equality is as easy to understand and accept as this one:

1/3 = 0.3333333...

Multiply both side by 3 and there you go.

 

[micromass]

The original equality is as easy to understand and accept as this one:

1/3 = 0.3333333...

Multiply both side by 3 and there you go.

Very true! But, after they saw this argument, some students even started to doubt that $\frac{1}{3}=0.333333...$!

 

[Millennial]

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 four-dimensional 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 10x-x=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 5-sphere 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.

 

[Vargo]

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).

 

[Millennial]

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, $\displaystyle \iint_{D}dy\,dx=\iint_{C}r\,dr\,d\theta$, which immediately reduces to $\displaystyle \int_{0}^{2\pi}\int_{0}^{a}r\,dr\,d\theta=\int_{0}^{2\pi}\frac{a^2}{2}d\theta=\pi a^2$ 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.

 

[Hurkyl]

My understanding is that when Riemann invented his non-Euclidean geometry, it did not, as far as he knew, correspond to any physical reality, it was just something that he found intellectually interesting.

Hrm. I was under the impression that the original development of differential geometry was for the purpose of studying shapes in Euclidean space in a way that didn't reference the ambient Euclidean space they are contained in.

 

[justsomeguy]

Very true! But, after they saw this argument, some students even started to doubt that $\frac{1}{3}=0.333333...$!

That may be an opportune time to (re)introduce primes and bases, so they can understand that the repetition isn't just a bit of an apple that always has "just a little bit more" than they can measure, but is just an artifact of the base they are working in. 1/3 "works" fine in a base-6 decimal system, but 1/5 does not, for example.

 

[Mark44]

That may be an opportune time to (re)introduce primes and bases, so they can understand that the repetition isn't just a bit of an apple that always has "just a little bit more" than they can measure, but is just an artifact of the base they are working in. 1/3 "works" fine in a base-6 decimal system

Or base-3, where 1/3₁₀ = 0.1₃.

, but 1/5 does not, for example.

 

[Mandelbroth]

Very true! But, after they saw this argument, some students even started to doubt that $\frac{1}{3}=0.333333...$!

I, too, doubt that one third is equal to zero point three (repeating) factorial.

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.

I am under the belief that a proof should be provided for any mathematical concept given in a classroom for the same reason that I think that statistics should be taught with calculus. If a formula or equation is simply given to you, the only thing that the student is learning is to apply a formula. Though the proof may stretch the minds of certain algebra students, it will be sufficient for them to believe and accept it if they are walked through a proof.

...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 $x$ in the domain such that $1-0.999...= x$. If you are working with a number system that does include infinitesimals...they aren't actually wrong if they say that 0.999... ≠ 1 .

 

[Vargo]

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.

 

[Vargo]

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 over-reactive and misinterpreted your comment.

 

[Hurkyl]

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 $x$ in the domain such that $1-0.999...= x$.

That won't convince anybody who thinks that 0.999... is an unspecified large, finite number of 9's. Or has a varying number of 9's. Or that decimal notation. Or that decimal notation has a final place.

If you are working with a number system that does include infinitesimals...they aren't actually wrong if they say that 0.999... ≠ 1 .

In every commonly used system with infinitesimals that I'm aware of, yes, they would be wrong.

e.g. in the hyperreals, 0.999... is still the decimal with a 9 in every place, and is still equal to 1 (and it had better be that way, due to the transfer principle). There are terminating decimals that have infinitely many -- but still hyper-finitely many -- 9's, but those are all strictly less than 0.999...

 

[lavinia]

Hrm. I was under the impression that the original development of differential geometry was for the purpose of studying shapes in Euclidean space in a way that didn't reference the ambient Euclidean space they are contained in.

Hurky

I don't know Riemann's actual thinking on this but from the history of non-Euclidean geometry and knowing that Gauss was his mentor, I think he was saying that there is an idea of space which has no intrinsic geometry and that geometry needs to be super imposed upon it as a separate structure. Until Lobachevskyan plane geometry was discovered, many felt that Euclidean geometry was intrinsic to the idea of space. When this second plane geometry was discovered Gauss immediately tried to measure large triangles on Earth to determine which of the two was correct. So he still retained the idea that some geometry was intrinsic to space but that which one needed to be determined by measurement. Riemann took this still classical thinking into the modern world by conceiving of a space where no geometry was intrinsic, that of a manifold. Then determining the world metric for him became an empirical problem for physicists to study and he proposed methods by which the true geometry of space might be determined. I so not thinkt that this was merely intellectually interesting to him. It was a thought experiment about measuring the real world.

 

[lavinia]

I Infinity is an operationally useful concept but does not actually exist.

How then does Achilles overtake the tortoise?

I think the examples of Achilles can be used to convince someone that .9999999 does exist.

But one can also just say that it is notation for the fact that no finite sequence of .9's every equals 1 yet there is no number less than one that is bigger than all finite sequences of .9's.

I sometimes tell people that the number 1 is the actual infinity because it is a concept that requires infinitely many predicates to understand. These predicates are among other things the infinite number of finite sequences of .9's. This is like the idea of the circle which implies all of its inscribed polygons - as mentioned above. Another example of this is the differential dx which implies all of the Δx's no matter how small.

To say that infinity does not exist in reality to me just says that there is no way to count infinitely many things (since the amount of time you have is finite). This primitive idea of infinity seems well replaced by the example of Achilles and its generalization to the idea of an actual infinity.

 

[Mandelbroth]

That won't convince anybody who thinks that 0.999... is an unspecified large, finite number of 9's. Or has a varying number of 9's. Or that decimal notation. Or that decimal notation has a final place.

Could you please specify what you are trying to say...?
Also, upon further consideration, it may be more understandable for a student to see that there is no nonzero number $x\inℝ$ such that $1-0.999...=0+x$. Then again, I may uncover some flaw in my thinking upon your specification.

In every commonly used system with infinitesimals that I'm aware of, yes, they would be wrong.

e.g. in the hyperreals, 0.999... is still the decimal with a 9 in every place, and is still equal to 1 (and it had better be that way, due to the transfer principle). There are terminating decimals that have infinitely many -- but still hyper-finitely many -- 9's, but those are all strictly less than 0.999...

The transfer principle doesn't apply here because we are talking about the assertion that there exists some theoretically infinitesimal number of the form 0.000...1 that can be added to 0.999... to get 1. Not all aspects of *ℝ are transferable to ℝ. For a pertinent example, infinitesimal elements of *ℝ are not part of ℝ. In fact, part of the premise of hyperreals is to add these infinitesimals to the domain. Thus, the notion of $\nexists x : x = \frac{1}{1+1+1+...+1}$ is true for reals but not true for hyperreals. In my experience, 0.999... means a number that falls infinitesimally short of 1, given we are working with hyperreals. Then again, not everyone agrees about everything.

 

[Hurkyl]

Could you please specify what you are trying to say...?
Also, upon further consideration, it may be more understandable for a student to see that there is no nonzero number $x\inℝ$ such that $1-0.999...=0+x$. Then again, I may uncover some flaw in my thinking upon your specification.

Those are all examples of ways in which students fail to understand what 0.999... means.

Some students have the idea that it just means a large unspecifed number of 9's. And so 1 - 0.999... is simply 0.000...1 with the correspondingly large unspecified number of 0's.

Other students have in mind that 0.999... is a "number" that "approaches" 1 -- I think they are imagining the sequence of partial decimals that have values $s_n = 1 - 10^{-n}$. So they would follow up with understanding 1-0.999... as a "number" that "approaches" 0 but never reaches it.

The transfer principle doesn't apply here because we are talking about the assertion that there exists some theoretically infinitesimal number of the form 0.000...1 that can be added to 0.999... to get 1. Not all aspects of *ℝ are transferable to ℝ.

Transfer goes the other way. 0.999... = 1 is a statement of the standard model, and therefore true in the standard model if and only if its transfer is true in the non-standard model.

Thus, the notion of $\nexists x : x = \frac{1}{1+1+1+...+1}$ is true for reals but not true for hyperreals.

There is some abuse of notation in your statement, which is obscuring what's going on.

In my experience, 0.999... means a number that falls infinitesimally short of 1, given we are working with hyperreals. Then again, not everyone agrees about everything.

Then you are writing 0.999... to refer to a terminating (non-standard) decimal with a large but unspecified number of 9's, rather than the repeating (non-standard) decimal.

0.999...9 would be a much better way to notate such a thing, so that notation is consistent.

 

[Mandelbroth]

Transfer goes the other way. 0.999... = 1 is a statement of the standard model, and therefore true in the standard model if and only if its transfer is true in the non-standard model.

[...]

Then you are writing 0.999... to refer to a terminating (non-standard) decimal with a large but unspecified number of 9's, rather than the repeating (non-standard) decimal.

0.999...9 would be a much better way to notate such a thing, so that notation is consistent.

I'm sorry. I really need to be more clear on these things. It appears I've started a completely tangent conversation based on my misinterpretation of your notation. Additionally, I thought you were trying to imply that transfer was going from *ℝ to ℝ, and thus thought to refute and correct your argument by explaining where this does not work with respect to the current topic.

How then does Achilles overtake the tortoise?

I like this idea of using Zeno, because it makes students either accept the concept of infinity or reject their current views of reality and motion.

 

[AlephZero]

I like this idea of using Zeno, because it makes students either accept the concept of infinity or reject their current views of reality and motion.

The danger there is that some ancient greek ideas about motion can seem more "obviously" correct than Newtonian mechanics. After all, if took about 2,000 years to decide that "Aristotelian mechanics" was actually wrong!

 

[lavinia]

The danger there is that some ancient greek ideas about motion can seem more "obviously" correct than Newtonian mechanics. After all, if took about 2,000 years to decide that "Aristotelian mechanics" was actually wrong!

Newton's laws assume a continuum and that the continuum is traversed smoothly. He agrees that Achilles can overtake the tortoise.

The alternative is to deny the continuity of motion or to assert that motion does not exist at all, that it is an illusion.

 

[coolul007]

Introducing students to 0.999.. = 1, is just something pulled out of one's butt, to confuse and screw with them. I have never encountered 0.999.. in any actual computation. It seems to be man made to screw with people.

 

[justsomeguy]

Introducing students to 0.999.. = 1, is just something pulled out of one's butt, to confuse and screw with them. I have never encountered 0.999.. in any actual computation. It seems to be man made to screw with people.

You've never encountered 1/3? Or 1/7? That's all 0.999... is about; decimal representation of x/y in a base where y has a prime factor that isn't also a factor in the base.

 

[coolul007]

You've never encountered 1/3? Or 1/7? That's all 0.999... is about; decimal representation of x/y in a base where y has a prime factor that isn't also a factor in the base.

Specifically, 0.9999... in base ten there is no decimal expansion a/b that produces that result. That is all I am saying. To introduce this to students is just a real waste of time. I was introduced to it as a Junior in High School. Other than a lot of discussion it has served no purpose. Decimal expansions is taken for granted in most early math courses. Very little time is spent examining it. Then they throw in 0.9999... and the rest is history...

 

[lavinia]

Specifically, 0.9999... in base ten there is no decimal expansion a/b that produces that result. That is all I am saying. To introduce this to students is just a real waste of time. I was introduced to it as a Junior in High School. Other than a lot of discussion it has served no purpose. Decimal expansions is taken for granted in most early math courses. Very little time is spent examining it. Then they throw in 0.9999... and the rest is history...

You seem to equate Mathematics with computation. It is true that a calculator can not perform a limit. But humans can. Mathematics is a conceptual subject that probes the meaning of ideas such as number, measure,motion, and shape. These ideas are not available to a calculator.

I have shown this decimal expansion to beginners and they have loved it. Many people find the idea of infinity inspiring.

A big problem ,in my opinion, in mathematical education is that it does not teach mathematics as a mental discipline but merely as a set of useful computational tools.

 

[coolul007]

You seem to equate Mathematics with computation. It is true that a calculator can not perform a limit. But humans can. Mathematics is a conceptual subject that probes the meaning of ideas such as number, measure,motion, and shape. These ideas are not available to a calculator.

I have shown this decimal expansion to beginners and they have loved it. Many people find the idea of infinity inspiring.

A big problem ,in my opinion, in mathematical education is that it does not teach mathematics as a mental discipline but merely as a set of useful computational tools.

Please do not read more into my comments. I am addressing 0.9999... The idea of limits and infinite series is interesting, 0.999... = 1.0 is not useful, as it is dropped from the sky into one's lap and then goes nowhere.

 

[uart]

Introducing students to 0.999.. = 1, is just something pulled out of one's butt, to confuse and screw with them. I have never encountered 0.999.. in any actual computation. It seems to be man made to screw with people.

I had the exact opposite experience. I can remember back at school (a very long time ago), being "confronted" with the layman's proof that 0.999... = 1 and finding it very interesting. In particular, it made me think about infinite series, it made me think about what it means for two real numbers to be equivalent. Stuff I had never even thought about before, and the process was very insightful. I'm very much glad that my teacher all those years ago chose to demonstrate this!

 

[hddd123456789]

Specifically, 0.9999... in base ten there is no decimal expansion a/b that produces that result

Another poster in this thread noted just such an example that produces 0.999...

The original equality is as easy to understand and accept as this one:

1/3 = 0.3333333...

Multiply both side by 3 and there you go.

Just a suggestion, but perhaps a good way to introduce 0.999...=1 might be to discuss the language of mathematics in a rudimentary but sufficient way. After all, what this is is just a language artifact left over from a conversion from (1/3*3) to a decimal representation, rather than any actual computation. (1/3) = 0.333... and (1/3*3) = 1, so clearly, 0.999...=1, not because of limits or infinitesimals or calculus for that matter. It is just a language artifact required to maintain a self-consistent language. If it weren't so, then it is tantamount to saying $(1/3*3)\neq1$ which is plain silly.

Edit: Oh wait, I just remembered someone else mentioned that this doesn't help because a student might not buy that (1/3)=0.333... in the first place :P Still, I think that language argument holds water since even 0.333... is in itself a language issue since it is a representation of a number. It should be possible to show to a student that 0.333... and 0.999... are representations of (1/3) and (3/3), respectively; with the latter one equaling 1 of course.

 

[coolul007]

Specifically, an a/b that produces 0.99999... does not exist. Yes, I can create series that add up to 0.999..., I can also multiply, however the original statement that no a/b exists to produce 0.999...

As simple trick I learned that placing any number of 9's will create a repeating decimal:

1/9 = .111..., 23/99 = .232323..., 3/9 = .333..., 9/9 =1, no repeat

 

[Hurkyl]

Other than a lot of discussion it has served no purpose. Decimal expansions is taken for granted in most early math courses. Very little time is spent examining it. Then they throw in 0.9999... and the rest is history...

Did you know there are a number of people who think numbers aren't "exact"? That people get the idea that there is no such number as "one third", with reasoning that decimal can only provide an approximate value of such a thing? Non-terminating decimals really need some amount of coverage, to properly connect decimal notation to arithmetic and algebraic knowledge.