Reasoning outcomes of simplifications involving infinity

There are some cases where final result looks like strange when the concept of infinity involves. My issue is how to describe these situations to our students.
Here are few examples ;

1. When you keep on multiplying 1's such as, 1×1×1×... can you say here we have product of infinite number of 1's or is this limit of product of n number of 1's as n tends to infinity? If so can we say this product is indeterminate? Then what about the product 2×2×2....... because this also can be written in terms of 1×1×1×....?
2. We all know, sum of two rational numbers is rational and if you add another rational, answer should be a rational. But when you consider the irrational number e , by its definition we have ,e = 1+1/(1!)+1/(2!)+1/(3!)+... .Here the right side is a sum of infinite number of rationals . What we have to say, is this because of infinity number of rationals involved ?
3. We all know tan(π/2) can not be defined and it is not infinity. But in definite integrals as an example, integral of 1/(1+x²) from x = 0 to x = ∞ We write tan inverse of infinity as π/2 . Is this because we consider interval of values and therefore x tends to infinity?
I know many students don't think about these issues if we do not mention, but it is much better if we have the most appropriate way of describing these issues to give much confidence of what students learn.
• These are great math questions. So many of your questions seem to be at the edge between math and math education. I think you'd get some great answers at the math.stackexchange site. Jul 7, 2022 at 18:18
• Still another example: Let $S_5 = 1 + 2 + 4 + 8 + 16 + 32.$ Then $2S_5 = 2 + 4 + 8 + 16 + 32 + 64.$ Therefore, we have $S_5 = 1 + (2S_5 - 64).$ Solving for $S_5$ gives $S_5 - 2S_5 = 1 - 64,$ or $S_5 = 63.$ Now let $S_{\infty} = 1 + (2 + 4 + 8 + 16 + 32 + \cdots).$ Then $2S_{\infty} = (2 + 4 + 8 + 16 + 32 + \cdots).$ Therefore, we have $S_{\infty} = 1 + 2S_{\infty}.$ Solving for $S_{\infty}$ gives $S_{\infty} = -1.$ So apparently $-1$ can be the sum of infinitely many different positive integers?!? Jul 7, 2022 at 19:36
• I don't understand the sentence "tan(π/2) can not be defined". You can, for instance, define $\tan(\pi/2)$ to be equal to $-17$. This would not be a particularly useful definition (and hence, nobody does this), but obviously one can do it. Jul 7, 2022 at 21:58
• @JanakaRodrigo: "because 2 = 2×1". It seems that the comments are increasingly about the mathematical content of the post rather than about the educational part of it, but anyway: I cannot see how the fact $2 = 2 \times 1$ can be used to express a product of the type $2 \times 2 \times 2 \times \dots$ by means of $1 \times 1 \times 1 \times \dots$ without using $2 \times 2 \times 2 \times \dots$ again. Jul 8, 2022 at 7:29
• Am I correct? Yes. This example also illustrates a situation that often arises in mathematics, where the calculation of something makes use of an assumption (that needs to be proved separately) that the thing exists. In my example the manipulations with $S_{\infty}$ assume that $S_{\infty}$ exists as something (real number, complex number, or whatever) that can be operated on by arithmetical operations. For another example, see the first "Surprising results" here, where the apparent paradox is due to manipulating something whose existence has not been shown. Jul 8, 2022 at 11:46

4 Answers

"How to describe these situations to our students" ...

At any level below calculus, tell the students: "You will cover this later."

This is a special case of: Do not try to tell your students about something you do not understand yourself.

[I resist the urge to provide off-topic commentary on the three questions in this forum.]

• Yes, do not try to explain to students something you yourself do not understand (well...) Admitting that "even the grown-up teachers" may fail to understand things is, in my opinion, very helpful to students in many ways. Just one of those ways is understanding better the genuine nature of mathematics. Jul 7, 2022 at 20:41
• @Gerald Edgar you suggest just to avoid challenging situations in mathematics. But I think good teachers becomes great in mathematics if you can encourage curiosity of students even by letting them to find out reason even if you are not able to do so . Sometimes it can help to find out who is the genius in your class too. Jul 8, 2022 at 1:04
• @JanakaRodrigo I think advanced level mathematics probably depend on the view you have of mathematics. I'm guessing you're talking about high school education, in which these topics would certainly be advanced level. But that's one of the reason you really want to be careful not to teach things you don't really understand. This is not meant to be disparaging at all. Leibnitz and Newton who invented calculus and were most likely complete geniuses made many errors exactly because they didn't have formalism. It is formalism that allows us to do as much math as we can now as normal humans.
– DRF
Jul 9, 2022 at 14:34
• I'd have been so disappointed as a student if my teachers had ever told me "You will cover this later." (if years later). What's wrong with (aside from probably having no damn time to do so), saying "You know, I don't really understand how it works either. But let's find out together."
– towr
Jul 9, 2022 at 16:35
• @towr A student could very easily ask a question which is generally not answerable until one has engaged with the appropriate material for quite a long time. "You'll learn about this later" is an entirely reasonable answer if the instructor in question knows the material well enough to know that it will be taught later. "I don't know, let's try to find out together" can also be a reasonable answer, but I would worry a little about the blind leading the blind. Jul 9, 2022 at 18:42

What resolved these questions for me was when a teacher explained that you cannot add or multiply infinitely many things together: all infinite sums are actually limits.

It can help to think of addition and multiplication as naturally binary operations that get extended to n-ary operations in a well-defined (by associativity) way, as opposed to thinking of them as n-ary operations. This will reduce the temptation to extend them to infinity-ary operations.

• Yes I too agree with all infinite sums are limits to address these issues, then you need to say limit is the sum . Jul 8, 2022 at 1:21

"When you keep on multiplying 1's such as, 1×1×1×... can you say here we have product of infinite number of 1's or is this limit of product of n number of 1's as n tends to infinity?"

In general, anything with "..." is imprecise, and it is meaningless if you cannot make it precise. I am sure that in your case it is meaningless, because...

"If so can we say this product is indeterminate?"

No. Your use of the term "indeterminate" reveals that you conflated values and expressions. For example, 2×3 = 6, but "2×3" ≠ "6". See this post for more.

"Then what about the product 2×2×2....... because this also can be written in terms of 1×1×1×....?"

Wrong. You haven't even defined "2×2×2......." precisely, and if you could then you would not be saying that it "can be written in terms of 1×1×1×....?".

"But when you consider the irrational number e , by its definition we have ,e = 1+1/(1!)+1/(2!)+1/(3!)+... .Here the right side is a sum of infinite number of rationals . What we have to say, is this because of infinity number of rationals involved ?"

No. Again. You shouldn't be saying that something happens because of "infinity [sic] number of rationals". You should instead be telling yourself that you cannot in general say that an infinite sum has the same properties as its partial sums.

Consider the geometric sequence 1, 1/2, 1/4, ... For every k∈ℕ, the sum of the first k terms is less than 2. But the sum of all of them (which needs to be defined as a limit) is not less than 2.

"We all know tan(π/2) can not be defined and it is not infinity."

No. You are free to define tan(π/2) to be whatever you like. Whether or not your definition is useful is another matter, but it's false to say it cannot be defined. In fact, it may be convenient in some (not too common) situations to define tan(π/2) to be the infinity of the projectively extended real line, but one has to learn basic real analysis before looking at extensions of the standard real line.

"But in definite integrals as an example, integral of 1/(1+x²) from x = 0 to x = ∞ We write tan inverse of infinity as π/2."

In proper mathematics, we don't write whatever we like. We give precise definitions and prove precise theorems about the objects that we have defined. It doesn't matter if we define tan to have domain excluding odd multiples of π/2. We are still free to define atan on the affinely extended real line such that atan(∞) = π/2. What properties we get for atan is another matter. Also, in basic real analysis we typically define integrals on a bounded interval, and define an integral on [0,∞) to be the limit of the integral on [0,x] as x → ∞. So in fact whether you define atan(∞) or not is irrelevant!

"I know many students don't think about these issues if we do not mention, but it is much better if we have the most appropriate way of describing these issues to give much confidence of what students learn."

If you first correct all your own errors in your understanding of the concepts, then you would be much better positioned to teach students and recognize when they make a conceptual mistake.

Understanding comes before teaching.

• Thanks for interesting explanation. What about if you denote 1×1×1×...and 2×2×2×...by limits such as limit of 1ⁿ as n tends ∞ and limit of 2ⁿ as n tends to ∞ ? Jul 8, 2022 at 18:00
• Are you able to precisely define what you mean? If the answer is "no", it remains meaningless. If the answer is "yes", what is your precise definition of "limit of 2^n as n → ∞"? Jul 8, 2022 at 18:02
• according to my knowledge that particular limit related to 2 is infinity and limit related to 1 is indeterminate. Jul 8, 2022 at 18:05
• @JanakaRodrigo: I did not ask you what you think the limit is. I asked you to define it. If you're unable to define what you mean, then sorry but it's just meaningless, and you need to study a proper textbook such as Spivak's "Calculus". And by standard mathematical definition (i.e. the one by Spivak), $\lim_{n∈ℕ→∞} 1^n = 1$, so even your unjustified guesswork is wrong. Jul 8, 2022 at 18:11
• It is NOT 1^∞. Stop. There is no point continuing wild guesswork. Start learning. Jul 8, 2022 at 18:57

This is a point where it's useful to explain to students that the simplifications of infinite phases we give are not always obvious from the related finite phrases. When dealing with this myself, I like to remind people that the laws of math were not pulled from thin air. People had to think them through, reason them out, and come up with answers which seem to be in agreement with the reality around them.

I agree with the sentiment that one really should not study such infinities until one is ready for calculus. But I do think marveling at them is fair game. It's simplifying things a bit much, but I like starting from Zeno's paradoxes to provoke thought -- to realize that this "infinity" thing is really tricky. It takes a lot of formalism to get it right. I'd argue that Zeno's most famous paradox was not truly laid to rest until Newton and Leibniz invented calculus and showed a rigorous way of dealing with infinities.

To show how odd infinities can be, I do like Hilbert's Grand Hotel. I like it because it's right on the edge of math. The solutions to adding more guests make sense numerically, but they run just afoul of our intuitions. And I point out that the math behind Hilbert's Hotel isn't the only approach possible, but it is the one that we have found to be most consistent while we explored it.

It's probably worth at least mentioning the concept of limits, as they are the rigorous tool mathematicians use to tame infinities. For example, we can say the sum of any number of rational numbers is rational... but the devil is in the details. By "any number" we really mean any integer number. Your sum that leads to e dies not have an integer number of terms, so it is not bound by the obvious inductive patterns. We can either speak of this as the limit of this sequence (and realize that limits change the game), or we can use transfinite numbers and speak of $$\omega$$ terms in a series. But by invoking transfinite numbers, we have to realize that rules that worked fine for finite numbers don't always work here... just as the rules for integers don't always work out when you expand them to the real numbers.

But all the details are not necessary. It's enough to realize that infinity is different in material ways, and have a few examples like Zeno's Paradoxes and the Grand Hotel to guide you. Learn how we rigorously handle these infinities when you are ready to learn how to rigorously handle these infinities, and not before.