Can infinity be divided by anything?

Question

I'm in ninth grade and I've been thinking about this for a while. It's related to a question that came to my mind, namely which is the highest number you can divide 11 by. Is "infinite" a number and is it the highest number by which 11 can divide it?

@user647486: the natural numbers are defined for instance by the Peano's axioms. — , Apr 12 '19 at 14:43
@Ilearnmathnotknowit Dividing by $11$ is multiplying by $\frac{1}{11}$. The multiplication of $\infty$ by any real number is defined as $\infty$. — user647486, Apr 12 '19 at 14:43
Sure, it depends how you define it. You can define it to be the number such that for any real number $x$ $x$ can be divided by infinity and also that for any real number $x$, $x < \infty$. Thus it would be the greatest number that divides 11. — famesyasd, Apr 12 '19 at 14:44
@user647486: what is your definition of infinity (which is not a number) ? — , Apr 12 '19 at 14:45
@user647486 I'm not sure what you are asking but the generally accepted definition of infinity is that it means undefined. Hence Sean Roberson's entirely correct comment. — user27119, Apr 12 '19 at 14:45
Welcome to MSE! As @SeanRoberson said, infinity is not a number. To understand infinity, you have to understand the concept ''limit''. Maybe it's too young for you to understand infinity but I encourage you to initiate your first exposure to calculus, any calculus textbook would help. — Bach, Apr 12 '19 at 14:45
In order to answer the question you need a precise definition for these concepts: number, division, biggest (highest) number and finally infinity. And no offense but if "grade 9" means "high school or less" I highly doubt such person will be able to grasp it. These topics (at least "number" and "infinity") are just soooooo deep and difficult and may mean different things in different context. There are whole books written on these subjects. Not the place from Math SE unfortunately. — freakish, Apr 12 '19 at 14:49
There is an arithmetic for the extended real numbers (real numbers with $\pm\infty.$) You can see its arithmetic here: https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations. — Adrian Keister, Apr 12 '19 at 14:50
@AdrianKeister Is the extended real line the only way to define infinity? No, it's not. Are all definitions equivalent? No, they're not. The question is too broad. — freakish, Apr 12 '19 at 14:52
@freakish: Yes, I'm aware of that. I just wanted to point the OP to one way of doing things that I've found helpful for developing an intuition about infinity. — Adrian Keister, Apr 12 '19 at 14:54
@QuantumPenguin Undefined is just an operation which hasn't been defined in some context. Infinity, are a name for many different numbers, in different areas of mathematics. For example, there is one that is the number obtained as the one-point compactification of the plane. This number is the result of many operations, for example $1/0$. There are two numbers $\pm\infty$ called infinity that are the end points of $\mathbb{R}$. These are, for example, the results of $\lim_{x\to0^+}\frac{1}{x}$ and of $\lim_{x\to0^-}\frac{1}{x}$. — user647486, Apr 12 '19 at 14:56
@QuantumPenguin Note how $\pm\infty$ are valid cases for L'Hospital for $f'/g'$ to imply the limit of $f/g$. 'Undefined' doesn't have that property. All that bullshit that infinity is not a number is only being said by people that are not mathematicians. They don't know what they are talking about. They are numbers, you can compute with them just fine. They have slightly different rules, but so does $0$. — user647486, Apr 12 '19 at 14:58
Your title asks about dividing infinity by a number, then in your question you start out asking about dividing eleven by something, then you go back to dividing something by eleven. Dividing something by eleven and dividing eleven by something are very different, and you need to more clearly distinguish between the two. — Acccumulation, Apr 12 '19 at 16:05
@YvesDaoust What about non-standard models of the Peano axioms? — user76284, Apr 18 '19 at 19:06
You said "the natural numbers are defined for instance by the Peano's axioms", but some of the models satisfying Peano's axioms contain "infinite elements" beyond the initial segment that is isomorphic to the standard naturals. — user76284, Apr 18 '19 at 19:17
@user76284: Standardness is an illusion. If ZFC is consistent, then ZFC has a model whose naturals are non-standard. So how do you know your naturals are standard? Of course you can't; everything is relative to the meta-system. — user21820, May 13 '19 at 05:55
@user21820 I get your point, but it seems to me that, when people think of natural numbers, they think of something that satisfies, say, Goodstein's theorem or the strengthened finite Ramsey theorem, which are independent of PA. So I think it's wrong to say PA defines natural numbers. "Describes" might be a better choice. — user76284, May 13 '19 at 16:15
@user76284: You are absolutely right about that, and it's worth expanding that remark of yours. Specifically, for ordinary mathematicians the natural numbers are not only a structure that satisfies PA but also the minimal such structure, which is the term model ${0,1,1+1,1+1+1,\cdots}$. This notion of course is relative to the meta-system MS, but at least from the viewpoint of MS this minimality uniquely defines the naturals. Of course, my comment was that this is still an illusion because MS itself has non-ω-models if it is consistent, but yes in a sense people think their MS is 'correct'. — user21820, May 15 '19 at 02:47

score 14 · Answer 1 · answered Apr 12 '19 at 14:55

Thinking and playing with infinity is a popular thing to do, especially for people your age. To do so correctly however, you need to understand what we actually mean by infinity and what we mean by a "number."

Note: For the remainder of this post, by "division" and "divided by" I will be referring to the usual division operator rather than the related concepts from number theory.

Infinity is not a "real number" (that is not to say it doesn't exist in some contexts, but it does not belong to the set of numbers known as 'the real numbers') so we are not even allowed to refer to operations involving infinity. It just flat out doesn't make sense in this context. There is no largest "real number" and so there is no "highest (real) number" that $11$ can be divided by.

Since you are talking about "dividing by infinity" then you are probably working in the extended real numbers rather than the real numbers. In such a context, yes infinity is the "largest" extended real number and $11$ can indeed be divided by it. In fact, every real number when treated as an extended real number instead can be divided by infinty and the result would be zero. In that context, yes... you are correct.

The extended real numbers however is not the usual context to be working in. If the question was asked "does there exist a largest number that $11$ can be divided by" without any additional clarification about context then the answer would be no.

(Note: infinity and negative infinity can not be divided by infinity)

score 6 · Answer 2 · answered Apr 12 '19 at 15:03

This was getting too long for a comment, but it isn't really an answer - more an encouragement to hang in there with the kind of question you are asking.

The question "what is a number?" has caused many mathematicians to reflect on the basic material they work with and how it is defined. It leads to rich and fundamental questions about the foundation of mathematics.

Combined with geometric insights we can extend familiar numbers/systems to obtain projective spaces or the Riemann Sphere, both of which accommodate notions of infinity.

Algebraically it leads to reflection on what properties we expect our numbers to have, and the kinds of mathematical objects which have those properties.

And you might want to look up Cantor's diagonal argument and reflection on how infinite sets don't always behave in accord with our intuition. (An infinite set can be put in one-to-one correspondence with a proper subset of itself - we can match the integers with the even integers by pairing $n$ with $2n$)

It is possible to create systems of numbers in which the existence of infinite numbers is consistent with arithmetical operations - JH Conway's "Surreal Numbers" are an example.

To return a bit to the question:

If you are working with non-negative integers, then there is no infinite integer, and you can't divide by something which doesn't exist.

If you were working within the rational numbers (fractions), you could divide by any rational number (including any integer) except for zero. That would give you a valid fraction as an answer.

The non-negative integers and the rational numbers are systems of numbers which are carefully constructed to obey clearly stated rules - the need for careful definition arises because of the kind of question you are asking.

+1. Your answer has the flavor of an encouragement. If you want to encourage the asker then you may also consider "liking" the question. [I am always amazed when several people give very long answers to a question that they do not "like." So far, I am the only person who has "liked" (+1d) the question.] — Michael, Apr 12 '19 at 15:36
PS: In the comment trail of the question, my son wrote his comment using my account name "Michael." I encouraged him to look at the question because I thought he would appreciate it. — Michael, Apr 12 '19 at 15:56

Peter Foreman · Answer 3 · 2019-04-12T18:20:45.140

5

The number $11$ can be divided by anything - real or complex. If you mean what is the greatest integer - $x$ - such that $$11\equiv 0\mod{x}$$ In other words, $11$ divided by $x$ leaves no remainder or fractional part, then the solution would be $x=11$.

edited Apr 12 '19 at 18:20

answered Apr 12 '19 at 14:41

Peter Foreman

19,947

4

small suggestion : you may want to rewrite 11 ≡ 0 mod x, OP may not understand that notation (at least I wouldn't on 9th grade) – Rod Apr 12 '19 at 17:18

score 1 · Answer 4 · edited Apr 12 '19 at 16:01

It all depends on the defintions of $\infty$, "number" and "divides". I will assume that you're asked what is the greatest integer number dividing 11 then basically in any sensible interpretation of words $\infty$, "integer number" and "divides" the answer to your question will be "11". As far as I know, $\infty$ is always interpreted distinct from the usual numbers (by usual I mean real numbers) (including integers), that is, you have an axiom (or a theorem)

$$\forall x \in \mathbb{Z} , x \neq \infty.$$ Hence $\infty$ could not possibly be the greatest integer number dividing 11 since it then would be an integer contradicting this axiom (or a theorem, whatever you wanna call it)

Joshua · Answer 5 · 2019-04-12T17:30:44.790

Julia thinks ∞ is evenly divisible by everything. Julia got ∞ by Π ℙ = ∞ (that is, the product of all prime numbers).

Anna thinks ∞ is evenly divisible by only by 1. Anna got ∞ by Π ℙ + 1 = ∞ (that is, one more than the product of all prime numbers).

∞ is not a number. Trying to treat it like one will eventually cause a headache.

(If you're having trouble getting the reference, there was this famous question I cannot find anymore that had five students giving five different answers to a "what is the next number in this sequence" question using five different fourth order polynomials.)

score -1 · Answer 6 · answered Apr 12 '19 at 18:12

-1

It is really great to be thinking of stuff like this.

You will find people all around you who will say infinity is not a number and they will stop there, but there are a set of math classes in your future (Calculus) that are all about dealing with numbers that are virtually infinite--they call them limits.

All those people who say that it's absolutely impossible to deal with infinity will find it perfectly fine to deal with "x as x approaches infinity" and you get the same results you would intuit for infinity (Division by infinity is still a bit tricky if I recall correctly)

Anyway, if you have ever noticed that math is often about describing things, calculus is how you describe strange (arbitrary) curves and such--Exactly how much water fits into your strangely shaped sink, or how long will it take to drain said water since the speed it drains is slower as there is less water in the sink.

This kind of thing is calculated by slicing up curves into infinitely (Approaching infinite anyway) thin slices and adding the slices together to get an exact result.

(PS, I haven't done much calculus in decades, if this is factually wrong please feel free to fix it rather than just blindly downvoting. Thanks)

answered Apr 12 '19 at 18:12

Bill K

137

I don't think it's a great idea to compare limits to infinity. Limits are (usually) real numbers with fairly easily definable properties, unlike infinity. – Jam Apr 16 '19 at 21:27
That's what confuses me. Why are people so reluctant to compare the two. How is a limit as it goes to infinity unable to simulate what would happen if you actually were able to act on infinity? Intuitively the results of limit calculations always seemed to match what would happen if you were able to drop infinity in it's place. There is always such a violent reaction to the concept of comparing them and it confuses the heck out of me why. Maybe I need to post a question about it :) – Bill K Apr 17 '19 at 16:52
@Bill_K Fwiw, my previous comment and downvote weren't intended as a violent response, simply an opinion of the pedagogical value of your answer OP's question. Downvotes are a dime a dozen on this site so please don't feel discouraged or devalued :). Per your question, there's a few different types of 'infinity' that are often muddled together. The first (and probably easiest) type of infinity is the infinity of calculus, which represents boundless processes and sequences. For instance limn→∞ represents what happens to the sequence x1,x2,x3,…,xn as n gets larger (1/4). – Jam Apr 17 '19 at 18:38
(cont.) This 'boundless sequence' infinity is the same one that is represented by $\int^\infty$, $\sum^\infty$ and in derivatives. which are a essentially special case of limits with $\lim_{n→\infty}\frac{f(x+1/n)−f(x)}{1/n}$. Clearly we can't do arithmetic on this infinity, since it's a process, not a value. Let's consider the second type of infinity, ostensibly represented by your definition $\lim_{n→\infty}x_n=L$. When we pick apart what this means with a rigorous definition of a limit, we see that it means that (2/4) – Jam Apr 17 '19 at 18:40
(cont.) We can make the terms $x_n$ as close as we want to $L$, as long as we go beyond a sufficiently high $n$. This is due to the definition of a limit. Let's assume (falsely) that at some point, our $x_n$ get close to a limit $L=\infty$. Then (again by definition), we should be able to get even closer to $L=\infty$ if we continue further into the sequence. But how can we be closer to $\infty$? Is $10$ closer to $\infty$ than $100$ is? No - any number we pick will be infinitely far away from $\infty$. Hence we've contradicted ourselves, so we can't get "closer" to $\infty$ (3/4) – Jam Apr 17 '19 at 18:41
(cont.) and we have no limit to the sequence, so we can't construct $\infty$ that way. Hence, I wouldn't compare $\infty$ with limits. Limits are what we hone in on, as a sequence keeps going. If we don't hone in, then there's no limit. When we use the shorthand $\lim_{x\to\infty}=\infty$, this does not mean the $\infty$ on the right-hand-side exists, it means means that for whatever $x$ we stop at, there's always an $x+1$ beyond it. However, we can construct infinities in other ways: e.g. https://en.wikipedia.org/wiki/Projective_line. (4/4) – Jam Apr 17 '19 at 18:56
@Jam I just found this summary in a very simplified text:"Why do we need limits? Math has “black hole” scenarios (dividing by zero, going to infinity), and limits give us an estimate when we can’t compute a result directly". This is all I meant to say, Calculus covers these impossible situations by simulation--it is weird that I get such strong reaction when I suggest that it lets you interact with numbers that are "Virtually infinite", Am I just phrasing something wrong? Maybe simulated would be better than virtual? https://betterexplained.com/articles/an-intuitive-introduction-to-limits/ – Bill K Apr 17 '19 at 20:29
The terminology of 'virtually' isn't the issue. The issue is that infinity isn't a number that you can construct with limits, as I've said in my previous comments. The real answer to $\lim_{x\to\infty}x$ is 'does not exist', not a number called $\infty$. As I've said before, the sequence $1,2,3,4...$ doesn't 'hone in' on anything in the same way that $0.9, 0.99, 0.999...$ does. The second sequence has a limit but the first one doesn't. The correct way of constructing infinity is not with limits. Instead of thinking of limits as 'simulations' or 'estimations', think of them as destinations. – Jam Apr 17 '19 at 20:44
I think I see what you are saying. It doesn't let you interact with a "Bare" infinity because that's absolutely pointless--but it might let you divide by ~infinity. Perhaps it's my use of the unqualified "Infinity". I guess a more correct statement might be that calculus lets you simulate interacting with Infinity and divide by zero in the cases where those operations would lead to sane (Real number)? results. – Bill K Apr 17 '19 at 20:59
Let us continue this discussion in chat. – Jam Apr 17 '19 at 21:09

Can infinity be divided by anything?

6 Answers6