# The concept of infinity for a 5 year old

My son, who just turned 5, has been interested in the concept of infinity since long. He asks me a lot of questions regarding infinity. For example, not accepting my infinity + any number = infinity, he asked me how old I will be when he himself becomes infinity years old.

How should I explain to him this concept that resonates with his existing understanding of mathematics. If it matters, he knows how to add and subtract very large numbers, knows about negative numbers and has figured out tables of any number less than 100.

UPDATE: He derived one possible answer a few minutes after I typed the question on this forum. So I have written it as an answer below instead of a comment. Along with many other useful answers and comments here, this will be helpful to future questioners searching for the same to satisfy their children. Furthermore, it would be interesting for some to look at the questioning child's own thinking process.

• he asked me how old I will be when he himself becomes infinity years old. This might be a perfect opportunity to talk about your (and his) mortality. Feb 13 '19 at 2:22
• Let me also encourage you to continue giving deep, correct answers to your son. Mine is 12 now and I have always tried to answer all questions he had to the best of my knowledge, without undue simplifications. This often was, of course, a learning experience for both of us, and thus a great common experience. Many advanced concepts are more accessible to children than one would think, sometimes more than to adults who have to overcome wrong preconceptions (like, "the electron is a round solid ball with a charge"). Feb 13 '19 at 12:50
• "infinity + any number = infinity" is blatant fallacy and sure to cause undue confusion. It's the same problem with saying "Infinity is the highest number there is", it's a gross misrepresentation of the concept. Feb 13 '19 at 16:42
• @NickC As important as understanding mortality is, I'm not sure the best response to "I don't understand math" is "Not to worry, Son. We're all going to die, anyway."
– Ray
Feb 13 '19 at 16:48
• My kid(4) asked my kid(6) to count down while she completed a task: kid6: "10, 9 .." Kid4: "No no. Start higher!" Kid6: "20, 19.." Kid4: "No. Start at the higest number!" Kid6: "Infinity, infinity, infinity . . " Feb 14 '19 at 22:34

This does not directly concern the $$\infty+1=\infty$$ issue and I am not certain that I understand what you mean by his previous understanding of mathematics, but I wanted to give the following suggestion:

1. Ask your child to name the biggest number he knows (besides $$\infty$$). (Let's say he answers $$1000$$);
2. Tell him to add $$1$$ to it;
3. Ask him again what is the biggest number he knows. (It should be $$1001$$).

Repeat the process a few times and he should realize at some point that he can do this indefinitely. He can just keep on adding $$1$$ for free. It doesn't matter if he can't name the numbers eventually, as long as he understands that the next number is one more than the previous one.

While this does not necessarily show the various types of infinities that might exist, I think the idea that you can "keep on going" is a fair definition of infinity for a 5 year-old. It's not too hard to understand and it illustrates that infinity is not a number like $$1$$, $$2$$ and $$3$$, but rather an idea: infinity is what you get if you keep on going forever. It is certainly better than the belief I had as a child that infinity was the biggest number; there's no such thing as the biggest if you can keep on adding $$1$$. I hope this helps in some way!

Edit: As requested, I should mention that I haven't had the chance to test this with a 5 year-old, but it did work with teenagers (12-16) who had the same question (what is infinity?), as they seemed satisfied with the answer.

I also reiterate that this approach does not treat all types of infinity and should sound quite incomplete to a mathematician. However, there must be some limit to what we can and can't explain to a 5 year-old without sacrificing rigor and without "burning" them out. (Also, they'll have the opportunity to improve their understanding of the concept as they grow up). This particular approach seems to me to be "viable" for a 5 year-old (especially one who "knows negative numbers" and "has figured out tables of any number less than 100", like OP's child).

In particular, this approach involves the child in his own learning: he should be the one to realize on his own, inductively, what infinity is. This is much more convincing than "being told" what infinity is and should help with the "resistance" issue OP had.

• To me as a programmer $\infty+1=\infty$ results in a type error because + is not defined for a left hand operand of $\infty$ and a right hand operand of a natural number; or if it is defined, it is overloaded for these types and does something different than the other overloads. The important thing is that $\infty$ is, from my programmer's perspective, "not a number" (literally a NaN ;-) ), and you cannot naively use it like a number. Feb 13 '19 at 11:26
• @Peter A. Schneider: Exactly what I think also. Thus, the issue is not "what is $\infty + 1,$" but rather (if the questioner persists along this line of reasoning) "what might be a reasonable way to define what we mean by adding $1$ and $\infty$". For example, we can talk about mixing colors, such as mixing red paint and blue paint, and thus think of this as adding "red" and "blue", but what might we mean by adding "red" and $1$"? Is this even a useful path of inquiry? FYI, the notion of addition of cardinal numbers is simply one way of going about this, not THE way (as useful as it is). Feb 13 '19 at 11:59
• In IEEE floating-point arithmetic, $\infty+1=\infty$. :) Feb 13 '19 at 13:18
• @PeterA.Schneider: I guess you don't work with floating-point very often? std::numeric_limits<double>::infinity() is a perfectly valid double in C++, or C +INFINITY. You only get a NaN if you do inf - inf, or inf / inf. inf + 1, inf + inf, inf - 1 are all not errors and give you inf. 1.0/inf evaluates to 0. (godbolt.org/z/oPWwzc) This is somewhat questionable, but it's considered useful to make stuff like 1/(1/x + 1/y) "work" even for x=0 or overflow in the sum. Feb 13 '19 at 13:48
• I appreciate the comments relative to the programming point of view; it hadn't come to mind when I wrote the post. I would just like to reiterate that the presentation I proposed does not take into account everything about infinity. (There are many types of infinities, and ways to think about it; a 5 year-old does not need to know them all. I chose one I thought was simple enough). I may actually have presented the induction principle, rather than infinity, but I think it can give a glimpse of what infinity "looks like". Feb 13 '19 at 15:23

First of all, regardless of age, people need to understand that "infinity" is not a number, and not a placeholder for a number, but an attribute of them (i.e. the fact that you can increase numbers without ever getting to an end).

For my children, the concept somehow came into their mind all alone due to the book "Guess How Much I Love You" by Sam McBratney. It plays with ever bigger numbers, and the kids can easily increase the distances used in the book on their own as soon as they learn that, for example, the sun is farther away than the moon, that stars are even farther, etc.. They had to increas their number, because otherwise I would increase mine, and "win" the contest laid out in the book.

At some age (I cannot remember if it was 5 or older), the kids figured out that they can make numbers ever larger - even larger than in the book - by adding or multiplying (doubling as in "there and back again"), or whatever operation they learned in school.

At that point, the concept of infinity seems to be represented by - literally - "in-finite", or "non-ending". I.e., they understand that there is no end to numbers, that you can go on and on adding ever more of them.

• I had the same discussion with my daughter about this book. However, at one point, she said that she loved me as much as everything that exists. This was kind of the end of it. You cannot really go +1 beyond everything that exists, or can you maybe? Feb 13 '19 at 12:34
• We stuck with distances, and got way past sleeping time whenever we really fought it out (the "... and back again" cop-out can be multiplied ad nauseam). :-) @Trilarion
– AnoE
Feb 13 '19 at 12:38
• @Trilarion Physically, likely no. Mathematically, sure. Feb 13 '19 at 15:47
• Infinity does seem to sometimes be used as a placeholder for a number, like in the notation for a sum to infinity.
– caf
Feb 15 '19 at 13:30
• >that "infinity" is not a number. In the extended arithmetic, it is. Feb 17 '19 at 22:05

On a piece of paper, he started with writing 10, then 100, then 1000, .... and he stopped after writing 40 zeros with 1. Then he came to me and said, "I understand infinity now; infinity is a number with infinite zeros."

The main point is that as most of you suggested, he has now registered infinity in his brain as a concept rather than a number, which is why he used the expression 'a number with infinite zeros'.

• This is seriously one of the best things I've seen on stack exchange :D
– Esco
Feb 17 '19 at 12:10

I'm not sure why the two basic things adults seem to say about infinity are "infinity is not a number" and "∞+1=∞", both of which are at best misleading. (Infinity doesn't name a number, but it does refer to a property some numbers can have. ∞+1 is nonsense, $$\aleph_0+1=\aleph_0$$, and $$\omega+1\neq\omega$$.)

The problem with talking about infinity with small children is that it's an imprecise term that covers multiple precise ideas that behave differently. Children that age aren't ready for that, so I don't think there are any particular concepts about infinity that it's useful to convey. I certainly don't think there's any reason to prioritize explaining the non-fact that ∞+1=∞ over other non-facts (say, that ∞+1>∞). Rather, I think the goal has to be to enjoy playing around with the concepts.

Your son's question - how old you'll be when he turns infinity - is a really good question, and points to the difficulty of dealing with the question: it's not at all clear what the right number system to discuss that in is.

• @Tim: Impossibility is not the same as meaninglessness; the fact that it would happen in the real world doesn't stop us from considering number systems in which it is sensible. But my point was that his son was making the right move, away from something underspecified ("what is addition with infinity") and towards something closer to being sufficiently specified. Feb 16 '19 at 18:33
• Actually, $\infty+1=\infty$ is not nonsense. It's not that $\infty$ is an umbrella term that covers both $\aleph_0$ and $\omega$, and one satisfies $x+1=x$ but the other doesn't. In fact $\infty$ represents a third concept, which unlike both of the above examples solves $2^x=x$. It also has an additive inverse $-\infty\ne\infty$. See here. (Oh, and since things get ever more complicated, see this point at infinity, which instead satisfies $-\infty=\infty$.)
– J.G.
Feb 17 '19 at 21:52
• @J.G. You are correct that the there are other notions of infinity besides the cardinals and ordinals, and that one of those happens to also use the symbol ∞. (Confusingly, your first link is to the wrong topic - the extended real line, which uses +∞ and -∞, not the unadorned ∞ symbol.) However context establishes that I was using ∞ in the conventional way as the undifferentiated infinity, for which "∞+1=∞" is nonsense; the fact that there are subfields which do use that notation does not cause ∞ to exclusively refer to that subfield. Feb 18 '19 at 3:04
• @HenryTowsner My link was fine: $\infty$ is an abbreviation for $+\infty$.
– J.G.
Feb 18 '19 at 6:02
• Son, there is this thing called ℵ0 you need to understand first... Feb 19 '19 at 6:37

Speaking as someone who was that kid, you might be able to explain $$\infty + 1 = \infty$$ via the Hilbert hotel.

Imagine a hotel that has an infinite number of rooms, one for every number. Imagine the hotel's full, and another guest shows up. You can make room for that guest by having the guest in room 1 move to room 2, the guest in room 2 move to room 3, and so on. So even though it's full, you can always fit more guests in, so you can always add one.

He's also not totally wrong to insist that $$\infty + 1 \neq \infty$$. Infinity isn't really number, but an idea, that you can apply in different ways. There are different types of infinite numbers, with different rules, and in some of them, he'd be quite right that $$\infty + 1 \neq \infty$$. The hotel doesn't have a room $$\infty$$, but if it did, the room next to it would be room $$\infty + 1$$, and would be a different room.

Five might be a bit young to understand the idea that different rules lead to different maths, but he can probably grasp the idea the idea that there are different types of infinite numbers, and it's good to tell him that he's not wrong.

• Young children will find this neat, but of course from a strict logical point of view this idea assumes that $\infty$ has meaning (or at least some of the meanings) we attach to the idea of a cardinal number. If we're considering ordinal numbers, then $\infty + 1 \neq \infty.$ And, of course, there are probably other ways of understanding the notion/symbol $\infty$ that result in different interpretations of what $\infty + 1$ could mean. I'm not saying that the different interpretations are useful or have even been studied. My point is that you need an interpretation before going anywhere. Feb 13 '19 at 12:10
• @DaveLRenfro Yes, I think I was trying to convey that. The more I think about what I wrote, the more I realise that the key point is that the rules of maths aren't set in stone, and there are different types of infinte numbers with different rules and different interpretations. I'll see if I can reword it to change the emphasis. Feb 14 '19 at 12:02
• At youtube.com/watch?v=Uj3_KqkI9Zo there's a TED-ed animation of the hotel. Feb 14 '19 at 16:14
• And there's a lovely picture book that uses the Hilbert Hotel idea: The Cat in Numberland by Ivar Ekeland. (bookfinder.com/search/…) Feb 17 '19 at 5:27

My son, also 6 yo, regularly talks about millions and billions and infinity. Obviously, large numbers have some attraction to children of this age.

I try to explain that infinity is not a number. Instead, infinity is an order of magnitude which has its own algebraic rules. Plus, minus, divison and multiplication do not work the way children learn in elementary school when applied to infinity.

My first explanation is that this has also an impact on how we use the words infinity and infinite:

Three meters. (works)

*Infinity meters. (completely wrong)

*Infinite meters. (sounds wrong)

Infinitely many meters. (works)

Another approach is that the concept of numbers does not work. Numbers grow. For every number, there is another number that is larger. The mathematical notation to this concept is $$\forall n \in \mathbb{N}: \exists m \in \mathbb{N}: m > n$$. If infinity was a number, then this statement would be false, because let $$n=\infty$$, then $$\infty + m > \infty$$ is false for all $$m \in \mathbb{N}$$ where $$m > 0$$. Surprisingly, children who already learnt addition upto, say, 100, understand this. They understand that 100 is not the largest of all numbers, neither is 1000, neither is a million, and so on. But the fact, that addition does not alter the "number", makes them understand that infinity is not a number.

In words that are better suited for children, you can also say: Adding any number to infinity does not change its size, because infinity expresses a magnitude, a size, rather than a number.

Admittedly, my daughter, 8 yo, understands this point better, because my 6 yo son has not yet learnt addition of numbers larger than his 10 fingers provide.

• The Phantom Tollbooth by Norton Juster.

The book is a rather surreal adventure trip through a Wonderland-style setting populated by mad grammarians and mathemagical wizards (among others). There is a section somewhere in the middle where the protagonist is sent on a quest to infinity, which is described in a couple of ways. The concept of infinity discussed here is, if I recall correctly, more akin to the infinity that occurs when you compactify the the real numbers, so it isn't quite the same idea as $$\infty + 1 = \infty$$.

In any event, the book is quite a lot of fun. Even if you aren't overly concerned with infinity, it is worth reading, and should be at about the level of a 5 year old (maybe a little advanced? I seem to recall having it read to me when I was in first or second grade, so maybe just a little older?).

• The Cat in Numberland by Ivar Ekeland.

This book might be a little advanced for a 5 year old, but maybe not–I think that it is intended for 3rd or 4th graders, but might be accessible with the help of a parent. In any event, the book deals with infinity as a cardinal. There is the basic "a new number arrives, everyone moves up a room" example, but my recollection is that the correspondence between rational numbers and the natural numbers is discussed, as well. I had difficulty getting a copy of the book several years ago, but it appears that it might be back in print(?).

• Just to add on - I'd also recommend the Number Devil, by Hans Magnus. Also may be too advanced (it's aimed at elementary school students, as I recall, and I think I read it when I was 7) but it has some excellent stuff on infinity, including a very good treatment of Hilbert's Hotel-style ideas. Feb 13 '19 at 15:41
• I haven't read the Number Devil, though it is one that I have seen recommended several times before. I should probably pick up a copy of it... Feb 13 '19 at 15:41
• I literally came to this page from HNQ, did Ctrl+F for Phantom Tollbooth, and happily upvoted this. Note there is a quite well-done cartoon movie for it, as well. I enjoyed that quite a bit as a kid. Feb 15 '19 at 2:41
• My favorite line from The Phantom Tollbooth is the directions on how to reach the land of Infinity: "Go down this hallway forever, then turn left..." Feb 18 '19 at 5:48

There is a well-known Christian hymn, Amazing Grace, whose last lyric captures the idea of (countable) infinity quite well, and may be more effective to a five year old because it includes a context in which the notion of infinity can be applied.

The lyrics goes:

When we’ve been there ten thousand years,

Bright shining as the sun,

We’ve no less days to sing God’s praise

Than when we'd first begun.

Obviously, this brings issues of religion into the matter. If you don't want a Christian hymn, you might rephrase the lyric in the framework of another religion or atheistically. If the child can grasp the notion of an infinite number of future years, then this lyric is "explaining" that $$10,000 + \infty = \infty$$.

• That's a good example. Though theologically I'd want to point out that it treats eternity as an infinitely long time dimension rather than as being outside of time and space . . . (Which makes it good mathematically, though.) Feb 13 '19 at 18:28

I would start by saying something along the following lines...

"You're asking some very grown-up questions for someone that's only 5. Are you ready to do some really, really, grown-up thinking about the answers?"

He will of course, answer, 'yes'. I would respond...

'Ok, but this is serious stuff. You need to be ready to take this thinking very seriously.' Only when you really have his attention, do all the 'infinity is not a number' stuff, and all the other things that people have suggested.

A 5-year old, thinking about infinity, if he's really thinking about infinity, is a very smart kid indeed. He is probably smart enough to confront the idea of types of thinking beyond what he's encountered.

But he doesn't yet realise how he needs to change gear. Teach him that, before you teach him about infinity, and 20 years from now, he'll thank you for all the other things that enabled him to do.

• How has this approach worked for you? Feb 14 '19 at 10:45

I suppose one problem is that your son looks at $$\infty$$ the same way he looks at $$10$$. But infinity is not a natural or real number, even though it has a symbol and can be used in "equations" like $$\infty + 1 = \infty$$. These equations do not have the same meaning and do not follow the same rules as with "normal" numbers — the reason being that at least one operand isn't one. Making clear that infinity is not a single number but is used to describe the unboundedness of number sequences should go a long way towards understanding and is totally within reach to a 5 year old.

Let me indulge in my computer science perspective. (I suppose it is correct in purely mathematical ways as well. Please correct me if that is not so.) How do we use "infinity" in math? For example we say that the result of some sum is infinite: $$\sum_{i=1}^\infty{f(i)} = \infty$$ for some $$f$$. Similarly for some integral, or simply the value of some function $$f(x)$$ when $$x$$ approaches a certain value. The essence is that we make statements of the outcome of procedures. $$\infty$$ is not a static "value"; it is a statement of what happens to a value which is computed procedurally, provided we never stop. Specifically, it is the statement that we cannot name a limit that this value will not exceed. In other words, infinity can be considered the opposite of a static, fixed value.

Making this crucial distinction will go a long way explaining why $$\infty + 1 = \infty$$ holds (even if I don't like writing it at all, as mentioned in a comment elsewhere in this thread): If I cannot put an upper threshold to a result, incrementing the result will still not yield an upper threshold, and that's all that $$\infty$$ means.

• @PeterCordes My point is that $\infty$ is emphatically not a value ;-). It is a statement about the properties of a series of values. Feb 13 '19 at 16:01
• @PeterCordes Not necessarily. As Henry Towsner explains in his answer, we need to define what infinity we're talking about before we can say whether those are equal. If it's an ordinal, then you can add one to it and get a different infinite ordinal. If it's a cardinal, you can add an element to the set that it's the cardinality of and get the exact same cardinal infinite.
– Ray
Feb 13 '19 at 16:12
• @TommiBrander What is formally correct should solve the understanding issues ;-). I specifically address the problems of the child mentioned in the question: "not accepting my infinity + any number = infinity". I could make that clearer. Feb 14 '19 at 10:45
• Unfortunately, formal correctness and understanding do not always go hand in hand in mathematics. Feb 14 '19 at 10:48
• @TommiBrander If you think of formal correctness not in terms of notation and other formalisms; but instead in terms of the essence of the used constructs and relations; then I believe the opposite is true. Formal correctness (at least, the absence of formal incorrectness, like "infinity is like 10") is a necessary prerequisite for understanding. Infinity and related concepts were for most of history not properly understood by the brightest minds on earth, until strict correctness was achieved. Often that correct view is simpler than the muddled and unclear previous attempts. Feb 14 '19 at 10:58

My children both learned about infinity at around four to five years old (now 5 and 7). For both of them it was fairly straightforward; it came about with my eldest when he was talking to other kids at school about the biggest number. We talked about trillion, quadrillion, etc.; as they were at a Montessori, it was easy to understand these. Then we talked about googol, and googolplex. Then we talked about a few other numbers, like Graham's number, and the idea of other extremely large numbers.

Finally, we talked about infinity. Because we were talking in the context of very large numbers, the first thing we learned - before anything else - was that infinity is a concept, not a number. A way of thinking about extremely, impossibly large numbers, without actually naming one. This brought a little confusion, until we went through the thought exercise of: "What's the largest number. Okay, add one to it." But ultimately they got the idea of 'concept' pretty easily.

Now, at 5 (almost 6) and 7, they get some of the other ideas pretty easily - like 1/0 approaches infinity, infinty/n is infinity, but 0/0 = undefined. Teaching infinity as a concept made it easy for them to understand these are basically just rules to follow, and that it's not the same as a number.

Of course, I had a lot of sympathy that week for the poor preschool teacher who had to deal with the arguments 'infinty is the biggest number' 'no, it's a concept, not a number' between my children and the other children...

• I'm not particularly familiar with this site's standards, so perhaps this isn't in line with what you expect (sorry if so), but typically showing how one taught something to a similar child is an answer to how should I teach my child?
– Joe
Feb 13 '19 at 16:55
• It's an answer. It shares an approach that's been actually used with actual children, and the results it had. Not all the answers do this (mine included). Feb 13 '19 at 18:44
• $1/0=\infty$ is not even undefined. It's sheer nonsense. At best, it leads to chaos, or trivia. Feb 14 '19 at 17:53
• @TommiBrander Yeah, fixed that, thanks for the correction. Everyone else, I think teaching limits is a bit over 5yo level, though I do make sure it's clear that things like 1/0 aren't actually infinity, because infinity is not a number, but instead a placeholder for convenience. And the +/- infinity ... will leave that out for now too.
– Joe
Feb 14 '19 at 18:17

I've no idea whether this would work, but would relating it to forever hekp? Infinity is like forever but for nunbers. Doing something for a week and then forever is the same as just doing it forever. When is forever? That's when you'd be "infinity years old", but there isn't a when because forever means "never stop" . . .

So if you're travelling to infinity, which is travelling forever, when do you get there? When you stop. When do you stop? Never! Because, forever says never stop. But maybe prepare for questions about whether infinity really exists . . .

I think forever a much more familiar concept to a five-year-old than infinity, so maybe you can start from that.

• PS I think I was about $8$ when my father explained involute gears to me in terms of a piece of string with a knot in it, and I followed that, so I don't think you should give up just because an idea is quite advanced. Though I was $8$, not $5$. Feb 13 '19 at 15:16

Before trying to explain $$\infty + 1$$, it'll help if he has an intuitive grasp of what infinity means. I recommend using the cardinals rather than the ordinals, because they can be constructed without needing to understand limits.

Below is a possible explanation that may help with that; I try to avoid using too much terminology, since he's 5, and in particular, when I say "number", I mean "non-negative integer" and when I say "infinity", I mean $$\aleph_0$$. You might mention briefly that when you say "numbers" here, you mean numbers like "3", but not numbers like "3 and a half".

Start with a set with just one number in it: $$\{1\}$$. We then add numbers to it one at a time, and we'll do it so that that the biggest number in the set is also how many numbers are in the set. So right now, there's $$1$$ number in the set, and $$1$$ is the biggest number in the set. Next, we'll add the number that's one bigger than the biggest number in the set. $$1+1=2$$, so now our set is $$\{1,2\}$$ and the biggest number in it is $$2$$. We continue as such. $$2+1=3$$, $$3+1=4$$, and so on. So for any number, we have the corresponding set containing every number from 1 to that one, and our number is how many numbers are in that set.

No matter how many numbers we add, there's always a biggest number in the set and that number is always how many numbers there are in the set. But there's also always a number that's one bigger than that one. So we can keep adding numbers to the set forever and never get all of them. There is no biggest number, and there are an unlimited number of numbers; no matter how many we have, we can always add one more. So if we want to ask "how many numbers are there altogether", we need a word for that that isn't a number. That's what infinity is.

So $$\infty + 1$$ doesn't mean anything, because $$\infty$$ just means "unlimited"; it's specifically the thing we can't get to by adding 1 to numbers we already have.

There are a few other kinds of infinity, but this is the easiest one to understand, so you want to make sure you really understand this kind before moving on to the others.

• This is basically matheducators.stackexchange.com/a/15215/667 with sets. I don't think that sets are a helpful concept for a 5-year-old (also, for almost no 5-year-old, "3 and a half" will be a number). Feb 13 '19 at 17:43
• @Jasper You can call them groups of numbers or collections of numbers if that helps. The idea I want to emphasize here is that we move from viewing infinity as a number (even one bigger than any other) to infinity as an answer to the question, "How many?", along with an understanding that there is no number that's bigger than every other number. I wouldn't have thought that infinity would be a helpful concept for a 5 year old, but if Qasim thinks he's up to it, this is the approach I'd suggest. (And perhaps he'll consider non-integers to be numbers once he's 5 and a half years old.)
– Ray
Feb 13 '19 at 18:21
• +1 especially for "numbers like "3", but not numbers like "3 and a half"" which is the right kind of "definition" for an audience below a certain age. Feb 13 '19 at 21:35
• @TommiBrander I've never tried explaining infinity to a 5 year old before, I'm afraid. Similar approaches (albeit ones using a much more technical vocabulary) have certainly worked when explaining infinity to young adults who only had a similar level of understanding as the OP's son seems to. I think this explanation is worth trying, but can offer no guarantees.
– Ray
Feb 14 '19 at 16:56
• Adding that into the post would improve it, I feel, since answers should be backed up by experience or reliable sources. Feb 15 '19 at 9:06

Have you considered geometry? Take two points and draw lines, one through each points.

• Two lines meeting at an obtuse angle meet "soon," that is a small number.
• Two lines meeting at an acute angle meet "far away," that is a large number.
• Two parallel lines meet "infinitely far away."

In a way that gets you to limits, which may be too complicated.

Note that this is an Alexandroff Extension or Riemann Sphere, depending on how you look at it. With slightly older students, the Riemann sphere is a good way to "visualize" infinity.

• Where the two lines meet, there will be both acute and obtuse angles. Just leave out the angle part. Feb 17 '19 at 5:37
• @SueVanHattum, I'm thinking of two points, two rulers or pencils. Or perhaps two points, one line on a graph paper, and a ruler. Change the angle from perpendicular to parallel and the angles from 90° to 0° become distances from zero to infinity.
– o.m.
Feb 17 '19 at 5:48
• Yep. Obtuse means greater than 90 degrees. I think you might want close for greater than 45 degrees and far for less than 45 degrees, perhaps. Feb 17 '19 at 20:38
• @SueVanHattum, what I had in mind when I wrote that was adjusting two sets of lines, not keping one constant. Starting with an isoceles triangle with an obtuse angle on top, then "stretching" it step by step.
– o.m.
Feb 18 '19 at 6:08
• How has this worked when you tried? Feb 18 '19 at 7:44

The thing is, saying that infinity + any number equals infinity is a bit imprecice.

There are two widely used concepts of infinity; they refer to cardinality and ordinality.

In finite numbers, Cardinality is the concept of "how many" of something there are -- 1 sheep, 2 sheep, 3 sheep. Ordinality is "what order" they come in -- 1st sheep, 2nd sheep, 3rd sheep.

With finite numbers they are highly tied to each other. You can just "count labels" in a sense.

With infinite sets the two concepts diverge.

We'll start with the natural numbers -- the set of all counting numbers. 0, 1, 2, 3 etc. That'll be our "first infinity".

If you take the first infinity, and add another element to it, you get the same cardinality. This is what people talk about when they say "infinity+1 equals infinity". More than that, if you take the first infinity, and double it, you get ... the same cardinality. You can even add an infinite number of infinities to it -- take "first infinity times first infinity" or "first infinity squared" -- and you get the same cardinality.

It isn't until you reach (assuming continuum hypothesis) 2^"first infinity" that you reach a new cardinal. This can either be described as the "set of all the first infinity" or "the set of functions that go from the first infinity to yes/no" (it shouldn't be hard to see they describe the same thing). This is a bigger cardinal than the first infinity, and is also the same cardinality as the real numbers.

The cardinality game continues from there.

So that is one branch.

The other is ordinals. In ordinals, we talk about ordering things. For any two things, you can say which is in front of the other in the order. And for any collection of things ordered, we can find the "least" element (the one "behind" all the others), including the entire collection of ordered elements (we normally call this element 0).

The "first infinity" in ordinals is ordering everything by the natural numbers. Everyone gets a tag that says "1st" or "1 million and 7th" or whatever.

Now, in ordinals, we can they have someone with the label "1st in 2nd lineup", and we can state that the 2nd lineup goes after every value in the first lineup. This is "infinity plus 1" in ordinals, and it is a distinctly different way of ordering people.

What more you can have 2 infinite lineups (where the 2nd goes after the first), or 3, or an infinite number of infinite lineups (where each lineup goes after the one before). These are all distinct ordinals -- they describe fundamentally different ways of ordering things.

And the game continues from there.

Now that I have disabused you of the notion that infinity+1 always equals infinity, how do we talk about it with a 5 year old?

You could talk about that split. Say "up to infinity the idea of ordering and counting is the same. At infinity they are different."

Then talk about infinite ordering and lineups.

The ordinal $$\omega$$ is a lineup that goes on forever. There is someone in front.

The ordinal $$\omega + 1$$ is two lineups. One that goes on forever, and one with a single person in it. That single person goes after the first lineup. It will get very boring for them.

The ordinal $$\omega +2$$ has 2 people in the second lineup.

The ordinal $$\omega + \omega = 2 \omega$$ has two infinitely long lineups. The second goes after the first.

The ordinal $$k \omega$$ has $$k$$ lineups, each infinitely long.

The ordinal $$\omega \omega = \omega^2$$ has an infinite number of lineups, each infinitely long.

The ordinal $$\omega^2 + 1$$ has an infinite number of lineups, each infinitely long, plus one person who gets to go after everyone else is done.

The ordinal $$\omega^2 + \omega$$ has an infinite number of lineups, each infinitely long, plus another lineup that goes after the previous infinite set of lineups are all done.

The ordinal $$2\omega^2$$ has a two collections, each with an infinite number of lineups, each infinitely long, with one going after the other.

The ordinal $$\omega \omega^2 = \omega^3$$ has an infinite number of collections, each with an infinite number of lineups, each infinitely long.

The ordinal $$\omega^\omega$$ has an infinitely long order of layers. In each layer, there is an infinite number of the next layer, all ordered.

You probably will break down before getting this far.

Also talk about cardinality. Here the other answers cover things really well -- things like the Hilbert Hotel and the Cat in Numberland are great resources.

A fun part of this is that every Cardinality has a whole bunch of Ordinalities associated with it. You can look at the stories, like Hilbert Hotel, and talk about how the Ordinality changed even when the Cardinality didn't.

And you can talk about how this doesn't work for "normal" numbers. You cannot change the fundamental ordinality without changing the cardinality.

Having two lines, one of 2 people, followed by a line of 3 people, is the ordinal 2+3, which is fundamentally the same as the ordinal 5. You can "just paste" the 3 people onto the end of the first line.

With infinite ordinalities, you cannot reach the end of the first line to paste the second line on. It is infinitely far away.

• How has this worked when you have tried? Feb 18 '19 at 7:45

If you are 24 years older than him now, the answer can be "I will be 24 years older than you."

• How did this explain infinity when you tried using it as an explanation? Feb 18 '19 at 7:45

Imagine you have a snack machine that contains infinite number of candies and gives one to you (for free) every time you press the button. So you have infinite number of candies.

Now your friend gives to you a candy of the same kind as a snack machine. Do you have more candies than you had before that? No as instead of your friend's candy you could just press a button and get it from there. The machine will continue to give them to you infinitly.

So infinity plus one is still infinity - no profit in extending infinite resource by the same thing from a finite source.

• How has this worked in your experience? Feb 18 '19 at 7:46

From what i know, infinity is a fun "number".

Let's say it takes one minute to climb on a rock. If you have a tower made of 100 rocks, it takes you 100 minutes to climb on top of it.

Now let's say you have a tower made of an infinite number of rocks. It's not just big, it goes higher that the sky, it goes higher than the sun, it never stops going higher than everything that isn't infinite too.

If you try to climb on top of it, you'll never achieve to, you'll keep climbing forever, and won't reach the top even after an infinite amount of time.

If you possess a tower made of an infinity of rocks, and someone gives you another rock, you are now possessing an infinity of rocks from the tower, plus the rock you just have been given. You possess infinity +1 rocks.

If you manage to put this new acquiered rock in the tower, it still takes as much time as before to reach its top by climbing on it: forever. The tower still is made of an infinite amount of rocks.

If you possess two towers, each made of an infinite amount of rocks, you possess two infinities of rocks. But if you somehow manage to put the two towers one "on top" of the other, it takes the same amount of time to reach the top of it as it is taking to climb on the former tower.

Now let's say you have an infinitely big hole, it's so big that you can put as many rock as you want in it, it will never get full.

Even if you put your infinitely big tower in it, there will still be room left for an infinite amount of rocks. Even if you put an infinite amount of infinite towers in it, it won't change that, there is still an infinite amount of room left for rocks.

If you share your infinite amount of rocks with poeple around you, you can give them rocks forever, they all get an infinite amount of rocks. Even if you share it with an infinite amount of poeple, you'll never stop giving them rocks.

If you build a square floor made of rocks, by putting an infinite straight line of rocks in front of you, and an infinite straight line of rocks aside every rock composing the first line, it's area will be infinity².

You can then take the rocks of your floor and build a tower with it, you will never stop building the tower, which means it would create an infinitely big tower. But the tower will take much more space than the floor, because it has an infinite amount of floors which areas are also infinity².

• How has this worked in your experience? Feb 18 '19 at 7:46
• Mostly, kid will try to make you add one more rock to the tower making you repeat that it still takes you the exact same "amount of time to reach the top" multiple times, before processing it. Once he gets that, the two towers "on top of each other" goes smoother. Infinite holes, infinite sharing, and infinite dimensions are most of the time near to impossible to understand for young childs, but it works for older ones. For clever ones, you sometimes have to explain the difference between an infinite hole and a hole that has the size of an infinite tower. Feb 19 '19 at 1:50
• About the infinite amount of time climbing not enough to reach the top, you might have to explain that the tower never stops, there is no top of the tower, and it's impossible to reach somewhere that doesn't even exist. Feb 19 '19 at 1:54
• Nice. Adding those details into the answer would improve it. Feb 19 '19 at 6:47

I don't think explaining the $$\infty + 1 = \infty$$ thing directly is very productive. It's meaning is pretty fuzzy, it is very counter-intuitive, and it doesn't give any new perspective by itself. If I wanted to explain how equality works different on infinities, I would rather try to show an example of how 2 infinities can appear both equal and unequal, at the same time. This might be e.g.:

• (natural) numbers vs even (natural) numbers. There is obviously the same amount of both, as every even number is just $$2n$$ for some $$n$$. But equally obvious facts is, that there are twice as many natural numbers as even numbers, because there is $$n+1$$ for each of them.

• (natural) numbers vs $$n+5$$ for each of them. Similar logic to the one above: there is obvious 1:1 relation between them, but also the first set has clearly 5 elements more.

This doesn't really prove anything like $$\infty + 1 = \infty$$, but it conveys a general fact of "equality for infinities is really strange, and definitely different than for numbers".

This is similar to the hotel example from a different answer, but I believe it is much better by having clearly separated sets. Instead of changing one set in some complicated way, we have two pretty simple things, lying next to each other, that we can compare in different ways.

I'm not sure whether this is a useful answer for 5-year-old. I guess other answers are much better in this regard. However, I myself struggled with these strange infinity eqalities, and examples like above were a key to get me to understand it, so I wanted to share. I believe they are the best way to explain where the "$$\infty + 1$$" paradox comes from.

• How has this worked when you have tried? Feb 18 '19 at 7:46
• I haven't really used that on anyone, but it was used on me. I struggled with this ∞+1 thing for quite a while, and seeing a similar example got me to finally accept it. I guess I was about the high school age than, so, as noted, I can't really tell how relevant it is for younger children.
– Frax
Feb 18 '19 at 21:08
• @TommiBrander I slightly modified the answer to make the intent clearer. I have to admit with some embarassment that it is still only borderline relevant. Yet I think this perpective (i.e. showing 2 sets that are both intuitively equal and unequal in size) was actually missing from other answers, so hopefully it will be useful for someone.
– Frax
Feb 18 '19 at 21:41

Short answer: I would say, infinity is larger than the largest number anyone can count.

• How has this worked in your experience to explain the concept to a young child? Feb 18 '19 at 7:46
• Never got a chance to do that. Certainly I think this is the right way to explain. A curious child would ask a follow up question to this answer, from which you can lead him up to the concept. Feb 18 '19 at 12:17
• Okay. Answers in this SE should be backed up by reliable sources or personal experiences, which explains the low score on your answer. We welcome your input on matters you do have experience with. Feb 18 '19 at 12:54
• @TommiBrander this is a new user, it would be best to help then understand the site and encourage them. Writing nearly the same phrase onto all of these answers isn't really helpful. They don't need to try it to write an answer, but I agree that explaining why they think it would work would improve the answer. Feb 19 '19 at 7:13
• Hi, welcome to Mathmatics.SE! Please check out the tour and help center page. This answer could be improved by explaining why you think this would be a good explanation. Feb 19 '19 at 7:13

Imagine a Steel ball the size of the earth. A fly lands on the surface once every hundred years. When the fly lands his feet wears away the surface of the steel ball by an amount that can only be described as next to nothing. When the whole of the Steel ball has been worn away to nothing, INFINITY has not even started.

• How has this worked in your experience to explain the concept to a young child? Feb 18 '19 at 7:47
• Hi, welcome to Mathmatics.SE! Please check out the tour and help center page. This answer could be improved by explaining why you think this would be a good explanation Feb 19 '19 at 7:11
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