# How to explain infinity to a $3^{rd}$ grader?

In my country in $3^{rd}$ grade in math kids learn the four basic arithmetic operation (addition, subtraction, multiplication and divison) up to $10 000$.

My sister this year goes to $3^{rd}$ grade and one day she was writing her homework in math and out of a sudden she asked me up to which number a know to add, subtract... I answered that I know to add, subtract... for every number and because there are infinite amount of numbers, I know to how to calculate up to infinity.

This concept of infinity was unclear to her. He couldn't go over the fact that there are infinite amount of integers, because she thinks that ultimately there must be a largest number, one that's bigger of all of them.

I told her that because there are infinity amount of numbers I can always say a greater number than one she can told. She start saying $600000, 1245000000, 99999999999$ and I easily just added $1$ and obviosuly that makes my number bigger, but still it didn't helped her. I thought that just adding $1$ to hers humber will make her feel that she's close to beating me, so I though to double the number she says, but again it came with no success, because she stubornly continued "fighting with windmills".

How can I exlpain the existance of infinity to a 10 years old kid?

• You may have done all that’s possible for now. You’ve prepared the ground; let it rest until she’s a bit older. Sep 30, 2013 at 19:26
• But the problem is that she almost every day, once od twice at days is telling me that she found even bigger number, but I easily beat her. This can help her, because she can become curious in maths and its beauty, but at the moment she seems to bothered by this problem and the fact that she can't understand this fact iritates her. Sep 30, 2013 at 19:37
• ‘I’m older and stronger, and I can stay awake longer, so I can keep adding numbers longer than you can!’ :-) Seriously, I think that you’ve done all that you can. I wouldn’t tell her outright that she’s too young to understand such abstract things; nobody likes to hear that. I’d simply try to suggest that she let it go for now and give the ideas a chance to sort themselves out. (And judging by your description, she probably won’t let it go right away.) Sep 30, 2013 at 19:55
• She has found that whatever number she thinks of, you can beat it, and whatever number you think of, she can beat it. That is a good thing to know. Why bring theology ("infinity") into the game? Sep 30, 2013 at 19:55
• While slightly off topic, this quote from Douglas Adams seems rather apt. "The car shot forward straight into the circle of light, and suddenly Arthur had a fairly clear idea of what infinity looked like. It wasn’t infinity in fact. Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity—distance is incomprehensible and therefore meaningless. The chamber into which the aircar emerged was anything but infinite, it was just very very very big, so big that it gave the impression of infinity far better than infinity itself." Sep 30, 2013 at 20:10

Bishop Berkeley then later Poincare, Kronecker, Weyl, and Wittgenstein had trouble understanding the infinite, so give the 10 year old a break.

• As I said in the upper comments I didn't even bother her, I don't even force her to be interested in maths, I just help her when she asks me, but the problem is that she constantly thinks that she can beat me. Sep 30, 2013 at 19:42
• If she doesn't understand that you can add 1 to anything she says, then she's not ready for any other explanation. Sep 30, 2013 at 19:43
• Maybe, it's best to tell her that's she's too young to understand such abstract things. Sep 30, 2013 at 19:48
• I wouldn't; nobody likes to hear that. Tell her this is a hard topic, that took mankind thousands of years to figure out, so it's okay if it takes her a long time too. Sep 30, 2013 at 19:51
• That's seems to be the best appoach, that's she'll get knowledge about this thing as the years come and maybe this will make the curiosity to last and who knows one day maybe she'll be interested in math and she'll finally understand this term. Sep 30, 2013 at 19:54

A simple way that may help is explaining that infinity is just a word to describe that the number line never ends. Numbers will continue to get higher and just when you think you know the highest number, there is one number higher than that one. At this point, a ten year old does not need a specific proof or theoretical definition, but most importantly need something to compare it to, so they can visualize it in their head.

• The problem is that infinity is abstract term so she can't quite understand it and I can't give her an everyday example that will convince that my statement is true. Sep 30, 2013 at 19:49
• Perhaps ask her to write down the largest number she can think of. Once she writes it down, ask her if you add the same amount to that number, wont it be a bigger number? Repeat this a few times until she becomes tired of adding such large numbers. Then explain that be adding on the same number repeatedly she will get a larger number and that this word "infinity" is meant to describe this process that numbers will continuously get larger. Sep 30, 2013 at 19:54

You can try to ask her to provide a disproof rather than doing it yourself. That is, if she comes and says "I've found the number $n$ which is larger than any number you've told me so far", then you could ask her "Do you think that is the biggest number possible or do you think I can come up with a bigger one?" If she says that she thinks it is the biggest number possible, press her, "Are you so sure that you will perform chore $x$ if I can think up a bigger number?"

If she is the one who has the responsibility of always producing a bigger number, she may grasp the infinitude of whole numbers more easily.

I also agree with others that if she just does not get it at this point, it is no big deal. It is a very subtle concept that the human brain is not particularly well equipped to grasp, something we forget after many years of mathematical training.

Say that "infinity is like higher." You can always go higher, and you can always count one more.

Of course the next step is the difference between potential infinity (you can always count one more) and actual infinity (the set of all integers). There have been a few arguments about this.

Then you can talk about different types of infinity. The traditional way is to ask if there are more fractions than integers and if there are more points on a line than there are integers.

Then comes Lebesgue measure, but the child should at least be in fifth grade for this.

• To be honest, I don't think she copes good with fractions, as long as I know she just know how to add fractions with same denominator, so adding fractions IMO will just make thinkg even more complicated, but as you said this is something for which she's too young and I'm just interested in explaining the infinity in the set of natural numbers. Sep 30, 2013 at 22:38
• @Stefan4024 He was just joking in the last paragraphs... Sep 30, 2013 at 22:53
• @chubakueno Yeah, fourth grade is easily high enough for the Lebesgue measure!
– rlms
Jan 7, 2014 at 16:29

I know you've accepted an answer but because this popped up on the home page... You could always try explain with something physical, like something relate-able (ish). I have two approaches that come to mind at the moment.

The approach with numbers of "there is always a bigger number" if I pick $n$, $n+1$ is bigger again, which was frequently mentioned is akin to the idea of having a path/road that stretches far into the distance, to the horizon say. You can't see the end of it but for physical reasons you expect it should stop or end somewhere. You just do. People post about "infinity" being non-physical or real and maybe that's just in-built initially into how we think.

Which brings us to approach two: What if we have a path that loops, in a circle say or a figure 8? If we keep walking in one direction and keep following the path (there could be arrows on the path?), will we run out of path? What if we mark a start point and count how many times we pass it? So we relate it back to numbers this way.

Just a thought, kind of like the whole 8 on it's side thing $\infty$.