Explaining infinity?

[thatsenough]

DS year 1 has developed an obsession with infinity, so far my explanations have been met with a blank face - I've tried a never ending road, space doesn't end etc.

Have any of you got a good explanation I can try?

Why not give him an elastic band and tell him to find the beginning and the end?

DS (aged 4 at the time) discovered infinity by thinking about numbers, and the fact that they go on and on and you can always get one more, even when you've already got a really big number.

He could see, when a little bit onlder, that there were different infinities that result from e.g. multiples of 10 rather than counting in 1s.

Have you tried talking about numbers? That is at least a 'real' example of infinity.

My 6 yr old is obsessed with infinity too. I blame Buzz Lightyear.

My 6 yr old reckons it comes after the number 162.

I shall read all responses with interest.

I agree that my example of infinity is on a smaller scale; but perhaps the larger-scale examples are difficult to grasp because of their sheer size. An elastic band is 'infinity in a nutshell', if you will. :)

I must see how DS thinks about it (he is 7). And also DD (5). We talk about infinity from time to time (makes us sound a bit weird unusual but hey). He did surprise me by knowing its symbol but maybe my mum abd dad (retired maths teachers) have been influencing him

Infinity is the place where parallel lines meet.
Infinity plus one is still infinity. So you can have different sizes of infinity.
You can't count to infinity.
If you divide any number by 0 (except 0, 0 divided by 0 is often "defined" as 1) you get the answer infinity. You can show this by using the subtraction method of finding the answer to a division question. eg 3 divided by 0: So 3-0=3, 3-0=3..... for ever.
0 times infinity can be defined as 1.
These are all concepts that we've discussed with the dc (age currently 10, 7 and 3)

Some great concepts there, DeWe. I especially love the paradox of the parallel lines - or at least the paradox that goes on in my head when I try thinking about it.

I have a maths background and a 5-year old who talks about infinity a lot. I use the definition I learned at university.

We started with "countable" infinity (the smallest infinite number).

You go one, two, three, four and keep going. And keep going. And keep going. You can never count to infinity. If you can say a number or write it down, then someone can say a bigger number or write down a bigger number just by adding one. Infinity is different to ordinary counting numbers because it's the one you just can't see at the end of the number line, where if you add one it doesn't make a difference.

Infinity plus one is infinity. Infinity times two is infinity. Infinity times infinity is infinity. All these numbers are the same size. There are infinitely many integers (counting numbers). There are infinitely many odd numbers. There are infinitely many even numbers. There are the same number of odd numbers and even numbers. There are the same number of odd numbers as there are integers (this is not a typo, this is how infinite arithmetic works, and it is counterintuitive at first, so don't tell your 5yo this :))

After the countable infinite numbers, come uncountable infinities. This is where the fun really starts, but I think you need to be comfortable with quite a level of abstraction to really get to grips with this concept, so I'd wait until 13 or 14 before really getting into set theoretic concepts at this level.

set up two mirrors opposite each other, look into one. at the right angle you can "see" infinity

Ask him to tell you the very biggest number he can think of.
Then ask him what happens if he adds one... no matter what number he thinks of he can always make it bigger.

I like the rubber band idea, maybe you could make a möbius strip. Just because they are fun :)

"There are the same number of odd numbers as there are integers (this is not a typo, this is how infinite arithmetic works, and it is counterintuitive at first," absolutely, blew my mind when I first heard that :o

maybe have miss-understood, why not just go outside, lookup at the sky and ask where they think it ends.

Love this - we're had this debate in our house too.
Interesting question - have any girls asked this question or does it tend to be boys? Boys in my house

My DD (4 on Weds) has asked me about infinity. So far I've just told her it's the biggest number there is, bigger than anyone can ever count to, even if they spent their whole lives doing nothing but counting.

Watch out though - my nightmares from about 7 to 13 were all themed around infinity. My 8 year old winds me up about it, trying to find the chink by "artlessly" talking about the extent and location of space, seeing if he catches me with my defences down.

Well I was a very mathsy child but was never particularly bothered by infinity (maybe because I hadn't seen toy story ) - I've always been more interested in the more concrete aspects of number iyswim

the way small children count is oddly similar to the way mathematicians enumerate sets, including infinite ones. If you give a two year old a bag of teddies and a bag of toy cups, and say "do we have enough cups for all the teddies?", he'll probably put the teddies in a line and give one cup to each teddy in order to answer the question, rather than count "one, two, three,..." for the teddies and cups.

Mathematicians enumerate sets by finding a way of matching each member of a set to a member of another set whose size we know. This is very useful for infinite sets too. If you can get them to understand it this way, it may help, rather than just explaining it using the counting numbers.

This has a nice description of infinite cardinals, and touches on transfinite arithmetic at the end, which is fun. You could probably paraphrase it to make it accessible to an interested 5-year-old.

In 13 years time, if they're still interested, I recommend this excellent book.

DeWe - Sorry, but there are a lot of wrong statements in your post:

"Infinity is the place where parallel lines meet"

No. Parallel lines don't ever meet. That is their definition. Besides, there is no such physical place as 'infinity', therefore nothing can happen there, let alone meeting of parallel lines.

"Infinity plus one is still infinity. So you can have different sizes of infinity."

There is only one infinity which does not have a size.

"If you divide any number by 0 (except 0, 0 divided by 0 is often "defined" as 1) you get the answer infinity. 0 times infinity can be defined as 1."

No, no, and absolutely not.

Dividing by 0 is not a legitimate operation in mathematics and it is not "often defined as 1". If it was so, then you could prove that 1=2 through the following:

Since 01=0 and 02=0, then
this must be true: 01=02
dividing both sides by zero gives: 0/0 1 = 0/0 2
if 0/0=1, then you can simplify the above to: 11=12
and therefore: 1=2
... which is false.

0/0 and 0/infinity are "indeterminate forms" with no single answer.

CoteDAzur there is certainly more than one size of infinity. Cantor proved this. Suppose (for a contradiction) that the set of natural numbers was the same size as the set of real numbers.

Then there would exist a bijection f from the natural numbers to the reals. List the elements of the reals as follows:

f(1)
f(2)
f(3)
f(4)
...
f(n)
...

now construct the real number whose first digit differs from the first digit of f(1), second digit differs from the second digit of f(2), and so on, with the nth digit differing from the nth digit of f(n). This number cannot appear in the range of f, yet is a real number. Contradiction.

Thus there are more reals than there are natural numbers. The size of the real set is described as uncountable infinity, whereas the size of the set of natural numbers is described as countable infinity.

I agree with you on the post about undefined answers though. It would help if DeWe used limit theory (with epsilons and deltas) to tighten things up in her definitions.

for non-mathematicians, a bijection is a one-to-one map, like pairing the teddies and the teacups :) Hope the rest is comprehensible.

This reminds me of when I discovered infinity for myself. I was reading one of those old-fashioned 'Peter and Jane' Ladybird books (I must have been in the first class at primary school). On the front, the picture had Peter and Jane holding the same book, with a picture of Peter and Jane on the front holding the same book ... I still remember the physical jolt with which I realised that it went on forever (not, obviously, in the picture as that was limited by the resolution of painting and printing, but I 'knew' in some way that the series of images must go on forever)

galois, non mathematicain here - but with a mathematically able child who I would love not to give the wrong answer to... It seems intuitive to both my son and myself that e.g. there are infinitely many multiples of 10, but this definition gives a 'different' infinity to the infinite number of counting numbers, and equally that the infinite number of numbers that can be expressed as decimals is a different infinity to both of these two.

Is this mathematical gibberish, or are we along the right lines?

you're along the right lines with the second point. But the first is a funny one. I will try not to lapse into mathematician speak.

Intuitively, it makes sense for there to be a smaller number of multiples of 10 than there are counting numbers. Intuitively, it would make sense if there were 10 times as many counting numbers as there were multiples of 10. But this is where infinite arithmetic blows your mind!!!

The way counting is defined, there are exactly the same number of multiples of ten as there are numbers altogether. There are exactly the same number of prime numbers as there are numbers altogether. There are exactly the same number of even numbers as there are numbers altogether.

The way we count things is the way that small children count: they match one set of things with another set of things by pairing them together. You give each teddy a cup and see if you have any teddies or cups left over at the end. If not, then bingo!, you have the same number of teddies as cups.

It's the same with infinite sets. You get one infinite set (the counting numbers) and another infinite set (the multiples of ten) and if you can find a way (any way at all) of unarguably matching each element of one set with a friend in the other set, then bingo! you have the same number of things in each set.

So for multiples of ten, and counting numbers I match them as follows:

match 1 with 10
match 2 with 20
match 3 with 30
...
match n with n x 10
...
and so on.

I have no counting numbers left without a friend at the end of this. I have no multiples of ten left without a friend at the end of this. Hence both sets are the same size. I have done a lot of set theory and this still blows my mind. It's a part of mathematics that I particularly love.

The number of decimal numbers (what we call the real numbers) is a much, much bigger infinity than the number of counting numbers. That's what I proved above. So you're right on that one.