My daughter and I have this recitation, a verbal game we've played since she was very young.

It's trivially simple and starts with one of us saying 'I love you', the other saying 'i love you more than that', 'i love you that times 1000', 'i love you that times infinity', 'i love you infinity + 1', 'i love you infinity times infinity', 'i love you infinity to the power of infinity infinity times' and so on, forever adding one, multiplying by 1000 and eventually just ending in a hug and a smile.

a little while back i said, 'you know, infinity is the biggest thing there is'. yesterday we played the same game and she said 'no dad, there is no infinity +1 it's just infinity'. hoist on my own petard.

then she asked me whether infinity was a number and i said 'i don't know, but i don't see how it can be because if infinity + 1 = infinity then it doesn't follow normal maths. it's used in maths, but not as a number. it's more like when this particular thing goes on forever to infinity, then this other thing over here gets to a number we know. blank face in response, but that's cool, she doesn't have to do calculus for many years yet.

my question is whether there is another way I could mathematically represent infinity to someone who won't do calculus for a while. is there a simple mathematical equation I could show her, perhaps describing a phenomenon familiar to her, where the answer or a key component of the equation was infinity?

Just keep going and don't do this part of the game.

Infinity + 1 = Infinity, but Infinity^Infinity (infinity raised to the infinity) is a bigger infinity.

A very nice analogy is Hilbert's infinite hotel.
The Wiki entry seems a little too technical (you can read it here), so I'll give a brief summary:

Suppose there is a hotel with infinitely rooms 1, 2, 3, 4, ... (a room for each positive integer). Now suppose it is all occupied. A person shows up and wants a room. Think about what the hotel clerk could do to fit him in. I'll post the answers to different scenarios in spoiler tags, I suggest you think for a while before reading the answers:

This example sort of illustrates that infinity + 1 = infinity.

Now what if there are 20 people that show up, what can the hotel clerk do to find vacancies for them.

This illustrates infinity + 20 = infinity.

It is easy to keep going and figuring out a way to solve the problem whenever there is a fixed number of vacancies needed. But what if an infinite bus with seats 1, 2, 3, ... (a seat for each positive integer) arrives? How do you fit infinitely many people. You can't just send person in room 1 to room 1 + infinity, because there is no such room. So where do you send people to create vacancies?

This illustrates that 2*infinity=infinity.

What if you have 11 infinite buses?

This illustrates 12*infinity = infinity.

You can easily extend this solution to take in a finite number of infinite buses. But what if you have an infinite parking lot for infinite buses. So the first slot in the parking lot has an infinite bus. Slot 2 has another infinite bus and so on. How can you fit all of the people in the parking lot into the infinity hotel?
This is much harder, but in the spoiler tag I give a sketch of what you do:

This illustrates infinity*infinity = infinity (or infinity^2 = infinity).

Turns out you can generalize this for infinite parking lots full of infinite buses and get infinity^3 = infinity. And you can keep going one by one, so infinity^n = infinity. The concept gets a littler weirder to grasp, but it is doable.

But it breaks down at infinity^infinity. This was one of the most important discoveries made by Cantor. The argument is for another day, but he shows that you cannot assign a room if you for everyone in that case. In fact, Cantor showed 2^infinity > infinity, so not only infinity^infinity is bigger but also 2^infinity.

Note: Since there are different infinities, as mentioned in this post. I should point out that whenever I say infinity in an equation, I mean the number (where I use the term number loosely) of positive integers, known as aleph_0 (the Hebrew letter). Surprisingly we've shown here things like the number of positive integers is the same as the number of even integers and even the same as the number of rational numbers. However, the number of real numbers is greater.

The age of your daughter is important. Doing mathematics which is reality based is appropriate for all ages such as counting toes and fingers. Again, how old is she.

The type of mathematics which posits a watch which travels at the speed of light and then comes back is unreality and shouldn't be taught to children. You can't come to an appreciation of "infinity" in her lunch bucket and being clever with infinity is stultifying.

I know, I'm being rather critical but perhaps an imaginative picture of the star filled cosmic drama would speak livingly to her inner being rather an inchoate abstraction such as "infinity".

Yea, one infinity is as good as another .

And right, a hug and a smile overshadows the rest.

Why not?

There's more involved but dealing in abstractions, though adult like, is a barrier to proper thought and thinking.

There is no reality to traveling at the speed of light for of course we would literally burn up, far before that supposed speed. this is not real, star treck notwithstanding.

If children are to mathematize, then the initial steps should be that of proper counting, such as fingers and toes and not a calculator/computer which separates the child from a living penetration into the mathematical world.

I suppose we could go on and what I'm attempting to display is that the child is within a feeling fantasy and she should be taught iwithin that realm.


A"feeling" for the starry cosmos or sunlight sky is the path to knowledge, not a measured/abstraction , devoid of life.

You could say that an artistic approach to education, abstractly speaking, will concur with the inner sprightliness of the little child. to use abstractions/intellectualization is weakening, especially in its effects in later life.

So, what have we got in our age; a plethora or overwhelming intellectualization /abstractionist culture. Can't fight city hall but I'll state it; the abstracti0ns of the material overcome the human condition, a death borne dyscrasis which forgot that the children needs and desire the loving reality of a living life.

Just pontificating here and know it but its the best i can do at present. I'm not an educator and so i'll refer you to the Waldorf School System.

http://www.waldorfanswers.org/Waldorf.htm

i'm intrigued by the infinite hotel. x^infinity > infinity?

what does that mean? is it a case of infinity stretching in a linear direction endlessly is one thing, but infinity stretching in multiple directions/dimensions creates an area or volume or some other x^x that is larger than a one dimensional infinity? or is it some other concept? what's the correct way to describe x^infinity > infinity?

Numbers can be mapped to the number of objects in a collection. Five corresponds to five beans in a pod. Captures the essence of fiveness.

Extend this to infinite collections. Take a bag with all the integers in {1,2,3,...}. How many items are in the bag. Infinite or aleph zero.

Now take a bag with all the real numbers, numbers like fractions, square roots phis numbers which do not have a finite decimal expansion. Now how many numbers in that bag. Infinite of course. However there is a clear reason why this bag should be considered bigger than the bag of integers.

Ask her what's the sum of 1+2+3+4+...forever

If she says -1/12, skip all grades and send her straight to college!

Sum(infinity) = -1/12 is basically proof that God exists and he's trolling us.

Can you have a bag smaller than that but bigger than the first?

Thats a cute way to have fun with https://en.wikipedia.org/wiki/Continuum_hypothesis

Too bad there is no such things as infinity...not even of the first kind.

A set that contains all rational numbers plus all irrational numbers that have a repeating sequence that becomes non-repeating after infinite repeats?

Then add irrational numbers that start repeating after an infinite sequence of non repeating digits?

Have no idea whether such numbers exist (or if they do whether the sets of such numbers would have the same cardinality as either rational or real sets) but short of imaginary numbers I can't think of any other possible number forms that aren't either repeating/terminating or irrational.

There is no such thing as infinite repeating that then.... You cannot define the "then". It never comes. Your number is simply a rational number (since it is repeating to infinity) as close as you want it to the rational number with the same repeating digits. Same for the other case, the number is simply the original irrational number. The second parts never happen because they are essentially 0. They are bound by something that is as close to 0 as you want.

so pi =ROUND(pi,infinite)?

there are no numbers on the line of real numbers that are simultaneously rational and irrational? it's the only medium size bag I can think of.

I think that the "then" moves into infinitesimals and surreal numbers.

Jebus, guys, a child needs to see infinity in action. Here's an exercise that a child that can do division and multiplication will be able to see: divide 1 by 7. The answer is .142857142857142857142857142857142857...
The decimal .142857 repeats to infinity.
Now multiply by 2. The answer is .285714285714285714285714...
The decimal .285714 repeats to infinity.
You can multiply times 3, 4, 5, or 6; you will see those same six numbers in order going out to infinity. And beyond!

But if I multiply by 7, I get 0.99999999999...

Shouldn't I just get 1???

I use the following when I teach about countably infinite sets.

Imagine you have two shepherds who each have a lot of sheep. They want to know who has more. But being uneducated shepherds, they can't count high enough to count all their sheep. So after some thought, they decide that they're going to take their flocks up to a fence and let them through one pair at a time, one from each flock. The one who runs out of sheep first must have a smaller number of sheep.

But now let's take this idea, but instead of sheep, let's use the natural numbers 1, 2, 3, ...

I'll let you have all of them, and I'll just have the even ones. Who has more?

You might think that you do, but you would be wrong. Because all we need to do is line them up like sheep. I'm not going to run out before you do, so I don't have fewer numbers than you do.

Hi Aaron. I knew nothing about this topic when posting the OP, but have since read through the wiki articles on CH, countable and uncountable infinite sets, cardinality and the bijection you speak of above, and remembered the real line from school.

It's been quite fascinating to work through this. It appears (to a lay person like me) that there are only two types of infinity - the 'traditional' one that extends to infinite orders of magnitude (rational numbers) and another that extends into infinite orders of complexity (all real including irrational numbers). I'm not sure if 'infinite complexity' is the right description for irrational number sequences, but it feels like that to me.

I think that's a concept I can talk to my daughter about. She's learnt about fractions and I can show her a circle and how pi is made and then compare repeating/terminating decimals in a fraction to the non-repeating endless complexity of irrational numbers and point to these on a real line. She may be able to comprehend ideas like magnitude and complexity in a broad sense initially, but then gain a deeper understanding when she starts working with pi in secondary school. (Thanks Wetdog for the suggestions on fractions above.)

Of course, like any belligerent first-timer I immediate sought to prove CH false and construct some mythical numbers that were part rational and part irrational to create a set with a cardinality between rational and real. Masque is ofc correct that they simply collapse to rational or irrational, but it was fun to try and did lead to a mention of surreal and hyper-real numbers which I've yet to investigate. It looks like there are some interesting GT applications there, although I'm probably starting from too far back to get my head around it

There was one other discussion in the wiki articles that seemed interesting. I'd always understood that the real line contained all non-imaginary numbers including fractions, pi, e and everything else. But there was some reason that the real line was proved to be an inadequate mathematical foundation prior to 1900 and that real numbers in now defined in some other way. I did wonder whether the differences between real line and the way real number sets are currently defined could provide some potential for a cardinality that sits between rational and real sets, and therefore a third type of infinity.

They are going to tell you that there is more than two. Don't pay too much attention.

You get to where infinity ends!

7 x 1/7 = infinity, according to you. And you teach math to college students? tch!

There are two main constructions of the reals from the rationals: Dedekind cuts and Cauchy sequences, both of which were developed before 1900.

Not sure what you mean by inadequate foundation, but Euclidean geometry was not fully axiomatized until 1899, by Hilbert.

You could say that the real line is the only complete ordered field, which is an algebraic characterization from the 1920s, but you still need to use e.g. Dedekind cuts or Cauchy sequences to show that such a field exists.

https://en.wikipedia.org/wiki/Ordered_field

It's not a linear versus multiple directions thing. It is a multiple directions thing (in terms of dimensions) versus infinite directions.

x^infinity > infinity if x >= 2 (maybe it works for x between 1 and 2, but I'm not sure what it would mean).

Let me explain what 2^infinity is (where we'll use the infinity as aleph_0, the infinity of the natural numbers).

To explain it, let's use some examples. Consider the set {1}. It has two subsets, the empty set and itself. Consider the set {1,2}. It has four subsets: {}, {1}, {2}, {1,2}. Now consider {1,2,3}. The subsets are {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. So it has 8 subsets. You might see the pattern. A set with n elements has 2^n subsets.
So 2^infinity is the number of subsets of {1,2,3,....}.

Cantor showed that this set cannot be enumerated. You can't assign a room in the infinite hotel to each subset of {1,2,3,...}.

To see it, suppose you could assign a room number to each subset. To illustrate, suppose you assign room 127 to {}, room 3 to {1, 3}, room 1234 to {2}, room 1 to {1, 4, 7, 9, 13}.

Now, consider the set of rooms that are not contained in the subset assigned to them (call this set B). For example, room 127 is not contained in {}, so 127 would be in B. 3 is in {1,3} so 3 is not in B. 1234 is not in {2}, so 1234 would be in B. 1 is in {1,4,7,9,13}, so 1 is not in B.

Now B is a subset of {1,2,3,...} so it should have a room assignment m. Now let's answer the question is m in B?
On the one hand if m is in B, then by the definition of B, m is a number that is not contained in the set assigned to it, so m is not in B.
On the other hand, if m is not in B, by the definition of B, then m is contained in the set assigned to Room m, so m is in B.
With either choice (m is in B or m is not in B) we reach a contradiction. Therefore there is no possible way to assign rooms to every element of the subset of {1,2,3,...}.

The Cantor proof is similar in spirit to Russell's barber paradox. Suppose you have a small town with one barber. Suppose everyone is clean-shaven. The barber shaves everyone who doesn't shave himself. Who shaves the barber? If the barber shaves himself, then he doesn't, because he only shaves people that don't shave themselves. But if the barber doesn't shave himself, then because he shaves everyone who doesn't have himself, he must shave himself. Both paths lead to a contradiction!

aren't they the same?