Math Circle Session 13: Infinity

Given the number of questions we’ve been asked by the children about infinity in recent weeks, I decided that today we’d have a session just on infinity.

I opened the session by asking the children whether they had any questions about infinity they’d like to try to answer in this session. I received the following suggestions:

• Why does infinity have the symbol $\infty$?
• What is infinity plus infinity?
• Why do we sometimes say infinity is a number and sometimes not?
• What are the rules for adding and subtracting with Aleph numbers? (From a child who had discussed infinite cardinals at home in the past.)

We began the session by one child answering the first question: “it represents going round and round and never stopping, which would also be a circle, but that’s zero.” I hoped the others would crop up during the discussion that followed.

We began by discussing prime numbers. All children had heard of primes, but only the older children could give a clear unambiguous definition. I had them list out some prime numbers, and asked how many there were. Opinion was divided: one child thought there would be an infinite number (after all, the session was on infinity!), while another child thought that this can’t be true “because most numbers are not prime, and there are infinite counting numbers, so there can’t be infinite primes too.” I then showed them Euclid’s proof that there are infinite primes.

Starting first with a hotel with a finite number of rooms, I then explained Hilbert’s  Hotel. For blog readers, there is an excellent animation of these ideas on TED. This seemed to get them quite excited, which is what I was hoping to see! We covered one new arrival and an infinitely large coach of new arrivals, each time stopping to see whether they could contrive of a construction that would allow the new arrivals to fit in the full hotel. The child who had previously seen infinite cardinals suggested a solution to one new arrival, but nobody came up with a solution to the infinitely large coach of new arrivals. However, it was great to see the thought going into this (most children wanted to move each person from their room to one infinitely further down the corridor.)

At this point, there were two directions I had thought we might go, given enough time: one on an infinite number of infinitely large coaches using a prime construction, given our proof of Euclid’s Theorem, and the other to look at a diagonal argument proof of the uncountability of the continuum, as a route into the final question the children asked. However, given we only had five minutes left, I decided it was better to just let the children’s discussion free-range, since they were still talkative after the hotel story. A variety of discussion points emerged involving infinity, and I just let them talk to each other: is space infinite, how can space not be infinite, are there an infinite number of life-supporting planets out there, what happened “before” the big bang, etc.! It was great to see this discussion, even if we hadn’t manage to answer all the questions posed at the beginning.