Student: In the Tortoise and the Hare, Cantor's Comb, and the Hilbert Curve activities, we are asked to think about what happens if we repeat the same thing " infinitely many times." Sometimes we are told to repeat " indefinitely." What does this mean?

Mentor: This is a very important idea from math, but it's not very precise when stated as "infinitely many times" or "indefinitely." Let's try to fix this up. First a new word: Repeating a set of rules or steps over and over is called iteration. Each step is an iterate.

Student: So when we ran the tortoise and hare race, each time we pressed the advance button we were iterating? And each time we pressed the next step button on the Hilbert curve we were iterating?

Mentor: Yes. Now, when did you stop on each of these?

Student: In the tortoise and hare race, I stopped when I realized that no matter how many times I advanced, the tortoise was always ahead and wasn't at the finish line yet. In the Hilbert curve, I stopped when the different stages were pretty much the same.

Mentor: So you stopped when you realized that you knew what would happen if you kept pressing the advance or next stage buttons forever?

Student: I guess so.

Mentor: So you could have just kept pressing the buttons -- iterating -- forever, supposing that you can live forever to do it. This is what we mean by "infinitely many times."

Student: That still isn't very precise.

Mentor: You're right. Let's fix this, too. We need another definition. Infinite means larger than any fixed counting number. That is very large! Here's a very large counting number:

Can you find one bigger?

Student: That's easy! Just add one:

Mentor: Great! So anytime I give you a counting number (mathematicians call them Natural Numbers) you can find one bigger. This is part of the idea of infinite. So when we say infinitely many times, we mean more than any counting number. There are other parts that are less easy to understand; in fact, it took thousands of years to get infinity right.

The ancient Greeks hated the idea that there could be a set of infinitely many things (like the set of Natural Numbers). Infinity was regarded as an impossible concept. People struggled with the idea for centuries. Even Leonard Euler -- a brilliant mathematician from the eighteenth century -- didn't quite get the idea. It wasn't until the dawn of the twentieth century with the work of Georg Cantor (as in the Cantor Comb) that the idea was understood very well. Here are some of the interesting things about infinity:

• Things that work for finite sets may not work for infinite sets.
• An infinite amount of stuff doesn't always take an infinite amount of space. Think about the Hilbert curve: It was "infinitely" long, but fit in the square.
• The sum of an infinite number of numbers can be finite. Think about the tortoise and hare race: The tortoise travels the following distances, one fraction for each time step: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ....... but never gets to the end of the race! So the sum above never gets past 1!

Student: So I can think of infinity as being larger than any counting number? And iterating infinitely many times is the idea of repeating the steps forever?

Mentor: For now these are good ways to think. Here is a more standard way to say "repeat infinitely many times:"

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