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 a pattern " indefinitely." What does this mean?
Mentor: This is an important idea from math, but it's not very precise when stated as "infinitely many times" or "indefinitely." Let's try to clarify this. 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 the computer wouldn't let me go any further, I think it was step 15. The tortoise was always ahead but wasn't at the finish line yet. In the Hilbert curve, I stopped when the computer froze, at the 7th stage.
Mentor: Unfortunately for our purposes, a computer's memory is finite and will only let you repeat the experiment a limited number of times. Were you able to repeat the experiments enough times to picture what would happen if you continued to repeat the experiment many more times?
Student: Yes.
Mentor: So if the computer would let you, you could have just kept pressing the buttons -- iterating -- forever, supposing 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. Here's a large counting number:
1,000,000,000,000,000,000,000,000,000
Can you find one bigger?
Student: That's easy! Just add one:
1,000,000,000,000,000,000,000,000,001
Mentor: Great! So anytime I give you a counting number, mathematicians call them Natural Numbers, you can find another number which is greater than the one I gave you. 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 the idea of 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. It wasn't until the dawn of the twentieth century with the work of Georg Cantor, who developed the Cantor Comb, that the idea was understood 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.
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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!
The mathematics of calculus deals with infinite sums and solves these types of problems. It is known as the mathematical concept called a "limit."
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:"
Let the number of iterations approach infinity.
Now you can go home today and tell your parents you learned something about the fundamentals of calculus!