Infinity, parts 3 & 4

Infinity, part 3

We talked about different types of numbers. First we have the counting numbers:

1,2,3,4,5,...

Then we include 0:

0,1,2,3,4,5,...

Then we include negative counting numbers to get the integers:

...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...

This gets us a "numberline" that looks like a bunch of disconnected dots. If we include rational numbers (all fractions), the numberline fills in. Here's the weird thing: although we can't see any holes in the numberline, there are holes nevertheless - infinitely many holes. We use the irrational numbers to fill in those infinitely many holes that we can't see. The real numbers are the rational and irrational numbers.

The integers are discrete: that is, there are big gaps between them. The rational numbers and the real numbers are continuous: between any two rational (or real) numbers, there is another rational (or real) number.

That was quite enough heavy thinking for one day, so we made Mobius strips.

Infinity part 4

An Infinity Bigger than Omega

We finally got to an infinity bigger than omega. First we did some warm-ups. I wrote down three 3-digit numbers:

123

234

235

and asked for a 3-digit number that was different from the first in the first place, from the second in the second place, and from the third in the third place:

123

234

235

One possible answer is

278

because the first digit, 2, is different from 1; the second digit, 7, is different from 3; and the third digit, 8, is different from 5.

If we do this with infinite decimals between 0 and 1, we can argue that it's impossible for these all to "hold hands" with the counting numbers. That means there are uncountably many real numbers - that is, more than omega-many. This is known as Cantor's Diagonal Argument.

Arc Length

Draw a squiggle on a piece of paper. Now measure that squiggle using only a straight ruler! It's hard to get it just right, but by measuring little lines you can make a pretty good guess. If you make the lines shorter, you get an even better guess.

Zeno's Paradox

Start out 10 feet from a wall. Walk halfway to the wall. Walk half of the remaining distance. And again. No matter how many times you walk halfway to the wall, there's still a distance between you and the wall. So you can never get to the wall! But somehow, you can walk over and touch the wall.

Cookies!

Start with a cookie. Cookie monster eats half your cookie. Then he eats half of what's left. If this goes on forever, how much cookie is left at the end of time?

Infinity, part 5

The first thing talked about today was why "infinity minus 1" doesn't make sense. Look at a list of numbers:

1,2,3,4,5,6,7,...

Adding 1 to a number means moving one number to the right in the list. Subtracting 1 from a number means moving one number to the left in the list. If we try to move one number to the left of infinity, where do we end up? We would have to land on a finite number, but which finite number? Gah! "Infinity minus 1" is undefined, because there is no answer that makes sense.

Then we talked about limits. The "limit" of a process can be thought of as what you end up with if you could actually get to the end of time. Here are several examples.

• Take a cookie. Eat half the cookie. Now eat half the cookie that remains. And eat half again. Continue this for all time. The limit is no cookie left!
• Inscribe a triangle in a circle. Then a square. Then a pentagon. Then a hexagon. Continue. The limit is a circle.
• Take a line segment. Draw dots to divide the line segment into thirds. Then erase the middle third (but keep the dots). With the two remaining line segments, draw dots to divide into thirds, then erase the middle third (but keep the dots). Continue. The limit is the Cantor Set.
• This one's easy to see, but hard to explain: the Koch Snowflake. And here's a good edible approximation.
• Draw a square. Divide into ninths, and remove the middle ninth. Divide each remaining square into ninths and remove the middle ninth. Continue. The limit is the Sierpinski Carpet. We didn't make Sierpinski cookies today, but I'm eyeing that for a future project.
• If you take the idea of the Cantor set (1 dimension) and the Sierpinski Carpet (2 dimensions) and go up to 3 dimensions, you get the Menger Sponge. Someone said this made them dizzy - I said it makes me a bit dizzy, too!

My favorite part of the class was when the kids figured out, all on their own, that the Cantor set contains no lines but infinitely many dots. Wow :)

We also had a discussion about how many sides a circle has. 0? 1? infinitely many? 2 (an inside and an outside)?