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?
Teacher Jesse, we are second year students of Mathematics Teaching Programme of the Universidad Catolica del Maule located in Talca, Chile. We are working on a research about mathematics teachers’ blogs. The search included the reading of other similar blogs but yours was one that had everything we were required to work on, specially to be an English blog about math, because we’re Spanish speakers. The question is, how useful, do you think, is the implementation of this tool in your pedagogical activities? Which were your main objectives when you began to use the blog? Have you reached them? We’re waiting for your soon answer. Thank you very much. Good bye!
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