Day 4, MAT115

Last Time

Next Time

Announcements:
- Reminder: please shut down your electronics.
- You will have a quiz Friday, and it might have a problem or two from your homework assignments. Keep tabs on that assignment page. I've added a new assignment for today.
Question of the Day: How do we use "rock groups" to define prime and composite numbers, square numbers, and triangular numbers?
Before we get to that, however, let's review what we did last time....
- We took a few more cracks at "counting by partition", or "primitive counting". In particular, we learned to start from the string (or "knotted cord" provided by the farmer to the king, and decoded it to write down the number of sheep that the farmer had.
- We ended the day by seeing a "counting card trick" based on counting. Your job was to figure out why it works.
  Will found this video on YouTube which explains the positions of the final three cards. (The original post, showing the card trick itself, is here).
  It doesn't really explain why those positions make sense. Any further observations?
Today we'll focus on chapter 2 (leaving chapter 25 for next time): rock groups. To explore, we'll use pennies (rather than rocks), and you'll work with a partner.
We want to understand each of these notions:
- prime,
- composite,
- square, and
- triangular numbers
while also thinking about the concept of the one-to-one correspondence.
So we're interested in arranging our pennies (i.e. rocks) in groups with certain properties.
All numbers of pennies can be arranged into a line (this is Humphrey's "Furry Arms" method):

Some numbers of pennies can be arranged into
- rectangles (of more than one row):
  
  (Note that this is how Humphrey said the number of fish -- he blocked the fish into two groups of fish.)
- squares
- triangles
If a number greater than one can be expressed as a rectangle of more than one row, then it is composite; otherwise, it is prime. The rows must stack to form a rectangle, which means that they can be put in one-to-one correspondence with each other (see the picture above).
Another way of saying that a number is composite is to say that it can be broken up into groups (each with more than one member) that can each be put into one-to-one correspondence with each other.
So 9 is an example, a square. It can be broken into three groups of three; so each group can be put into one-to-one correspondence with any other group.
Notice that we said greater than one in the definition above. The number 1 is special, and considered neither prime nor composite. This is emphasized in chapter 25, and we'll come back to it next time.
Let's look at examples of each.
1. Using your pennies, make (and document) all rectangles for pennies up to 15. Then answer me this: "What's special about 12?"
2. Find and illustrate the triangular numbers until you run out of pennies!
There is an interesting observation made by the author. It was about odds and evens. What do he notice about the sum of
- odd and even
- odd and odd
- even and even
Let's make a table to illustrate that.
Rock groups suggest formulas for representing even and odd numbers:

Evens: 2*n
Odds: 2*n+1
How do we say the formula suggested at the top of page 10? How would you say what's happening in words? What do you think of the proof of the "theorem" at the bottom of the page?
1. Do you see the theorem expressed in the rocks?
2. Can you see the form "2n+1" in the rocks?
3. I see a new theorem: T₅+T₄=5² (In words, "the sum of the fourth and fifth triangular numbers is the fifth square".) By playing with rocks we discover relationships. It's probably how primitive civilizations did it, too!
The story of the "The Housekeeper and the Professor" is interesting, and modelled on a famous mathematical event -- the story of child prodigy Carl Friedrich Gauss:

Once again, rock groups give us a formula for representing triangular numbers. So let's go through the reasoning for the housekeeper's answer.

Website maintained by Andy Long. Comments appreciated.

Evens:	2*n
Odds:	2*n+1