Day 14, MAT115

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Announcements:
- Sorry -- your homework isn't graded yet. Our Babylonian story competition will begin next week, when I get those back to everyone. Folks are free to participate in this, even if they didn't do the extra credit.
- How ironic: I just had a terrible website meltdown....
- Reminder: you have an assignment due to be handed in Wednesday.
  Note: I've attempted to rebuild the homework assignment, but if you have already started let me know, and I'll accept either version of that!
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: Egyptian division. I want to look at one more example, of both methods for arriving at the same answer:
Let's do the example 9/13.
Then we'll move on...
Today we'll focus on chapter 2: 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
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 three fish --
  fish fish fish,
  fish fish 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 (that is, each element in one group has a partner in another -- and just one partner. Perfect for the dance....).
So 9 is an example, a square. It can be broken into three groups of three; so each group of three can be put into one-to-one correspondence with any other group of three.
Notice that we said greater than one in the definition above. The number 1 is special, and considered neither prime nor composite.
We've already heard this important rule, which you learned at some point in your mathematical education:

Every natural number is either prime, or can be expressed as a product of primes in a unique way.
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 Strogatz notice about the sum of
- odd and even
- odd and odd
- even and even
Let's make a table to illustrate that:

+ Odd Even

Odd

Even

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? (By the way, add one more row to the top of the equations near the top of the page - what would the row be?
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.

+	Odd	Even
Odd
Even

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