Day 7, MAT115

• Your quizzes are returned.
  • key
  • Median score: 9 -- keep that up!

• The quiz this week will be over Yanghui's (and Pascal's) triangle: its properties, and its uses; also the class agendas, and the readings.
• You have a new assignment, which is to translate the documents you receive today into our numbers.

• We went over some more properties of Pascal's triangle, in particular the tetrahedral numbers.
• The entries in the triangle are also known as "combinations", and each entry can be thought of as a number of the form \[ C^n_p = \frac{n!}{p!(n-p)!} = {n \choose p} \] The number of ways of choosing \(p\) objects from \(n\) distinct objects. By "\(n!\)" we mean the product \[ n\cdot(n-1)\cdot(n-2)\ldots 3 \cdot 2 \cdot 1 \] That is, the product of the first \(n\) counting numbers. Interestingly enough, \[ C^0_0 = 1 \] which means that \(0!=1\) (this is by definition -- it doesn't make any sense if you think of it as multiplying counting numbers!).

Having this formula allows us to calculate combinations for things far down Pascal's triangle: e.g. the number of different poker hands possible, which is \[ C^{52}_5 = \frac{52!}{5!(52-5)!} = {52 \choose 5} = 2,598,960 \] It works for anything! Even cream pies....

We're going to do three things today: solve two mysteries, and celebrate our hero Zero.

• Remember this t-shirt?
  

  Now you should get the joke. And I'm hoping that, by the end of the day, you'll have come up with some more joke shirts!

• We begin by playing Sherlock Holmes in Babylon

  The author, Prof. Buck, guesses that "the document" (actually a clay tablet) comes from the city of Nippur, in what is now Iraq:

  

  Buck says this "Confronted with an artifact from an ancient culture, one asks several questions:

  1. What is this and what are its properties?
  2. What was its original purpose?
  3. What does this tell me about the culture that produced it?
  So: this is a representation of a clay tablet, with markings in it.

  Your job: to describe the markings, and investigate the patterns on this tablet.

  • What can we deduce about
    • the purpose of the table?
    • the way (and reason) Babylonians wrote their numbers the way they did?
    • the Babylonians and place value?
    • the Babylonian's missing number(s)?
    • the Babylonian influence in our world today?

  • How would the Babylonians write these numbers:
    1. 47
    2. 573
    3. 1,60, and 3600

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• The frequently unsung hero of the Hindu-Arabic numbers is the digit 0 -- a symbol for nothing! (The Babylonians can't distinguish the numbers 1 from 60, without some additional context -- they had no hero).

  Well let it be unsung no more! I hope that you'll enjoy this rendition of

  Zero, My Hero

  (favorite lyrics: "et cetera, et cetera; ad infinitum; ad astra¹; forever and ever")

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• Now to the mystery of the Mayans.... Same questions:
  • What can we deduce about
    • the Mayans and place value?
    • the purpose of the table?
    • the way Mayans looked at numbers and their world?
    • the Mayan influence in our world today?

  • How would the Mayans write these numbers:
    1. 47
    2. 573
    3. 3600
    4. 14001

• Hexagonal Paper