Day 17, MAT115

This session will be captured on Zoom, if I remember to turn it on, and record it.

Our page of zooms and the play-by-plays.

• The exam
• A key
• A small curve (add 3.5 points to your score)

So: if you fell, can you get back up? If so, then stick with me; if not, you might want to bail.

• The Bridges of Konigsberg problem: "is it possible to set off and walk around Konigsberg crossing each bridge exactly one?"
  • 

    As you can see, the Konigsberg bridge graph has 4 vertices of odd degree.

  • There is only an Euler path if a graph has either 0 or 2 vertices of odd degree. If there are two odds, then you will start the path at one, and end it at another; if there are no odd vertices, then you will start and end at the same vertex (and create what is called an Euler cycle).

• Definition: Planar graphs are graphs that can be drawn such that no two edges intersect. Which of our complete graphs are planar?

  • Exercise: $K_4$ is planar: it can be drawn without false intersections.
  • Exercise: $K_5$ is not planar:
  • The three utilities and three houses graph is not planar:

  • Kuratowski's theorem: Any non-planar graph has a copy of one of these two graphs in it somewhere, as a subgraph:
    

But our question of the day is this:

• By way of review, we'll start with some puzzles:
  1. can you find a way to pass through each door in these houses without passing through any door twice? (Or going through a window -- that's my comment to Bradley, before he suggests it!:)
  2. Euler's formula is the focus of the second page, and it refers to planar graphs. It turns out that there's a nice relationship between the number of edges, vertices, and faces of a planar graph. (We need to define faces, and I'll use my old friend the tetrahedron to do it.)
     • The graphs depicted do not appear to be planar -- or are they?
     • Does the tetrahedron satisfy the formula?
     • How about Euler's Konigsberg graph?

• In our new topic, we dig deep into history and "Location, Location, Location!" to discuss Babylonian Number representation
  • Let's reflect on some of the things that we discover in chapter 6 of our text, entitled "Location, Location, Location!".

    Here are a few of my musings:

    1. How cumbersome are the Roman numerals! How would you write your own birth year? (Mine is MCMLXI -- now you know how old I am!)

    2. "biology is deeply embedded" in number systems (10 "digits" in our system; tallies of 5; 5 and 10 in Roman numerals, and then multiples of 5 and 10; 10 in Babylonian; 5 and 20 for the Mayans)

    3. Babylonians based their system on -- its prime factorization gives it lots of factors. Our author Strogatz calls 60 "an exceptionally pleasant number." I hope that you agree!

    4. We need no new symbol for 10 in our Hindu-Arabic numerals: we use only the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 -- and place value (hence the name of the chapter: "location, location, location").

    5. Place value suddenly makes arithmetic possible. "Just master a few facts", our author says.

       My own musing: it's interesting that we seem to be abdicating our arithmetic to machines, and imply that it's not important to do arithmetic anymore. I would say that basic numeracy is tremendously important -- but oftentimes, more important than the exact answer is a good estimate. We could work harder to develop that skill.

    6. 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 -- as we'll see today!)

       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")

    7. Binary, or base 2 math: a place-value system with just two symbols, 0 and 1. Perfect for computers, for Binary Card Tricks, and for primitive counting!

    8. Cornell built the first telegraph line (Baltimore to Washington, D.C.): 1844. I know that the telegraph displaced the pony express, which only ran for one year, at the beginning of the Civil War (1860-1861).

  • Now let me introduce the Babylonians (or, rather, their mathematics).
    • Babylon, then (source)
      

    • Babylon, today

    • I want to introduce you to their mathematics via a "detective story": I hope that you all like detective stories.... The story is called Sherlock Holmes in Babylon.

      Today we're going to play Sherlock Holmes, and solve a mystery.

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

      

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

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