Day 20, 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.

• I hope you had a good and relaxing Spring break.

• You have an IMath assignment, which will be due this Friday, 3/18

• This from NKU's President, RE Covid:

  Effective Tuesday, March 8, masks will be optional on campus, with these exceptions:

  • Masks are required during class meetings in classrooms and lab spaces, through Monday, March 21.
  • Masks are required in NKU Health, Counseling and Student Wellness facilities.
  • Masks are required for at least five days following diagnosis of or exposure to COVID-19, per CDC policy.
  • Individual employees may continue to require masks in individual private offices where they meet with students, other employees or visitors.
  So one more week we wear the masks in class; and then I think that I, at least, will be maskless. You will be welcome to mask or not in class, as of 3/21.

• Babylonians:

• Mayans:

  What questions do you have?

  Let's do a couple of quick examples!

We will begin with Egyptian multiplication.

(The great news is that it reduces to the Fraudini trick!) It seems that the Egyptians didn't mind doubling things: that was easy for them; and that's the key to Egyptian multiplication.

• Let's start with a look at the history of a western understanding of Egyptian Mathematics.

• At the heart of Egyptian multiplication is the "Fraudini fact", or the "binary factorization":
  Every natural number is either a power of two, or can be expressed as a sum of distinct powers of two in a unique way.

  (I underline "distinct" because you cannot repeat powers: otherwise you could write, for example,

  3=1+1+1 rather than 3=2+1 (which is the unique binary factorization).

• Again this is parallel to prime factorization (and another factorization which is to come!):
  Every natural number (other than 1) is either prime, or can be expressed as a product of prime numbers in a unique way.

• We'll start with a simple example: multiply 23*42.

  1	42
  2
  4
  8
  16
  32	Too big!

  Now add up those rows marked with an asterix (*), and you'll get the answer (966).

• Let's show that we can do a multiplication in either order, by checking the product 42*23. (What is "too big" in this case?)

• Let's try a longer multiplication. Consider, for example, 321*112:

  1	321
  2
  4
  8
  16
  32
  64
  128	Too big!

  Now add up those rows marked with an asterix (*), and you'll get the answer (35952).

• Examples: Try these:
  1. 43*16
  2. 21*79

• Links:
  • Mayan Calendar, on Wikipedia