Day 5, Mat305/594, History of Mathematics

Last time: Sections 2.1, 2.2

Next time: Section 2.5

Today:

• Announcements
  • Your problems (for chapter 1) are due today
  • Worksheet for section 2.5 (for next time).
  • Pronunciation guide for mathematics (local copy)

• Several things today (a lot, actually!):
  • More lessons from Sections 2.2
  • Commercial -- Ashley (Next time: Katie)
  • Lessons from Section 2.3
  • (There is no way that we will get to) Lessons from Section 2.4

• Mathematical Elements, lessons and problems from section 2.2
  • Multiplication
  • Whole number division
  • When fractions arise
  • Multiplying with fractions
  • Dividing with fractions

• Mathematical Elements, lessons and problems from section 2.3: Problems of the Rhind papyrus
  • Problem #22 -- "completion"
  • Problem #24 -- false position
  • Fibonacci's egg problem -- false position (how many eggs?)
  • Double false position:

    Robert Recorde, born 1510; author of Grounde of Artes:

    'finding the true result from the chance answers of "such children or idiots that happened to be in the place."'

  • Problem #28 -- "think of a number"
  • Problem #79 -- sum of a geometric series?

• From the Rhind: "Knowledge of all obscure secrets" comes down to multiplication and division.
• Multiplication is accomplished through binary decomposition (successive doublings).
• Unit fractions -- the key and the curse
• The Rhind papyrus contains the Table of unit fractions multiplied by 2, which is critical for doublings (this table constitutes 1/3 of the Rhind papyrus! -- p. 40).
• An example problem from the Moscow Papyrus
• What's with 2/3?
  

• The splitting identity --
  

  allows us to represent all rational numbers as sums of unit fractions.

• Fibonacci also developed a method for doing so.
• A fantastic site on Egyptian Fractions (including a calculator to turn your rationals into sums of unit fractions).
• Another
• Ancient Egyptian Science: A Source Book (by Marshall Clagett)