- Return of your homework problems (for chapter 1)
- Section 1.2:
- #4. Your "simple rules" could generally be simpler: Tom or Spencer, what's the rule?
- #6d: I abandoned this one because of misinterpretation. I get 903,002,644. How?
- Please show work whenever possible. Show me your scratch work -- it's okay!
- Please use more paper, those of you who did all of your assignment on one face of one piece of paper. (By the way, it's always guys!;)
- Section 1.3:
- #16 was tough!
- Section 1.2:
- Worksheet for sections 3.1-3.2 (for next time).
- Lessons from section 2.5
- Commercial (Jennifer -- who's up next?)
- Lessons from section 2.6
- Technology had its limitations (What's wrong with clay?).
- Babylonians are credited with an abstract, theoretical approach to mathematics, outstripping the Egyptians (encumbered as they were by their unit fractions)
- Ancient -- and erroneous -- idea of perimeter/area: (p. 67)
- armies estimated size by perimeter
- Island area by circumnavigation
- Real estate cheats (as late as Polybius, 207-125 BCE)
- But the Babylonians got that figure out, based on the tablet at the bottom of p. 67.
- They solved quadratic ("rectangular") equations; even special cubic equations (p. 63/64)
- Tables of reciprocals were missing numbers whose reciprocals
produced non-terminating decimal expressions (in sexagesimal).
What decimals in sexagesimal are terminating that don't terminate in base 10?
- Grand calculations were carried out (due to astronomical calculations? - p. 65)
- Quadratic formula (p. 65):
.
But how did they get it? Look at the table of p. 67, and "substitute patience for brilliance...."
- Various "problems" involving quadratic equations, including
- Transforming equations into "normal form" (p. 69)
FYI: turning "negative" normal form into "positive" normal form
- "Extraordinary example" of Negative numbers as RHS (p. 71)
- Negative roots neglected (p. 70)
- Multiple solutions absurd (p. 71)
- Transforming equations into "normal form" (p. 69)
- Plimpton 322 (1900-1600 BCE): a tablet containing "Pythagorean triples" (many years before Pythagoras, 6th century BCE)
- How were the triples found?
- Sexagesimal system dictated the choice of the values of m and n
- Many tablets excavated from Susa (including Hammurabi's Law)
- From the Cairo papyrus (circa 300 BCE):
- Egyptians were aware of several Pythagorean triples.
- Examples of the ol' ladder problems
A ladder of 10 cubits has its foot 6 cubits from a wall; to what height will it reach? (p. 78)(Babylonian version: "A beam of length ...." p. 78)
- Solving problems by the addition of a special form of
zero:
- Babylonians: "Square plus sides" method produces