Discussion and Homework
First Reading Assignment

Questions for Discussion

Reading 1

1. To get used to sexigesimal notation, verify that 1;24,51,10 (sexigesimal) is indeed appoximately equal to 1.414 (decimal). (See pages 11 and 19.)
2. In the table of reciprocals on page 17, verify the pair 54 and 1,6,40.
3. Think about decimal representation and irrational numbers. For sexigesimal representation, what "takes the place of" irrational numbers? Do the Babylonians approximate these numbers? Find an example in the reading.
4. Verify that the solution to x y = 1, x + y = b given on page 22 is correct.

1. How are multiplication and division usually introduced in elementary school? Can you relate this to the Rhind papyrus?
2. Can every positive integer be written as a sum of powers of 2? Uniquely?
3. Read carefully about "false position" on pages 29-31. Use the technique to solve
   2/3 x - 1/2 y = 500, x + y = 1800.
   First assume that x = y = 900.

Homework Problems

Reading 1

1. Multiply 237 by 18 using the Egyptian method.
2. Divide 242 by 11 using the Egyptian method.
3. Divide 19 by 8 using the Egyptian method.
4. Find a unit fraction representation for 9/20 and 4/15.
5. Use the Egyptian method to approximate Sqrt[42/121].
6. • a. Verify the identity 1/2 (x² + y²) = [(x + y)/2]² + [(x - y)/2]².
   • b. Use part a. to solve the system of equations
     x - y = a
     x² + y² = b
     using techniques similar to the Babylonian method.
7. Look up the area of a regular n-gon and write the formula in terms of the length of a side, sn only. Calculate an approximation using the same n values as on page 25. How does the formula compare to the Babylonian area formulae on page 25?
8. • a. Show that taking v + u = 2;24 (2 2/5) leads to line 1 of Plimpton 322.
   • b. Show that taking v + u = 1;48 (1 4/5) leads to line 1 of Plimpton 322.
   • c. Find the value for v + u that leads to line 6.
   • d. Find the value for v + u that leads to line 13.
9. The Babylonians had a table that gave whole number values of n, n², n³ and n² + n³. Consider the cubic equation x³ + a x² + b x + c = 0. Let y = x + s and substitute. What assumption on s is needed to reduce to a cubic of the form y³ + A y² = D? Make this substitution. Make another substitution of the form y = something to reduce to a form where the Babylonian table described above can be used, that is, get to the form n³ + n² = something else.

   
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