MONT 108N – Mathematics Through Time Makeup Midterm Exam, November 3, 2010 Your Name:

Directions

Do all work on the sheets provided (if you use the back of a sheet, please place a note telling me to look there). There is an extra sheet of scratch paper at the end. You may detach that if you like, but please hand it in with your exam.

I. Compute “the Egyptian way”

A) (10) 51 × 187

. B) (10) 1332 ÷ 47 (find the quotient and the remainder)

A) (10) Express the base 8 number (246)_{8} in base 10.

B) (10) Express the base 2 number (111000111)_{2} in base 60.

III. A Babylonian problem text asks for the side of a square if the area of the square minus the side is the base 60 number (14 : 30)_{60}. The tablet says to do this to solve the problem (all numbers in base 60, of course!): “Take half of 1, which is 0.30, square that to get 0.15, add the 14 : 30 to get 14 : 30.15. The last number is the square of 29.30. Now add the 0.30 to get 30, which is the side of the square.”

A) (15) Solve this problem using modern methods.

B) (5) Explain how the Babylonian method of solution is essentially the same as using the quadratic formula for an equation of the form x^{2} − px = q.

IV. Short answer: A) (2.5) What base 10 number is represented by these Egyptian hieroglyphs?

Answer: B) (2.5) What base 10 number is represented in the Babylonian way by these symbols?

Answer: C) (2.5) Approximately when was the Old Babylonian period in Mesopotamian history? Answer: D) (2.5) The name for the writing system used in part B is Answer: E) (2.5) What are the surviving records of Egyptian mathematics? Answer: F) (2.5) Most of the geographical area of the Old Babylonian kingdoms is contained in

what present-day country? Answer: G) (2.5) What was special about the Egyptian way of dealing with fractions? Answer: H) (2.5) Which Old Babylonian tablet apparently contains a table of Pythagorean triples? Answer:

V. Essay. (20) “The distinguishing feature of Babylonian mathematics is its algebraic character.” Of the historians we have mentioned, who would agree with this claim, and who would disagree? Explain using the the interpretations your historians would give for the YBC 6967 problem of solving the equation x = 60/x + 7.

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