Egyptian and Babylonian Mathematics

Exercises on Egyptian Mathematics.


  1. Compute $623\times 255$ using the Egyptian binary algorithm.

  2. Compute $25\times 31$ using the Egyptian binary algorithm.

  3. Compute $7112\div 127$ using the Egyptian binary algorithm.

  4. Show that if $n$ is a multiple of five, $\dfrac{2}{n}$ can be broken into the sum of two unit fractions, one of which is a third of $\dfrac{1}{n}$.

  5. Solve $x+\frac{1}{2}x=16$ using the mathod of false position. Be sure to express the fractional answer alá the Egyptians. (This is Proposition 16 of the Ahmes.)

  6. Solve the following problem using the method of false position: If thrice a heep plus a fourth more is 247, what is the heep?

  7. Devise a method by which the Egyptians may have found the formula for the exact volume of a pyramid? (Remember, the formula can be computed by calculus, but the Egyptians did not have that tool.)

  8. The amount of bread to be distributed to four persons, $A,\;B,$ and $\;C$ are in the continued proportions MATH If there are 1200 loaves of bread to be distributed, how much does each get?

  9. Solve the following problem using the method of false position: If thrice a heep plus a fourth more is 247, what is the heep?

  10. Using the Egyptian method find the area of a circle of diameter 11.

  11. Pedagogy: Explain the issues of teaching mathematics by example, ála the Ahmes Papyrus. What aspect make teaching this way simpler? harder? What kind of graduate is the result of such instruction? (For example. Are graduates capable? Are graduates flexible?)

  12. Complete the checklist for Egypt by filling in the scaled numbers and justifying by essay answers the values you have given. You may cite comments or problems from the text or lecture notes; you may also contribute your own observations.

Exercises on Babylonian Mathematics .


  1. Express the numbers 76, 234, 1265, and 87,432 in sexagesimal.

  2. Compute the products. Use the product form MATH

    1. $1,23\times 2,9$

    2. $2,4,23\times 3,34$

  3. Convert the following numbers into sexagesimal and perform the computations in sexagesimal. Perform the computations. Use here the form MATH where MATH and all the MATH For multiplication use the form MATH. For example, in decimal MATH. Remember the Babylonians had a table of squares up to $59$.

    1. MATH

    2. $12+37$, $120-98,$ $3200+420$

    3. $23\times 12,$ $210\times 52$

  4. Determine $17\,\div \,24\,$ as a sexagesimal number. Here you need to use this form: MATH

  5. A problem on one Babylonian tablets give the base and top of an isosceles trapezoid to be 50 and 40 respectively and the side length to be 30. Find the altitude and area. Can this be done without the Pythagorean theorem?

  6. Solve the following system ála the Babylonian "false position" method. State clearly what steps you are taking.
    MATH
    (The solution is (200, 400).)

  7. Generalize this Babylonian algorithm for solving linear systems to arbitrary linear systems in two variables?

  8. Generalize this Babylonian algorithm for solving linear systems to arbitrary linear systems?

  9. Modify the Babylonian root finding method (for $\sqrt{2}$) to find the square root of any number. Use your method to approximate $\sqrt{3}$. Begin with $x_{0}=1.$ Compute five iterations.

  10. Explain how to adapt the method of the mean to determine $\root{3} \of{2}$ .

  11. Show that the general cubic $ax^{3}+bx^{2}+cx=d$ can be reduced to the normal form $y^{3}+ey^{2}=g$.\ Determine Modify the Babylonian root finding method (for $\sqrt{2}$) to find the square root of any number. Use your method to approximate $\sqrt{3}$. Begin with $x_{0}=1.\medskip $

  12. Consider the table:
    MATH


    Solve the following problems using this table and linear interpolation. Compare with the exact values. (You can obtain the exact solutions, for example, by using Maple: evalf(solve($x^{3}+x^{2}=a,\ x$ )). Here $a$ $=$ the right side.)
    MATH

  13. Derive the approximate value of $\pi $ as determined from the data at Susa.

    That is, show how the perimeter identity
    MATH
    is used to derive the approximation for $\pi $.

  14. Factor the other quadratic forms ála the Babylonian methods. You may use variables, but not general formulas.

  15. Complete the checklist for Mesopotamia by filling in the scaled numbers and justifying by essay answers the values you have given. You may cite comments or problems from the text or lecture notes; you may also contribute your own observations.

  16. Pedagogy: Explain the issues of teaching mathematics when clay tablets are used for written communication. What difficulties do you perceive in teaching the sexagesimal system? Using tables to solve nonlinear equations requires a working knowledge of interpolation. Explain why interpolation is a naturally occurring task in everyday life, even today.Write a lesson plan wherein you show students how to

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