MAT336: History of Mathematics

Spring 2026

Study Guide: Quiz 3

Ancient Egypt

Note: This is a study guide. The quiz will consist of three or four questions covering the material below. If you understand the ideas and facts in the non-computational questions and can work through the computational problems, you will be well prepared.

Part 1: Quiz-Style Questions

Using ancient Egyptian techniques, multiply 22 by 70.
Using ancient Egyptian techniques, divide 132 by 12.
Using the method of false position: A quantity and its 1/7 added together become 19. Find the quantity.
Here is a problem similar to one from the Rhind Papyrus: A circular field has diameter 18 khet. What is its area? The solution was: "Take away thou 1/9 of it, namely 2; the remainder is 16. Make thou the multiplication 16 times 16; becomes it 256; the amount of it, this is, in area 256 setat." (A khet is a unit of length. A setat is a unit of area, equal to one khet squared.) Find the formula for the area of the circle that the scribe would have obtained by starting with a circle of diameter d instead of diameter 18. (Hint: Start by taking away 1/9 of the diameter, that is d/9.) Give your final formula in terms of d only.
We know the area of a circle is π times the radius squared: Area = πr². Use your work in Problem 4, remembering that d = 2r (the diameter is twice the radius), to interpret the Egyptian calculation as an approximation of π. Express this approximation as a fraction. (You do not have to perform the final multiplications. For instance, 23/(66×34) is an acceptable format.)
What two arithmetic operations form the basis of most Egyptian computations? Give an example of how each is used in Egyptian calculations.
Explain in 2–3 sentences why the doubling-and-adding method works. (Hint: any whole number can be expressed as a sum of distinct powers of 2.)

A part of the Rhind Mathematical Papyrus table of division by 2 follows:

2 / 11 = 1/6 + 1/66
2 / 13 = 1/8 + 1/52 + 1/104

The calculation of 2 / 13 is given as follows:

1	13
1/2	6 + 1/2
1/4	3 + 1/4
1/8	1 + 1/2 + 1/8
1/52	1/4
1/104	1/8

Since 1 + 1/2 + 1/4 + 1/8 + 1/8 = 2, we have 2 / 13 = 1/8 + 1/52 + 1/104.

Following this method, perform the calculation for 2 / 11 to verify that 2 / 11 = 1/6 + 1/66.

(Problem adapted from Victor J. Katz, A History of Mathematics)

Using ancient Egyptian multiplication techniques and the 2/n table, verify that 11 × (1/6 + 1/66) = 2.
(Problem adapted from Victor J. Katz, A History of Mathematics)
Give an approximate timeline (century is fine): When was the Rosetta Stone created, and when was it deciphered?
Name at least two of the three types of Egyptian writing shown on the Rosetta Stone, and explain why its discovery was important for history.
What is the Rhind Mathematical Papyrus? In which century was it written, and what was its purpose?
Judging from the Rhind and Moscow papyri, how was mathematics learned and transmitted in Ancient Egypt?
Recall that Problem 79 of the Rhind Papyrus, which involved adding 7 + 7² + 7³ + ... + 7⁶, can be interpreted as theoretical. What makes this "theoretical" rather than "practical"?
The following is a paragraph from the book "The Universal History of Numbers" by Georges Ifrah: "Another example of high numbers is from a statue from Hieraconpolis, dating from 2800 BCE, where the number of enemies slain by a king called KhaSeKhem is shown as X by the following sign." You need to find X, that is, the number of enemies slain from the hieroglyphic below. It will come in handy to read how the book continues: "Early examples show rather irregular outlines and groupings of the signs," given that our example shows such an irregular grouping.

Hieroglyphic inscription from Hieraconpolis (c. 2800 BCE)

Part 2: Reflection Questions

Why powers of 2 in the Egyptian algorithm?
The Egyptian multiplication method relies on writing the multiplier as a sum of distinct powers of 2.

Now try to imitate this idea using powers of 3, but restrict yourself to sums of distinct powers of 3 (so each power may be used at most once).

(a) Try to compute 18 × 6 using this restriction. (Can you write 6 as a sum of distinct powers of 3? What about 18?)
(b) Explain clearly why this “powers of 3” version does not give a uniform algorithm for all multipliers.
(c) What extra feature would you need to add to make a base-3 version work for all numbers?
(d) State the key advantage of using powers of 2 in the Egyptian method.
Write your own definition of "measuring a segment."
The "scissor congruence" slides hint that the area of polygonal shapes can be defined by cutting and rearranging the resulting pieces. What does this tell you about area?
The cube-vs-pyramid demonstration question: Does sand-filling count as a proof? Explain what would have to be added (if anything) to make it "mathematical."
Your view on Problem 79 of the Rhind Papyrus—practical or recreational? What would convince you it's still practical, and what would convince you it's recreational?
Compare Egyptian multiplication with the multiplication you learned in school. Which is easier or harder for students to learn, and why?
Looking at the Rhind and Moscow papyri problems: they're practical (bread, beer, pyramids, grain). What does this tell you about who was doing mathematics and why?

Part 3: Challenge Problems

Imagine you are an Egyptian scribe. Write clear step-by-step instructions (in English) to compute the volume of a square pyramid with side 9 and height 9. (Hint: Follow the ideas of Problem 14 of the Moscow Papyrus.)
Deduce algebraically the volume formula of a truncated square pyramid and then think about the tools and concepts you use. Which of those appear (or do not appear) in the Egyptian sources we discussed?

Quiz Problem Rubric

Points	Criteria
3	Correct answer with reasoning/work shown
2	Partially correct with some reasoning shown
1	Correct answer without reasoning/work OR significant attempt with some understanding
0	Incorrect or blank

Notes

For computational problems: "reasoning/work" = steps shown
For conceptual problems: "reasoning" = explanation given
Round partial credit up when in doubt