Faculty of Mathematics
Waterloo, Ontario N2L 3G1
Centre for Education in
Mathematics and Computing
Grade 6 Math Circles
November 1st /2nd
Egyptian Mathematics
Ancient Egypt
One of the greatest achievements of the ancient Egyptian civilization was their sophisticated
development of mathematics. As early on as 2700 (BC), perhaps earlier, a base 10 number system had already emerged for agricultural and religious reasons. In addition, they
pioneered some of the most robust techniques of their time to solve very practical problems.
Though their use of mathematics was extensive, they had little regard for formulas or complex
mathematical relations. They often employed trial and error and used close approximations
as oppose to exact answers for their calculations.
Luckily, the ancient Egyptians left behind the Rhind Papyrus which explains how they
did arithmetic and geometry. It gives explicit demonstrations of how multiplication and
division was performed at that time and provides insight on their knowledge of unit fractions,
composite and prime numbers, arithmetic, geometric, and harmonic means, and quadratic
equations.
In this week’s circles, we will be exploring some the techniques the Egyptian implored in
their daily lives.
Egyptian Multiplication
The ancient Egyptians had an interesting way of multiplying of two numbers. Unlike us,
the ancient Egyptians did not possess nor memorize their multiplication table. They were,
however, capable of multiplying and adding by 2. In other words, they only knew their 2
times tables. However, with that ability alone, they were able to multiply any two numbers!
They key was to reduce all multiplication problems into a problem that involved multiplying
by 2.
1
Example. Let’s multiply 21 by 18 (using the notation we see today).
21
18
1
2
4
8
16
18
36
72
144
288
1. Create a chart with two columns, with the the first and second number on top of the
first and second column respectively
2. In the first column, write down all the powers of 2 that are smaller or equal below 21
3. In the second column, starting from 18 write down multiples of 18 until you reach the
last row
4. Starting at the bottom number of the first column. determine which powers of 2 add
up to the 21.
To figure out which powers of 2 that are add to 21. We include the biggest power of 2 that
is less than 21, so 16. We then subtract 21 − 16 = 5 and find the next largest power of 2
that is equal to or less than 5. In this case it is 4. Repeating this process, with 5 − 4 = 1,
the next power of 2 equal to or less than 1 is 1 itself. So we cross out all powers of 2 that
we did not use and the numbers in the column next to it. The remaining numbers should
be except 16, 4, and 1 and the numbers in the column beside it
21
18
1
18
2 36
4
72
8 144
16 288
We add the remaining numbers on the right column 18 + 72 + 288 = 378. So we have that
21 × 18 = 378
2
Exercise. Evaluate the following using Egyptian Multiplication.
1. 22 × 21.
2. 33 × 24.
Egyptian Division
The Egyptians did division in a very similar manner as multiplication.
Example. Let’s divide 25 by 4.
1. Create a chart with two columns, with the the dividend i.e. 25 on top of the first
column and the divisor i.e. 4 on top of the second column
2. Below the divisor (in this case 4), keep doubling 4 on each row until you reach the
largest possible number that is less than 25.
3. Below 25, starting with 1, keep doubling until you reach the same row as the largest
multiple of 4.
4. Subtract the largest possible number from the right column from 25. If you have
something remaining, repeat this procedure until all the numbers on the right column
is larger than what you have left over.
3
25
4
1
2
4
4
8
16
Now we subtract 25 with the largest possible number on the right column. Afterwards, we
repeat the process with what is left over until we can no longer subtract. First, 25 − 16 = 9.
Since we are left we 9, the biggest number we can subtract from the right column is 8, so
9 − 8 = 1. We are left with 1, with no possible others numbers to subtract from since all
numbers on the right column, 4, 8, and 16 are greater than 4.
Then we cross off the remaining terms that aren’t used.
25
4
1
2
4
4
8
16
To obtain our final answer we add the remaining numbers on the left column 2 + 4 = 6. So
we have a remainder of 1 with a quotient of 6.
Exercise. Using the Egyptian Method evaluate 132 ÷ 12
4
Egyptian Fractions
The Egyptians of 3000 BC incorporated the use of fractions in a very interesting way. They
1 1 1
exclusively used unit fractions (a fraction where the numerator is always 1) i.e , , .
2 4 7
2
3
They did NOT write fractions like or like we do today.
5
4
Egyptian Notation
The Egyptians used the hieroglyph
above a number to represent a fraction with a numerator of 1. For example, we have that
How do they represent other fractions? It turns that they represent all other fractions
as a sum of different unit fractions.
Example.
3
1 1
= +
4
2 4
However,
2
1 1
= +
3
3 3
was not allowed since it was the sum of the same unit fraction.
Why use Egyptian Fractions at all?
Motivating Problem. Suppose you are in ancient Egypt and you have 5 loaves of bread
to share among 8 workers. How would you ensure that you distribute 5 pieces of bread into
8 equal portion? Writing it as 58 does not help, that just simply restates the problem. If we
convert 85 into a decimal, we get 0.625, but how much is 0.625 of 5 loafs of bread anyway?
Think about this for a second. How would you divide 5 pieces of bread amongst 8 people?
Draw out your solution below.
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Exercise. How about 13 loaves share among 12 people? Can you use what you have seen
in the previous answer to help you out?
The next question naturally should be, how did the ancient Egyptians break a fraction into
a sum of different unit fractions? Secondly, is there one way to break into different unit
fractions? Let’s make some interesting observations first.
Pattern 1. Every Unit Fraction is a Sum of Two Distinct Unit Fractions
Evaluate the following. Did you notice a pattern?
1.
1
1
+
4 12
2.
1
1
+
5 20
3.
1 1
+
3 6
4.
1
1
+
7 42
5.
1
1
+
6 30
Can you use the pattern you observed to evaluate
The general pattern is:
6
1
1
+
20 380
3
1
1
3
Problem. We already know that
= + . Is it possible to express
as a sum of 3
4
2
4
4
distinct fractions? 4 distinct units fractions? Express your answer below.
3
as a sum of n distinct unit fractions, where n is the
In general, is it possible to write
4
number of terms or fractions? Justify your answer.
Pattern 2. Egyptian Fractions with Even Denominators
Evaluate the following expressions below. Can you spot another pattern?
1.
1
1
+
8 24
2.
1
1
+
6 12
3.
1
1
+
10 40
The general pattern is:
7
Fibonacci’s Greedy Algorithm
a
a
, where
is not a unit fraction i.e the numerator is
b
b
a
not 1. Fibonacci developed a method to express as a sum of distinct unit fractions. This
b
a
particular method involves finding the biggest fraction we can take from . Afterwards, we
b
repeat the process with what is left over.
Suppose we have a fraction where
7
as a sum of unit fractions.
15
4
Exercise. Using the Greedy Algorithm, express as a sum of unit fractions
5
Example. Using the Greedy Algorithm, express
The greedy algorithm always works, but sometimes, we may end up with horrendously large
denominators.
If we use the Greedy Algorithm to reduce
5
121
into a sum of unit fraction we will get:
1
1
1
1
1
5
=
+
+
+
+
121
25 757 763309 873960180913 1527612795642093418846225
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Problem Set
1. Use the Egyptian Method of Multiplication to evaluate the following
a. 28 × 24
b. 15 × 8
c. 12 × 13
2. Use the Egyptian method of Division to evaluate the following
a. 25 ÷ 6
b. 123 ÷ 8
c. 177 ÷ 25
3. Can you come up with a sum of distinct Egyptian Fractions that add up to 1.
Hint: It may help to know that 6 = 3 + 2 + 1
4. What is the largest unit fraction that is less than the following Egyptian fractions?
2
5
7
(b)
9
26
(c)
100
3
(d)
10
(a)
5. Express the following unit fractions as a sum of two distinct unit fractions.
1
1
1
b.
a.
8
10
11
6. Use Fibonacci’s Greedy Algorithm to express the following fractions as a sum of distinct
unit fractions.
11
2
a.
b.
12
5
7. Rhind Papyrus Problem 28 when translated into English reads as follows:
Think of a number, and add 23 of this number to itself. From this sum subtract
value and say what your number is. Suppose the answer is 10. Then take away
this 10 giving 9. Then 9 was the first number thought of.
1
its
3
1
of
10
The scribe, Ahmes, then gives the following argument:
If the original number was 9, then 32 is 6, which added makes 15. Then
which on subtraction leaves 10. This is how you do it.
1
3
of 15 is 5
If you read this carefully, you will find that Ahmes was trying to prove the equation
below:
2n
1
2n
1
2n
1
2n
n+
−
n+
−
n+
−
n+
=n
3
3
3
10
3
3
3
9
(a) Use n = 9 and verify that the left and right side of the equation do equal to each
other
(b) Algebraically simplify the left side of the equation and show that it is indeed equal
to the right side for all values of n.
Hint: Notice that a certain expression in brackets appear multiple times. If you
simplify it once you can use it throughout the whole expression.
8. The Inheritance Puzzle A man who had 12 horses and 3 children wrote on his will
1
to Same. However, just after he died
to leave 12 of his horses to Pat, 13 to Chris, and 12
one of his horses died too. How would they divide the remaining horses so as to fulfill
term of the will?
9. Suppose we have a shelf of identical books. If we lay them down one on top of another,
what is the maximum overhang we can achieve? Can the top book ever completely
overhang the bottom book?
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