Download 1 EGYPTIAN ARITHMETIC Bill Richardson Multiplication Egyptian

1
EGYPTIAN ARITHMETIC
Bill Richardson
Multiplication
Egyptian multiplication is based on a property of the sequence of integers
1, 2, 4, 8, 16, 32, 64, ... .
It is the fact that any integer, not in this sequence, can be written as the sum of two or more
of these integers. (Since the terms of this sequence are merely the powers of 2, it is the fact
that every integer has a binary representation.) Whether the Egyptian scribes were aware
of this mathematical fact is not known, but it is certain that they made use of the property.
For example, to multiply 24 by 13 the scribe would write the equivalent of the following.
√
1
24
2
48
√
96
√ 4
8
192
13
312
The left column stopped at 8 since the next double is 16 which is bigger than 13. The scribe
then identified the numbers in the left column (1 + 4 + 8) that summed to 13 and put a check
beside the required numbers. He then added the corresponding numbers (24 + 96 + 192) in
the right column to get the desired product 312.
If he preferred, he could have multiplied 13 by 24. In this case he would have written
√
√
1
2
4
8
16
24
13
26
52
104
208
312
since 8 + 16 = 24
We give one more example. Compute the product 21 × 47.
√
1
47
94
√ 2
4
188
376
√ 8
16
752
21
987
Since 1 + 4 + 16 = 21 we have 47 + 188 + 752 = 987 as the desired product.
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Division
The method for division was very close to that for multiplication. In fact, division problems
were posed as multiplication problems. Instead of asking, “How many times does 9 divide
225?”, the scribe would ask, “By what must I multiply 9 to get 225?” He would then write
1
2
4
8
16
9
18
36
72
144
Here he would stop multiplying by 2 (doubling) for the next doubling in the right column
would exceed 225. He would now identify the numbers in the right column which sum to
225.
√
1
9
2
18
4
36 √
8
72 √
16
144
25
225
Finding that 9 + 72 + 144 = 225, he would then add the corresponding multipliers 1, 8, and
16 to get the desired number, 25.
NOTE: 9 + 72 + 144 = 9 × 1 + 9 × 8 + 9 × 16 = 9 × (1 + 8 + 16) = 9 × 25 = 225.
What if the division did not yield a whole number? Could the Egyptians handle this? If the
previous problem were changed to divide 227 by 9, all would be the same except that 225
is the closest one could get to 227 adding numbers in the right column. The answer would
then be “25 plus a remainder of 2” or 25 29 in our notation. How did the scribes treat this
problem? Just like we do–with fractions; however, their treatment of fractions was a little
different from ours.
Fractions
The Egyptians did not hesitate to use fractions. In fact, the approach they used influenced
the treatment of fractions for the next 4000 years. Even the great Greek mathematicians
copied the Egyptians. Except for 23 and possibly 34 , the Egyptians worked with fractions
with numerators 1 (that is, the reciprocals of the positive integers). In about 1650 BCE
the scribe Ahmes made a papyrus roll 18 feet long and 13 inches high from writings made
about 200 years earlier (about 1850 BCE). This is known as the Rhind papyrus. The first
six feet of the roll (often referred to as the RECTO) is a table of the division of 2 by the odd
numbers from 3 to 101. The remainder of the roll consists of 87 mathematical problems.
Except for 2 divided by 3 all the divisions were represented as a sum of fractions with numer1
ators 1. For example, 2 divided by 5 was not written as 52 or as 15 + 15 , but instead as 13 + 15
.
3
If at all possible, the original divisor was not used as the denominator of any of the fractions.
Time does not permit a complete discussion of the Egyptian method of working with fractions. For those interested in pursuing the topic, a more complete discussion can be found
in Gillings[4].
The scribes were aware of many fascinating rules for working with fractions. One such rule
1
1
kn
1
= , provided m is an integer given by m =
.
that was often used was +
n nk
m
k+1
For example,
1
1
1
48
+
= , since 48 = 3 · 16 and 12 = .
16 48
12
4
1
1
1
20
+
= , since 20 = 4 · 5 and 4 = .
5 20
4
5
1
1
24
1
+
= , since 24 = 2 · 12 and 8 = .
12 24
8
3
On the other hand,
1
1
+
cannot be reduced in this way, since 32 = 2 · 16, but 2 + 1 = 3 does not divide 32.
16 32
It appears that the Egyptian scribes were aware of many arithmetic properties of numbers
and they made good use of them. How these properties were discovered is still a mystery.
Most of the papyrus rolls are themselves copies of earlier rolls not extant. Thus, we can only
speculate on the actual knowledge of the Egyptians. We can however note that the scribes
made very few arithmetic errors. We should be so lucky!
Some of the techniques used by the Egyptians may seem very strange to us, but remember,
we are examining the work from the perspective of our own number system which is very
different from the Egyptian system.
For example, one technique used quite often in the Rhind Mathematical Papyrus is the computation and use of 32 as a multiplier of complicated fractions. The scribe had a table of 23
times various numbers and he then used these values in other problems.
If we were asked to find
1
3
7
of 1 12
using the Egyptian notation, we might write
1
1 + 13 +
1
4
7
(= 1 12
)
1
3
1
3
1
12
(each term above multiplied by 31 ).
+ 19 +
1
We would then have to determine what 13 + 91 + 12
should reduce to–not an easy problem
for the Egyptians. However, in Problem 32 of the Rhind Mathematical Papyrus, the scribe
works as follows:
4
So instead of finding
took 21 of this!
1
3
1
1 + 13 +
2
3
1+
1
18
(from his table).
1
3
1
2
1
36
(=
+
1
4
7
(= 1 12
)
19
36
in our notation).
of the value directly, he first found
2
3
of the value using his table and
As an example, let us compute 7 78 × 12 23 as the Egyptians might have dome. First we note
that the problem would be to multiply 7 + 12 + 41 + 81 by 12 + 23 . We will look at two possible
methods.
Method 1. We will compute the product 7 87 × 12 23 by first multiplying 12 23 by 7 and then
by 12 + 41 + 18 and then sum the two results. Note: The material in parentheses represent
steps the scribe would not write down–we put them in for clarity.
√
1
12 + 32
(2
√
√
24 + 1 + 13 )
2
25 +
1
3
4
50 +
2
3
7
88 +
2
3
For the second part we have
√
√
√
1
2
+ 14 +
1
12 +
2
3
1
2
6+
1
3
1
4
3+
1
3
1
8
1 + 12 +
1
12
1
8
10 + 21 + 13 + 16 +
1
12
1
1
1
But 12 + 13 + 16 = 1. Thus, 10 + 12 + 13 + 16 + 12
= 11 + 12
. Hence the result is 88 + 23 + 11 + 12
1
1
1
or 99 + 23 + 12
. Now 23 = 21 + 16 , so 23 + 12
= 12 + 16 + 12
= 12 + 14 . The answer is then 99 + 12 + 14
(or, in our notation 99 43 ).
Method 2. If we make use of the Egyptians’ ability to find
7 + 12 + 41 + 18 by 12 and then by 23 and add the results.
2
3
of a number, we can multiply
5
1
√
14 + 1 + 12 +
1
8
1
4
)
2
15 + 21 +
1
4
(4
30 + 1 + 12 )
(2
√
7 + 12 + 14 +
1
2
4
31 +
(8
62 + 1)
8
63
12
94 +
1
2
Next the scribe would write
1
( 23
7 + 12 + 41 +
(4 + 32 ) + ( 31 + 16 +
2
3
1
8
1
))
12
5+
1
4
He would then be able to add these two results to get
1 1 1
1
1
1 1
2
× 7+ + +
= 94 +
+ 5+
= 99 + + .
12 +
3
2 4 8
2
4
2 4
It is reasonable to guess that the Egyptian scribe would have used Method 2 to compute the
required product.
Bibliography
1. Boyer, Carl B. A History of Mathematics, John Wiley, New York, 1968
2. Bunt, Lucas N.H., Jones, Phillip S., and Bedient, Jack D. The Historical Roots of Elementary Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey, 1976
3. Eves, Howard An Introduction to the History of Mathematics, 4th ed., Holt, Rinehart,
and Winston, New York, 1976
4. Gillings, Richard J. Mathematics in the Time of the Pharaohs, Dover, New York, 1982
5. Struik, Dirk J. A Concise History of Mathematics, Dover, New York, 1948