If I ask you what the product
of 13 and 7 is, you will answer immediately: it is
91. And what if I ask you how you got it? I am
pretty sure you are going to laugh at me, because the
question is too trivial.
13
x 7
91
Now, consider it is 2000 BC in Egypt and I ask you
the same question. Will the answer of “how” be the
same? It makes you think twice, doesn’t it?
Ancient Egyptians were quite advanced in
mathematics and knew how to multiply two numbers,
but they used alternate methods of multiplication
than we do today.
For example, to multiply two numbers such as 13
and 7, Egyptian scribes first identified which number
to use as the multiplier and which number as the
multiplicand. In this instance, let’s use 13 as the
multiplicand and 7 as the multiplier.
The next step is to write these numbers in columns,
with the multiplier in the second column and
doubled in each row. In the first column, write the
numbers 1, 2, 4, 8, 16, 32, 64, … until you exceed
the multiplicand. Here the multiplicand is 13, so we
would write 1, 2, 4, 8, but not 16, because that would
exceed the multiplicand 13.
So, the scribe would have written something like this:
1 7
2 14
4 28
8 56
Here is the twist: the Egyptians at that time were
smart enough to know that in order to express any
number from 1 to 2n, we can add numbers from 20,
21, 22 … 2(n-1). This is the system of binary arithmetic,
which is used by computers.
The next step would be to identify the numbers from
the left column that add up to the multiplicand
(here 13).
Doubtful? Try it with any number! You can also prove
it easily by mathematical induction if you use the
strong induction hypothesis.
Here is an example: if a trader wants to give his
customer any weight between 1 and 255 kg, what is
the minimum number of weights he needs?
The answer is 8, and the weights are 1, 2, 4, 8, 16,
32, 64, 128. Using them, the trader can measure any
weight between 1 to 255 kg. For example, to weigh
153 kg, the trader will use the following weights:
128 + 16 + 8 + 1 = 153.
You can try any other weights.
So, back to our multiplication question,
1 + 4 + 8 = 13.
To get the product of 13 and 7, the scribe would
take 1st, 3rd and 4th row and add the numbers in
the second columns of those rows. In our case, the
numbers would be 7, 28 and 56. When we add those
we get
7 + 28 + 56 = 91.
Bingo! We have indeed got the correct 3
If I ask you what the product
of 13 and 7 is, you will answer immediately: it is
91. And what if I ask you how you got it? I am
pretty sure you are going to laugh at me, because the
question is too trivial.
13
x 7
91
Now, consider it is 2000 BC in Egypt and I ask you
the same question. Will the answer of “how” be the
same? It makes you think twice, doesn’t it?
Ancient Egyptians were quite advanced in
mathematics and knew how to multiply two numbers,
but they used alternate methods of multiplication
than we do today.
For example, to multiply two numbers such as 13
and 7, Egyptian scribes first identified which number
to use as the multiplier and which number as the
multiplicand. In this instance, let’s use 13 as the
multiplicand and 7 as the multiplier.
The next step is to write these numbers in columns,
with the multiplier in the second column and
doubled in each row. In the first column, write the
numbers 1, 2, 4, 8, 16, 32, 64, … until you exceed
the multiplicand. Here the multiplicand is 13, so we
would write 1, 2, 4, 8, but not 16, because that would
exceed the multiplicand 13.
So, the scribe would have written something like this:
1 7
2 14
4 28
8 56
Here is the twist: the Egyptians at that time were
smart enough to know that in order to express any
number from 1 to 2n, we can add numbers from 20,
21, 22 … 2(n-1). This is the system of binary arithmetic,
which is used by computers.
The next step would be to identify the numbers from
the left column that add up to the multiplicand
(here 13).
Doubtful? Try it with any number! You can also prove
it easily by mathematical induction if you use the
strong induction hypothesis.
Here is an example: if a trader wants to give his
customer any weight between 1 and 255 kg, what is
the minimum number of weights he needs?
The answer is 8, and the weights are 1, 2, 4, 8, 16,
32, 64, 128. Using them, the trader can measure any
weight between 1 to 255 kg. For example, to weigh
153 kg, the trader will use the following weights:
128 + 16 + 8 + 1 = 153.
You can try any other weights.
So, back to our multiplication question,
1 + 4 + 8 = 13.
To get the product of 13 and 7, the scribe would
take 1st, 3rd and 4th row and add the numbers in
the second columns of those rows. In our case, the
numbers would be 7, 28 and 56. When we add those
we get
7 + 28 + 56 = 91.
Bingo! We have indeed got the correct product of 13
and 7. Now, isn’t that interesting?
