Ancient Egyptian Numbers

Egyptian Numbers --- Introduction --- Addition --- Multiplication --- Fractions

These are the symbols for Egyptian numbers. There is a description of what they mean here.


1	10	100	1000	10000	100000	1000000

Adding Egyptian numbers is easy, but multiplying them harder. Imagine the seven times table:

	x		=
	x		=
	x		=
	x		=
	x		=

Not very easy to learn! What's more, when we multiply long numbers in our number system, we multiplying numbers by ten or a hundred, which with our numbers is very simple - you just put the right number of zeroes on the end. This wasn't as easy for the Egyptians. They would have had to change all their symbols to the next one, or more, so the finger became the tadpole, and so on. However, the Egyptians had another way of multiplying which was very cunning (and worked). All you had to do was multiply and divide by 2. This was easier for them than for us, since half of eight of a symbol is four of a symbol - all you have to remember is that half a tadpole is five fingers! This is how you multiply 36 by 57 using this method in our number system, so you can follow the technique. You divide one number (here: 36) by 2 several times until you reach one. Sometimes you can't, of course, if you have an odd number, so you then subtract 1 before halving it. You multiply the other number by 2 the same number of times. Every line where you had the spare 1, you note what the doubled number has come to, and you add these numbers together, ignoring the others. This, believe it or not, is the same as the two numbers multiplied together, without using multiplication tables at all.

Halve first number	Odd number	Double second number	Doubled numbers on odd halved numbers
36		57
18		114
9	subtract 1	228	228
4		456
2		912
1	subtract 1	1824	1824
Total of relevant doubled numbers			2052

The following explanation is rather technical:
The reason this works is that you have converted the first number to binary, and then done the long multiplication using this fact.
When we do long multiplication with our current number system, we rely on the fact that you can break down one number into various multiples of ten. So 4,235 = 4000 + 200 + 30 + 5. Now it is true for all numbers that a x (b + c) = a x b + a x c. This is called the Distributive Rule. So if you want to multiply a number by 4,235, you can multiply the same number by 4000, then by 200, then by 30 and finally by 5, and add these together. This is easy for our system, since it is easy to multiply by a multiple of ten or a hundred.
The Egyptians did not like multiplying by 10, but they could halve and double. They didn't realise that they using binary, but they were! In the above example, 36 is 100100 in binary (you read the 'odd' column from the bottom to the top). One advantage of multiplying in binary is that the binary times table is trivial (0x0=0, 0x1=0, 1x0=0, 1x1=1, and that's it!) This means that you don't need to add in any doubled number on the 'zero' lines (since a number multiplied by zero is zero). The numbers of the 'one' lines are multiplied by one (which leaves them the same). All that then needs to be done is to add these numbers to get the answer.

Explanation of binary, including binary multiplication.