I will be showing the method used by the Ancient Egyptians as seen in the Text - " A History of Mathematics" by Jeff Suzuki. This is an excellent text and should be read by those with a great interest or love of mathematics and or history.

To do this multiplication we can use Suzuki's Pseudo-Hieratic notation. This involves writing a number such as 23 as 20,3. Another example would be writing 156 as 100,50,6.

So I will use the example Suzuki gives in the text. Namely, that of multiplying 13 by 27 or in Pseudo-Hieratic as 10,3 by 20,7. This may be considered a form of binary arithmetic because of the doubling. This method is also known as the Method of doubling and halving.

I will first show a step and then explain what is happening.

Step 1.

Col. 1 Col. 2

1 20,7

2 50,4

4 100, 8

8 200, 10, 6

Pick one of the numbers in this case 20,7 (27). Now, write a separate 1 term in a column to the left. Now double the row.That makes 1 go to 2 and 20,7 go to 50,4 (27*2 = 54).Doubling again makes turns the 2 into a 4 and 50,4 into 100,8 (54*2=108). Doubling once more 4*2 = 8 and108*2 = 216.The doubling does not go on forever of course. We do this until with some combination of the factors present in column 1 can make up the other original multiplying term. In this case 1,4, and 8 make up 10,3 (i.e 1+4+8 = 13).

Step 2.

Col. 1 Col. 2

\1 20,7

2 50,4

\4 100,8

\8 200,10,6

In step 2 we simply highlight the rows where the single digits in column 1 add up to 13. So Row's 1,3 and 4 have single digits 1,4,and 8 which sum to 10,3 (i.e 13).

Step 3.

20,7 + 100,8 + 200,10,6 = 300,50,1. ( 27 + 108 + 216 = 351).

Now, in step 3 we are looking at the rows of interest pointed out in Step 2. From there we add the terms in Column 2.