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Egyptian multiplication

In this video, Dr. Yossi Elran explains the principles behind Egyptian multiplication and also shows a 'cheat' to multiply numbers by 11.
So, the ancient Egyptians used Hieroglyphics and to understand about ancient Egyptian symbology you have to understand that the Egyptians mainly used numbers for counting for instance, how to divide a loaf of bread into equal parts for five workers or how to measure the area of a field of corn but they didn’t really need to use multiplication so there was really no meaning to numbers like 37 times 3 or 46 times 9 or division They did everything by looking at addition so lets take a look, we’ve adapted this to this ancient Egyptian symbology. From the Karnak temple we took 4 lines that really represent a multiplication problem and we can understand from this how the Egyptians looked at multiplication.
So, these are four very interesting lines. The first line, well, by now you should know a bit about Hieroglyphics, This is 1, and let’s see, 1,2,3,4,5,6,7,8 of these, and 2 of these, so this is really the number 2801. Now, the number below it, there are 5 flowers, and, 1,2,3,4,5,6 of these “snails”, and 2 rods, so this is 5602 and then we have over here, we have a new thing over here and this will give us 11,204 Now, notice this is exactly - the 5,602 is exactly double 2,801 and 11,204 is exactly double 5,602 so, what we have here is a list, this is, we can out the number 1 over here and the number 2 beside here because this is twice what is written in the first row and the number 4 over here because this is twice in the second row or four times the number written in the first row so we have a number and then a number times 2 and then a number times 4 Now here in the third row, this is actually just the sum of all of these 3 because you see over here, we get the number Well, this is 19,000 right, we have the 10,000 over here and then there are 1,2,3,4,5,6,7,8,9 of these 19,000 and 1,2,3,4,5,6 of these…
600 and 1,2,3,4,5,6,7 19,607 which is just - 1 + 2 + 4 is 7 zeroes - all zero, 8 + 6 + 2 gives us 16 1 + 5 + 2 + 1 gives us 9, and we have the 1 left over, so you can see, yes, this is 19, 607 so it’s just the add up of 1, 2 and 4 - what is this number? 19,607 is really 7 times, 7 times the first number and also, 1 + 2 + 4 gives us 7. so we have 7 times the first number over here so this is how they were doing multiplication. They take a number and they double it… and double it - that’s something they could do.
They knew how to add a number to itself. So, in fact, they are just taking powers of 2 writing down all the powers of 2 and then as the multiplicand, just choosing… from all of these, those that are relevant for the multiplication. So if for instance, they wanted to do 2,801 times 3, they just take this and this row and add it up. And if they wanted to do 2801 times 5 they just take the first row and the third row over here. 1 plus 4 gives us 5 so if you add 2801 to 11,204 you’re going to get the first number 2801 times 5. So, this is the way they turned addition into multiplication.
So all you need to do… …in fact we don’t need Egyptian numbers for this… - I’ll put this down here - we can do this with any number. Suppose you want to - I don’t know - you give me a number! Ah! You can’t! You’re over there. So I’ll take a number. I’ll make up a number by myself, Let’s take the number 173, and I want to multiply this, say by, 11. OK. So if I want to multiply this by 11, Actually I can do this - I’ll tell you afterwards - there is a trick to do this - but suppose we don’t want to use the trick, we’ll do it the Egyptian way.
Let’s take the 173 and double it. So, 173+173, we’re just going to get… 346. Right? Three two’s are 6. Two sevens are 14, 1 carried over. Two ones are 2 and one is 3. Now I can add 4 to it, double that… so to double that again I do 346 times 2… so six 2’s are 12… Well, I should really just add it to itself - right - I’m cheating by doing this but but that’s OK. We have 9 over here, and 6 over here so we have 692. And now let’s look at 8 because we are going to need the 8 as well to get to 11. So 8 again - we double 692.
So to double 692, we get 13,084 and now, all we need to do to get 11 is to choose the correct… numbers over here from the powers of 2. So this is really sort of a binary way of doing it, right? So, if we know that 8 plus 2 plus 1 will give us 11, Then, we need to take these 3 rows and add them up. So, 3 plus 6 that’s 9 plus 4 that’s 13 8 plus 7 is 15 plus 4 is 19 plus 1 is 20 3 plus 3 is 6 plus 1 is 7 plus 2 is 9 and we have the 1 over here, so we get 1903. Now - which is 173 times 11.
Now, I promised to show you a ‘cheat’. Here’s a cheat way of multiplying a number by 11. You see, take any 2 digit number , but it can be 3 digits as well, all you need to do if you want to take, let’s say 31 and multiply that by 11 - well - you take this digit over here, this digit over here, throw the 3 here, the 1 here, and write down in the middle 3 plus 1, 4 and that’s it! 341 Try another number - 52 You take the 2 over here, the 5 over here, 52 times 11… 5 plus 2 - put in the middle - 7 52 times 11 is 572.
So 31 times 11 is 431 and 52 times 11 is 572 What happens if the number here is larger than 9? If the number is larger than 9, you need to use the carry. So if we have for instance 65 times 11 we put the 5 over here, we put the 6 over here, and in the middle you put 6 plus 5 that’s 11 so you put down 1, carry 1 over here so that gives us… 715. And if you want to do 173 times 11, well it’s just the same. You put the 3 over here, you put the 17 over here, 17 plus 3 is 20, so you put the 0 here, carry the 2 over, you get 1903.
So if the Egyptians knew that trick, they would be able to do multiplication by 11, much quicker.

The Rhind papyrus teaches us a lot about how the Egyptians thought about the different math operations. There wasn’t a simple known way to multiply numbers. In fact, multiplication was seen to be a form of addition. Watch this video to see the principles behind Egyptian multiplication and on the way I’ll show you a ‘cheat’ to multiply numbers by 11.

Do you know any other ways to multiply numbers? Click Here for BBC Four’s exploration into Ethiopian multiplication. Please share your ideas about multiplication methods in the discussion below!

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