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