hey this is press tow Walker thousands of years before computers or calculators ancient civilizations were Computing square roots by hand this Babylonian clay tablet represents one of the most important mathematical texts of the ancient Mesopotamian civilization the Babylonians used a Bas 60 number system which has enduring Legacy in the 60 minutes we have in an hour they represented numbers by kunoor numerals if you study the tablet closely you may be able to make out some of the numerals that are carved in and you can convert them into the numbers that we use in our everyday life what did these particular numbers represent studying other clues in the tablet the archaeologist were able to convert these numbers from the base 60 that the Babylonians used into the regular numbers that we use in particular these numbers were coefficients of a mixed number where we had a whole part and then fractions of powers of 60 so these particular numbers all represented one number that was equal to 1 + 24 over 60 + 51 over 60 ^2 + 10/ 60 cubed notice that these numbers are written along the diagonal of a square think about a square which has a side length that's equal to one what's the length of the diagonal even the Babylonians knew this was equal to the square OT of two so this particular number was an approximation for the square root of two converting the Babylonian numbers into a modern decimal number we end up with the approximation that < TK of two is equal to 1. 4142 1 these are all accurate digits of the square root of two so the Babylonians were correct to five decimal places their technique for computing square roots is so efficient that essentially modern computers use the same algorithm what's even more remarkable is the technique is pretty simple to learn in this video I'm going to teach you how to do square roots like the Babylonians did and it's so easy you might even be able to compute these in your head so let's get started with an example of the square root of 17 we get started by referring to a table of squares of whole numbers even Babylonians started out with such tables we want to take the square root of 17 we look for the square number that's closest to 17 this will be the number 16 which is equal to 4 squar so imagine we were taking the square root of 16 that would be equal to four so theare < TK of 17 must be a little bit larger than four that'll be a starting point so the square root of 17 is approximately equal to 4 but we know that 4^2 is equal to 16 and we want the square OT of 17 which is a larger number so we know we're going to need to add a little bit to get to the square OT of 17 so we will add some fraction to make an adjustment the numerator will be calculated as the difference between 17 and 16 17 - 16 is equal to 1 and that's the number in the numerator for the denominator we take this number four and all we do is we multiply it by two we double this number 4 * 2 is equal to 8 so we now have our approximation the square < TK of 17 is approximately equal to 4 + 1 8 this works out to the decimal number 4. 125 how close is this estimate the actual value of the square root of 17 begins 4123 so this estimate is accurate to two decimal places and it's a very simple calculation so let's do another example what is the square root of 69 we start out with our table of squares which square is closest to 69 this will will be 8^2 which equals 64 so we start out that the square < TK of 69 is approximately equal to 8 we now need to add an adjustment the numerator will be the difference of 69 and 64 and that is equal to 5 for the denominator we just take this value 8 and we multiply it by two to get 16 so we have theun of 69 is a approximately equal to 8 + 5 16 which equals 8312 the actual value is 8307 so again we have a very accurate approximation let's do another example of the square root of 111 111 is closest to 100 which equal 10^ SAR so the square < TK of 111 is approximately equal to 10 plus some adjustment the numerator will be 111 minus 100 which equal 11 and the denominator will be double 10 which is equal to 20 so the square Ro T of 111 is approximately equal to 10 + 11/ 20 which equals 10.
55 the actual value starts out as 10536 again a very accurate approximation now let's do a slightly different example in the sare root of 23 what square number is closest to 23 this will be 25 which equals 5^2 so the square OT of 23 is approximately equal to five but unlike the previous examples we know that this is too large of an estimate because 5^2 is equal to 25 will this method still work let's illustrate it power so we need to add an adjustment the numerator in this case will be 23- 25 and that will be equal to -2 so we have a negative adjustment then the denominator will be equal to 5 * 2 and that will be equal to 10 so we have theare < TK of 23 is approximately equal to 5 + -2 over 10 this works out to be 4. 8 the actual value of the sare root of 23 starts out as 4. 79 six so the technique works and is accurate even if the adjustment is a negative number let us now return to a final example of estimating the square OT of two using the Babylonian method we start up by asking which square number is closest to two the best we can do is one which is equal to 1^ 2 so the < TK of 2 is approximately equal to 1 we now need to add an adjustment the numerator will be equal to 2 - 1 which is equal to 1 and the denominator will be equal to 1 * 2 which is equal to 2 so what does this method give us it gives us an estimate that the square of 2 is approximately equal to 1 + 12 which equals 1.
5 but the actual value of the square root of two starts out as 1. 414 so this is not a very impressive approximation it is not accurate to five decimal places as we saw in the very first clay tablet so how did the Babylonians actually get very accurate values of the square root of two there is another trick in this method we have come up with an estimate of 1 5 for the < TK of two but there's no reason we have to stop at this point we can take this result and repeat the whole process with this as our initial guess so let's see how that works so we start out that theare < TK of two is approximately equal to 1. 5 what are the steps in the process now we need to add an adjustment the numerator will be 2 minus the square of 1.
5 1. 52 is equal to 2. 25 2 - 2.
25 is equal to -25 so the numerator is -25 the denominator will now be double the initial estimate so we take 1. 5 multiplied 2 which is equal to 3 therefore the < TK of two is approximately equal to 1. 5 Plus -25 over 3 this works out to be 1.
