 So let's do another example of Al-Qashu's method of square roots, and let's see why it works. So let's try to approximate the square root of 151,245. Since we're looking for square roots, we'll break the number into two-digit cycles. Starting with the first cycle, the largest number whose square is less than 15 is 3. And so we'll subtract 3 squared from 15, leaving 6. Adding the gas gives us 6, and we shift. So remember we should read this number as 60, and this number we can bring down the next few digits and get 612. Now for our next digit, we want the largest number that when multiplied by the sum of 60 and the number is less than 612. So we try 9, and we find 9 times 60 plus 9, which is too much. So we try 8, and we find 8 times 60 plus 8, so 8 works. And so we subtract, we add our gas again to get 68 plus 8, 76, and we shift. And again, this column should be viewed as having two digits, so this number should be 760 something. And if we copy down the last few digits, we have the number 6845. And so once again, we want the largest number that when multiplied by the sum of 760 and the number is less than 6845. So we try 9, and find 9 times 760 plus 9 is too much. And we try 8, and find 8 times 760 plus 8 is, and so 8 works. We subtract. Now to get the fractional part, we add 8 again to get, and add 1 to get 777, the denominator. And the remainder is the numerator. And so the square root of 151,245 is approximately 388 and 701,777. To understand the algorithm, let's start with a different question. How do we know if we have a good approximation? We can't compare the approximation to the actual value, since if we had the actual value, we wouldn't be trying to find an approximation. Instead, if we've approximated the root by some number n, then the difference between the radicand and n squared will tell us how accurate our approximation is. So let's see why this works. We'll break it down into two parts. First, the whole number part, and second, the fractional part. Now for the whole number part, we might begin by noting that 300 squared is less than 151,245, which is less than 400,000. In other words, our square root is 300 something. Now we started by subtracting 3 squared or 9, but this was in that third cycle. And in fact, that 3 is really 300, and 300 squared is 90,000. And notice that if we split this number into cycles of two digits, we see that we really only need to subtract a 9 from the number in the left-most cycle. And this leaves 61,245. Now we note that 380 squared is less than 151,245, which is less than 390 squared. In other words, our square root is 380 something. And if we find that remainder, it's really 151,245 minus 380 squared. Now that's really the same thing as subtracting 300 plus 80 squared. And if we expand this term, the first thing to recognize is that we've already subtracted 300 squared. So we don't need to recompute this first part. Meanwhile, the remaining terms have a common factor of 80, so we could remove that. And 2 times 300, well, that's 600. And notice that since we've already subtracted 300 squared, we only need to subtract 80 times 600 plus 80. But if we multiply that out, and breaking this product into cycles of two digits of piece, we see that this is the second subtraction we did. And again, we note that 388 squared is less than our radicand, which is less than 389 squared. And again, if we find the remainder, and again, we've already subtracted 380 squared. So we only need to subtract 8 times 760 plus 8. But when we multiply that out and break our number into cycles of two digits, we see that that's the last subtraction. What about the fractional part? So we computed the difference between 151,245 and 388 squared to be 701. So let's consider the difference between 389 squared and 388 squared. Now, because this is a difference of squares, we can compute it very easily. And the way to interpret this is that increasing the root by 1 is going to increase the square by 777. Now, since we want to increase the square by 701, so we increase the root by 701,777. And so that's the source of the fractional part of the approximation to the root.