How do you figure out the square root of 7? - Answers

There are a number of ways (other than using a calculator).BracketingFind two integers, x and y such that x2 < 7 < y222 = 4 < 7 < 9 = 32 so sqrt(7) is between 2 and 3.Next find two integers between 20 and 30 such that x2 < 700 < y2262 = 676 < 700 < 729 = 272 and so sqrt(7) is between 2.6 and 2.7Next find two integers between 260 and 270 such that x2 < 70000 < y22642 = 69696 < 70000 < 70225 = 2652 and so sqrt(7) is between 2.64 and 2.65As you can see, you get one additional decimal place at each step. Continue until the desired level of precision is reached.Newton-Raphson methodThis is a more efficient method. If you want to know more about the method, look it up on Wikipedia.Define f(x) = x2 - 7 so that f(x) = 0 when x = sqrt(7).Then the derivative of f(x), is f'(x) = 2xStart with a value of x, say x0 that will make f(x) very approximately 0.Then calculate x1 = x0 - f(x0)/f'(x0).Repeat until you have the desired level of accuracy.Here, for example, if you start with x0 = 2 you will getx1 = 2 + (4-7)/(2*2) = 2.75x2 = 2.75 + (2.752-7)/(2*2.75) = 2.6477x3 = 2.6477 + (2.64772-7)/(2*2.6477) = 2.64575which is accurate to 5 decimal places.Incidentally, for finding a square root, a totally outlandish starting point will not matter. Even if you start with 10 as an estimate for sqrt(7), x3 will be accurate to approx one in a thousand; x4 to less than one in a million.There is also a method resembling long division, but this site is not particularly suited for explaining it.