Getting the Square Root of Any Number
The most common method people learn in school is long division style, but in practice it is slow and error prone. What actually works faster on paper or in your head is the Babylonian method, also called Heron's method. You pick a guess, divide the original number by that guess, average the result with your guess, and repeat. That is it. The numbers converge quickly even if your first guess is way off. Take the number you need the root of, say 144. Pick a starting guess. If you know your squares you might start at 12. If you do not, just pick something reasonable like 10. Divide 144 by 10, which gives 14.4. Average 10 and 14.4 to get 12.2. Divide 144 by 12.2, which is about 11.803. Average 12.2 and 11.803 to get 12.0017. One more iteration and you are essentially at 12. Done. Now try a number that is not a perfect square, like 50. Start with 7. Divide 50 by 7 to get 7.143. Average with 7 to get 7.0715. Divide 50 by 7.0715 to get 7.0710. Average again and you are at 7.07107. The true value is 7.0710678..., so you are accurate to five decimal places after two iterations.
I have used this method manually for years, especially when I needed a quick root without pulling up a calculator. One edge case I ran into recently was finding the square root of 0.000487. People tend to second guess themselves with decimals and small numbers. My workaround was straightforward: I multiplied by 10,000 to shift it to 4.87, found the root of 4.87 using the same iteration method, then divided the result by 100. That gave me approximately 0.022068, which checks out when squared. Here is a counter-intuitive point most beginners miss: your initial guess does not need to be close. The method is remarkably stable even if you start with a terrible guess. I once started with a guess of 1 for the square root of 200 and it took about six iterations to converge. Still faster than doing it by hand any other way. The other thing people overlook is that each iteration roughly doubles your number of correct digits. That is quadratic convergence and it is what makes this method so efficient compared to alternatives. There is a practical limitation though. If you are working with extremely large numbers or need more than about twelve decimal places of accuracy, manual iteration becomes tedious and mistakes creep in. In those cases you are better off using a scientific calculator, a spreadsheet function like =SQRT(), or a programming language's math library. Manual methods are fine for estimates and mental math but they do not scale to precision engineering work.
Another thing worth noting is that the Babylonian method fails or behaves badly when your number is negative because real square roots do not exist there. You will hit an issue during iteration when your guess crosses zero and the division flips sign. If you are working in a domain that requires complex results, you need a different approach entirely, usually involving polar form or a complex number algorithm. For perfect squares up to 10000, memorizing the base squares from 1 through 100 saves the most time. Between 100 and 200, the roots fall between 10 and 14.14, so you can narrow it down without any iteration. For rough estimates under time pressure, I usually round to the nearest perfect square and adjust linearly. The square root of 50 is close to the square root of 49, which is 7, and since the derivative of the square root function at 49 is about 0.0714, the estimate comes out to roughly 7 plus 1 times 0.0714, giving 7.07. That is basically correct and takes about ten seconds. If you want a tool for repeated calculations, most desktop operating systems have a built in calculator that handles this instantly. Windows calc in scientific mode, macOS calculator, or any online math tool will give you the result in milliseconds. I still use manual iteration when I am away from a computer and need something reliable.
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