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There are several. Here is one that's quite fast. For example, to calculate the square root of 2:

Make an initial estimate. Let's say that your initial estimate (or the result of a previous step) is 1.4. Divide 2 by this number: 2 / 1.4 = 1.428... Take the average between 1.4 and 1.428, which is 1.414.

Repeat, using the result of the previous step. Repeat as often as you want, until you have the desired precision. (In the previous step, the square root of 2 must be somewhere between 1.4 and 1.428, so by taking the average, the error can't be greater than 0.014.)

Each step doubles the number of correct digits, approximately.

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Q: What is the algorithm for square roots?
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