Thanks for you answer!

I think to the question it is unrelevant how physical calculators do it since physical calculators are not used much anymore. We are interested how to calculate square root i.e. different algorithms (=methods) used to calculate square root. It’s unrelevant what specific method calculators use.

I state here two mathematical facts used in the answer:

• for any positive numbers m and n, where m < n and m, n > 0, the following is true:
  • m < (m + n) / 2 < n,
    • where (m + n) / 2 is the average of numbers m and n
• for any numbers x > 1, it is true that:
  • x < x * x = x^2,
    • where x^2 is the square of x

Although partial sums of Taylor series use basic arithmetic operations so one could post an answer that calculates partial sums of Taylor series. We could then compare that with the average method described by other answers.

I'm not OP but thanks for your amazing answer!