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FearlessFaa t1_j248t9v wrote

Reply to comment by mfb- in ELI5 How do calculators get the square root of random numbers? by Mikinak77

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
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mfb- t1_j24bs0w wrote

x < x * x is only true for x>1, but that's not relevant here.

What we do need (for x>0 and y>0): if x < y then y/x > 1 or equivalently y^2 / x > y. Same for the other direction, if x > y then y^2 / x < y. Here y is the square root we are looking for. This inequality makes sure one of our numbers is always too small while the other one is too large, i.e. the square root is in between.

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