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posted by on .

how do you find the square root of any number with long division and not a calculator? I don't like calculator readings. They are not always easy to figure out, plus math principles can't be learned with a calculator.

  • algebra - ,

    You echo what I think also.

    However, for square roots, it is a little more complicated (but not difficult) than long division if you need many figures for your answer.

    For doing it by hand, you can try:

    Better still, most square-roots can be calculated mentally, based on Newton's method.

    The more squares of integers you can memorize, the more accurate your estimate of the answer will be.

    If you need the square root of 37, you know the closest perfect square is 36.
    So take 6 as a first estimate. Divide the residue of (37-36)=1 by twice the estimate to get 1/12=0.083. So the estimate of √37 is 6.083.
    The answer to 6 places after the decimal is 6.0827625.

    To calculate the square-root of 11, if you know by memorization that 33^2=1089, then the first estimate is 3.3 (since 3.3²=10.89)
    Divide the residue of 11/10.89=0.11 by twice 3.3 gives 1/60, or 0.01667.
    This gives our first estimate as 3.31667.
    The exact value to 6 decimal places is 3.316625

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