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How accurate is the square root of a number?

  1. Jun 6, 2015 #1
    This may sound like a silly question but: How accurately has the squareroot of numbers like 2,3,5 etc. been measured?
    When you type it into a calculator it gives you an answer with a certain amount of decimal points,
    the calculator is of course software programmed by a group of people who can't possibly
    know the square root of 2 to the last descimal point, so is it
    correctly assumed that you will never be able to define the square root of a number like 2,3,5 to the last
    decimal point because the number of descimal points of 2,3,5 goes to infinity?
    So the square root of 2 is undefined to the last descimal point.
  2. jcsd
  3. Jun 6, 2015 #2


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    Gold Member

    You are correct. The square root of some numbers is an infinite string of digits and thus literally cannot be known to "full" precision. It isn't something that is, as you stated, "measured", it is calculated and it can be calculated to as many digits as you wish but that's a waste of time.
  4. Jun 6, 2015 #3
    These questions don't really make sense in a mathematical context - in maths we don't "measure" anything in the sense you are using the word.

    A calculator is programmed with an algorithm that can calculate a representation of √2 to as many decimal places as you want, so we do not say that anything is undefined, however unlike 1/3 = 0.333... or 1/7 = 0.142857142857... whose decimal representation also "goes to infinity" there is no pattern to these digits: we use the word irrational to describe such a number.
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