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How Do You Find The Exact Value Of Square Root of 3, 5, 7, 11?

  1. Mar 31, 2013 #1
    Is there any method to find the exact value of the square root of 3,5,7,11,13,14,15,17,18, etc.?

    Thank you
  2. jcsd
  3. Mar 31, 2013 #2


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    Arithmetic algorithms can approximate these square roots, but because they are all irrational, the decimal representations are non-repeating and non-terminating.
  4. Mar 31, 2013 #3


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    The exact values of the square root of 2, 3, 5 ,7, 11, etc are [itex]\sqrt{2}[/itex], [itex]\sqrt{3}[/itex], [itex]\sqrt{5}[/itex], [itex]\sqrt{7}[/itex], [itex]\sqrt{11}[/itex]. That's the best you can do. As SteamKing said, all of those, and, in fact, the square root of any integer that is not a "perfect square", are irrational- they cannot be written as a terminating decimal, they cannot be written as a repeating decimal like "0.14141414...", and cannot be written as a fraction (integer over integer).

    (I added "2" to the beginning of your list. I am surprized you did not have it.)
  5. Apr 1, 2013 #4


    Staff: Mentor

    Might as well add 6, 8, 10, and so on to the list, since none of these is a perfect square, and consequently does not have a square root that is rational.
  6. Apr 1, 2013 #5


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    If instead of a infinite decimal expansion you would accept some other infinite expression then you can express the square root of 2 as an infinite continued fraction.
  7. Apr 1, 2013 #6
    The infinite fraction representation is a really nice one because it exhibits a lot of regularity. In the decimal expansion of ##\sqrt{2}##, there is no way to know which decimal comes next. But the infinite fraction is very straightforward and exhibits a nice pattern.
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