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

  • Thread starter mymachine
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  • #1
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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
 

Answers and Replies

  • #2
SteamKing
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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.
 
  • #3
HallsofIvy
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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.)
 
  • #4
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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.
 
  • #5
lavinia
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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.
 
  • #6
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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.
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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