## What is the square root of 5?Algebra Radicals and Geometry Connections Simplification of Radical Expressions ## 2 AnswersAviv S. Mar 14, 2018 The square root of#5#can't be simplified father than it already is, so here is#sqrt5#to ten decimal places: #sqrt5~~2.2360679775...# Answer link George C. Mar 14, 2018 #sqrt(5) = 2+1/(4+1/(4+1/(4+1/(4+1/(4+...))))) ~~ 2889/1292 ~~ 2.236068#is an irrational number. ## Explanation:All positive numbers normally have two square roots, a positive one and a negative of the same size. We denote the positive (a.k.a. principal) square root of#n#by#sqrt(n)#. A square root of a number#n#is a number#x#such that#x^2 = n#. So if#x^2 = n#then also#(-x)^2 = n#. However, popular usage is that "the square root" refers to the positive one. Suppose we have a positive number#x#which satisfies:
Then multiplying both sides by#(2+x)#we get:
Then subtracting#2x#from both sides we get:
So we have found:
SInce this continued fraction does not terminate, we can tell that#sqrt(5)#cannot be represented as a terminating fraction - i.e. a rational number. So#sqrt(5)#is an irrational number a little smaller than#2 1/4 = 9/4#. For better rational approximations you can terminate the continued fraction after more terms. For example:
Unpacking these continued fractions can be a little tedious, so I generally prefer to use a different method, namely the limiting ratio of an integer sequence defined recursively. Define a sequence by:
The first few terms are:
The ratio between terms will tend to#2+sqrt(5)#. So we find:
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