What is the square root of 5?
2 Answers
The square root of
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
A square root of a number
However, popular usage is that "the square root" refers to the positive one.
Suppose we have a positive number
#x = 2+1/(2+x)#
Then multiplying both sides by
#x^2+2x = 2x+5#
Then subtracting
#x^2=5#
So we have found:
#sqrt(5) = 2+1/(2+sqrt(5))#
#color(white)(sqrt(5)) = 2+1/(4+1/(4+1/(4+1/(4+1/(4+...)))))#
SInce this continued fraction does not terminate, we can tell that
For example:
#sqrt(5) ~~ 2+1/(4+1/4) = 2+4/17 = 38/17 ~~ 2.235#
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:
#{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 4a_(n+1)+a_n) :}#
The first few terms are:
#0, 1, 4, 17, 72, 305, 1292, 5473#
The ratio between terms will tend to
So we find:
#sqrt(5) ~~ 5473/1292 - 2 = 2889/1292 ~~ 2.236068#