How do you find the square root of 20?
1 Answer
Find approximation:
#sqrt(20) ~~ 2889/646 ~~ 4.472136#
Explanation:
The prime factorisation is:
#20 = 2^2*5#
Hence:
#sqrt(20) = 2sqrt(5)#
This is an irrational number between
#4^2 = 16 < 20 < 25 = 5^2#
It is not expressible as an exact fraction, but we can find rational approximations...
Since
In fact, we find:
#9^2 = 81 = 80+1 = 20*2^2 + 1#
which is in Pell's equation form:
#p^2 = n q^2 + 1#
with
That means that we can deduce the continued fraction for
#9/2 = 4+1/2 = [4;2]#
Hence:
#sqrt(20) = [4;bar(2,8)] = 4+1/(2+1/(8+1/(2+1/(8+1/(2+1/(8+...))))))#
To get a good approximation for
For example:
#sqrt(20) ~~ [4;2,8,2,8,2] = 4+1/(2+1/(8+1/(2+1/(8+1/2)))) = 2889/646 ~~ 4.472136#
From a calculator:
#sqrt(20) ~~ 4.472135955#