How do you find the square root of 20?
1 Answer
Find approximation:
sqrt(20) ~~ 2889/646 ~~ 4.472136√20≈2889646≈4.472136
Explanation:
The prime factorisation is:
20 = 2^2*520=22⋅5
Hence:
sqrt(20) = 2sqrt(5)√20=2√5
This is an irrational number between
4^2 = 16 < 20 < 25 = 5^242=16<20<25=52
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 + 192=81=80+1=20⋅22+1
which is in Pell's equation form:
p^2 = n q^2 + 1p2=nq2+1
with
That means that we can deduce the continued fraction for
9/2 = 4+1/2 = [4;2]92=4+12=[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