How do you find the square root of 20?

1 Answer
Aug 31, 2016

Find approximation:

sqrt(20) ~~ 2889/646 ~~ 4.4721362028896464.472136

Explanation:

The prime factorisation is:

20 = 2^2*520=225

Hence:

sqrt(20) = 2sqrt(5)20=25

This is an irrational number between 44 and 55 since:

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 2020 is roughly halfway between 4^242 and 5^252, its square root is approximately 9/292 - halfway between 44 and 55.

In fact, we find:

9^2 = 81 = 80+1 = 20*2^2 + 192=81=80+1=2022+1

which is in Pell's equation form:

p^2 = n q^2 + 1p2=nq2+1

with n = 20n=20, p = 9p=9 and q = 2q=2

That means that we can deduce the continued fraction for sqrt(20)20 from the continued fraction for 9/292...

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 sqrt(20) truncate this continued fraction early, just before an '8'...

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