How do you find the square root of 20?

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Answer 1 · 2016-08-31T20:59:55

Find approximation:

#sqrt(20) ~~ 2889/646 ~~ 4.472136#

The prime factorisation is:

#20 = 2^2*5#

Hence:

#sqrt(20) = 2sqrt(5)#

This is an irrational number between #4# and #5# since:

#4^2 = 16 < 20 < 25 = 5^2#

It is not expressible as an exact fraction, but we can find rational approximations...

Since #20# is roughly halfway between #4^2# and #5^2#, its square root is approximately #9/2# - halfway between #4# and #5#.

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

That means that we can deduce the continued fraction for #sqrt(20)# from the continued fraction for #9/2#...

#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 #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#