What is the Square root of 21?

1 Answer
Oct 15, 2015

Answer:

#21 = 3*7# has no square factors, so is not possible to simplify #sqrt(21)#

#sqrt(21) ~~ 4.583# is an irrational number whose square is #21#

Explanation:

#sqrt(21)# is not a rational number, so it cannot be expressed as #p/q# for some integers #p, q# and its decimal expansion does not repeat.

#sqrt(21) ~~ 4.58257569495584000658#

It is expressible as a repeating continued fraction:

#sqrt(21) = [4;bar(1,1,2,1,1,8)] = 4 + 1/(1+1/(1+1/(2+...)))#

To see how to calculate this see http://socratic.org/questions/given-an-integer-n-is-there-an-efficient-way-to-find-integers-p-q-such-that-abs-#176764

We can get a good approximation for #sqrt(21)# by truncating the continued fraction.

#sqrt(21) ~~ [4;1,1,2,1,1] = 4+1/(1+1/(1+1/(2+1/(1+1/1)))) = 55/12 = 4.58dot(3)#

This is a good approximation because #55^2 = 21*12^2 + 1#