# What is the Square root of 21?

Oct 15, 2015

$21 = 3 \cdot 7$ has no square factors, so is not possible to simplify $\sqrt{21}$

$\sqrt{21} \approx 4.583$ is an irrational number whose square is $21$

#### Explanation:

$\sqrt{21}$ is not a rational number, so it cannot be expressed as $\frac{p}{q}$ for some integers $p , q$ and its decimal expansion does not repeat.

$\sqrt{21} \approx 4.58257569495584000658$

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 ${55}^{2} = 21 \cdot {12}^{2} + 1$