# What is the square root of 5?

Mar 14, 2018

The square root of $5$ can't be simplified father than it already is, so here is $\sqrt{5}$ to ten decimal places:

$\sqrt{5} \approx 2.2360679775 \ldots$

Mar 14, 2018

$\sqrt{5} = 2 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \ldots}}}}} \approx \frac{2889}{1292} \approx 2.236068$ is an irrational number.

#### Explanation:

All positive numbers normally have two square roots, a positive one and a negative of the same size. We denote the positive (a.k.a. principal) square root of $n$ by $\sqrt{n}$.

A square root of a number $n$ is a number $x$ such that ${x}^{2} = n$. So if ${x}^{2} = n$ then also ${\left(- x\right)}^{2} = n$.

However, popular usage is that "the square root" refers to the positive one.

Suppose we have a positive number $x$ which satisfies:

$x = 2 + \frac{1}{2 + x}$

Then multiplying both sides by $\left(2 + x\right)$ we get:

${x}^{2} + 2 x = 2 x + 5$

Then subtracting $2 x$ from both sides we get:

${x}^{2} = 5$

So we have found:

$\sqrt{5} = 2 + \frac{1}{2 + \sqrt{5}}$

$\textcolor{w h i t e}{\sqrt{5}} = 2 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \ldots}}}}}$

SInce this continued fraction does not terminate, we can tell that $\sqrt{5}$ cannot be represented as a terminating fraction - i.e. a rational number. So $\sqrt{5}$ is an irrational number a little smaller than $2 \frac{1}{4} = \frac{9}{4}$. For better rational approximations you can terminate the continued fraction after more terms.

For example:

$\sqrt{5} \approx 2 + \frac{1}{4 + \frac{1}{4}} = 2 + \frac{4}{17} = \frac{38}{17} \approx 2.235$

Unpacking these continued fractions can be a little tedious, so I generally prefer to use a different method, namely the limiting ratio of an integer sequence defined recursively.

Define a sequence by:

$\left\{\begin{matrix}{a}_{0} = 0 \\ {a}_{1} = 1 \\ {a}_{n + 2} = 4 {a}_{n + 1} + {a}_{n}\end{matrix}\right.$

The first few terms are:

$0 , 1 , 4 , 17 , 72 , 305 , 1292 , 5473$

The ratio between terms will tend to $2 + \sqrt{5}$.

So we find:

$\sqrt{5} \approx \frac{5473}{1292} - 2 = \frac{2889}{1292} \approx 2.236068$