# How do you find the square root of 7/2?

May 16, 2018

$\sqrt{\frac{7}{2}} = \frac{1}{2} \sqrt{14} \approx \frac{13455}{7192} \approx 1.8708287$

#### Explanation:

It depends what you mean.

We can simplify $\sqrt{\frac{7}{2}}$ as follows:

$\sqrt{\frac{7}{2}} = \sqrt{\frac{14}{4}} = \sqrt{\frac{14}{2} ^ 2} = \frac{\sqrt{14}}{\sqrt{{2}^{2}}} = \frac{1}{2} \sqrt{14}$

$\sqrt{\frac{7}{2}}$ is an irrational number a little smaller than $2 = \sqrt{4}$.

If we want to find rational approximations to it there are at least $25$ different ways.

One of my favourites is to construct an integer sequence the ratio of whose consecutive terms tends to a value linearly related to the one we want.

For example, consider a quadratic with zeros $15 \pm 4 \sqrt{14}$:

$\left(x - 15 - 4 \sqrt{14}\right) \left(x - 15 + 4 \sqrt{14}\right) = {x}^{2} - 30 x + 1$

We can use this to define a sequence recursively as follows:

$\left\{\begin{matrix}{a}_{0} = 0 \\ {a}_{1} = 1 \\ {a}_{n + 2} = 30 {a}_{n + 1} - {a}_{n}\end{matrix}\right.$

The first few terms of this sequence are:

$0 , 1 , 30 , 899 , 26940 , \ldots$

The ratio of successive terms of this sequence converges rapidly towards $15 + 4 \sqrt{14}$. Hence we find:

$\sqrt{\frac{7}{2}} = \frac{1}{2} \sqrt{14} \approx \frac{1}{8} \left(\frac{26940}{899} - 15\right) = \frac{13455}{8 \cdot 899} = \frac{13455}{7192} \approx 1.8708287$