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

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Answer 1 · 2018-05-16T16:43:08

$sqrt(7/2) = 1/2sqrt(14) ~~ 13455/7192 ~~ 1.8708287$

It depends what you mean.

We can simplify $sqrt(7/2)$ as follows:

$sqrt(7/2) = sqrt(14/4) = sqrt(14/2^2) = sqrt(14)/sqrt(2^2) = 1/2 sqrt(14)$

$sqrt(7/2)$ 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 $15+-4sqrt(14)$ :

$(x-15-4sqrt(14))(x-15+4sqrt(14)) = x^2-30x+1$

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

${ (a_0 = 0), (a_1 = 1), (a_(n+2) = 30a_(n+1)-a_n) :}$

The first few terms of this sequence are:

$0, 1, 30, 899, 26940,...$

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

$sqrt(7/2) = 1/2sqrt(14) ~~ 1/8(26940/899-15) = 13455/(8 * 899) = 13455/7192 ~~ 1.8708287$