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#