How do you approximate #2/sqrt3#?

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Answer 1 · 2015-07-06T19:42:00

Use an iterative method to get a good rational approximation for #sqrt(3)# then use that to calculate #2/sqrt(3)# to get (say)

#2/sqrt(3) ~= 2/(97/56) = 112/97 ~= 1.155#

Start with a reasonable approximation #a_0 = 2# for #sqrt(3)#.

Then iterate using the formula:

#a_(i+1) = (a_i^2+3)/(2a_i)#

#a_1 = (a_0^2+3)/(2a_0)#

#=(2^2+3)/(2*2)#

#=7/4#

#a_2 = (a_1^2+3)/(2a_1)#

#=((7/4)^2+3)/(2*7/4)#

#=(49/16+48/16)/(7/2)#

#=97/56#

We will stop here, but if you want more accuracy, just iterate again.

In general, to find the square root of #n#, pick a reasonable first guess as #a_0#, then iterate using:

#a_(i+1) = (a_i^2+n)/(2a_i)#