# How can you calculate #sqrt(2)# in your head?

##### 1 Answer

In your head, you probably want to stop at about:

#sqrt(2) ~~ 577/408 ~~ 1.4142#

#### Explanation:

One way of calculating rational approximations for

Your basic Newton Raphson method for finding approximations for the square root of a number

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

That's fine, but the fractions can get a bit messy and distracting.

So I prefer to split

#{ (p_(i+1) = p_i^2+ n q_i^2), (q_(i+1) = 2 p_i q_i) :}#

So for

Then:

#{ (p_1 = p_0^2 + 2 q_0^2 = 3^2+(2*2^2) = 9+8 = 17), (q_1 = 2 p_0 q_0 = 2*3*2 = 12) :}#

So if we stopped after one iteration, our approximation would be

Let's do another iteration to get more accuracy:

#{ (p_2 = p_1^2 + 2q_1^2 = 17^2+(2*12^2) = 289+288 = 577), (q_2 = 2 p_1 q_1 = 2*17*12 = 408) :}#

This is probably as many iterations as you want to do in your head, since you have to work with double the number of digits each time.

So it remains to long divide

#sqrt(2) ~~ 577/408 ~~ 1.414216#

Not bad - it's actually closer to