What is the square root of 30?
You can't get an exact answer by hand, but you can estimate it using some tricks.
so it must be between
What you can begin with is assuming that because:
... the square root of
or more specifically,
Actually, it's not a bad bet. The actual square root is about
You can calculate an approximation by hand as shown below...
I explained my favourite method (a sort of Newton Raphson method) for approximating square roots of integers in an answer to the following question:
Given an integer
Then iterate using the formulas:
#p_(i+1) = p_i^2 + n q_i^2#
#q_(i+1) = 2 p_i q_i#
If the resulting values of
The successive pairs
For our example, let
#p_1 = p_0^2 + n q_0^2 = 11^2 + 30*2^2 = 121 + 120 = 241#
#q_1 = 2 p_0 q_0 = 2 * 11 * 2 = 44#
This would give
#p_2 = p_1^2 + n q_1^2 = 241^2 + 30*44^2 = 58081 + 58080 = 116161#
#q_2 = 2 p_1 q_1 = 2*241*44 = 21208#