How do you approximate #-sqrt30#?
Let's forget the
So the answer must be between
My first approximation would be
The calculator answer is
Use Newton Raphson method to find a sequence of approximations for
The Newton-Raphson method, specialised to finding square roots boils down to choosing a reasonable first approximation
In our case let us choose
Notice that the number of significant digits roughly doubles on every iteration.
Actually, if we want to use rational approximations,
it is nicer to write
That avoids some of the messy fractions.