State of the System: a real number called y below.
Rule of Evolution: 
.
Motivations: This method provides numerical solutions of the
equation 
. Suppose that 
gives an approximate value of
, you can try to find a better approximation 
.
To find 
, you have to solve 
. If you neglect the 
in this equation
and solve for you
obtain the evolution rule given above. Incidently, this was invented by Newton.
Main Result: If 
is a positive number, approaches
rapidly the positive solution of , namely y=1.41421.. .
This solution is a fixed point of the evolution: if 
exactly, then 
, because
at each iteration.
Important Concepts: Fixed Point, rational number, irrational number.
Rational numbers are ratios of integers. If is a rational
number, then 
is also a rational number. However, the sequence 
converges to a fixed point which is not a rational number
and is called an irrational number. We call the limit
of the above sequence.
Exercise 1.1:
Starting from a rational value 
for
, show that
is also a rational number.
In other words, write as the ratio of two integers which can be expressed
in terms of 
and 
.
Using this result, explain why if is positive, then is also
positive.
