A sequence of approximations 
to 
can be derived by factoring
 |
(1)
|
(where 
is possible only if 
is a quadratic residue
of 
).
Then
 |
(2)
|
 |
(3)
|
and
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Therefore, 
and 
are given by the recurrence relations
 |  |  |
(7)
|
 |  |  |
(8)
|
with 
.
The error obtained using this method is
 |
(9)
|
The first few approximants to 
are therefore given by
 |
(10)
|
This algorithm is sometimes known as the Bhaskara-Brouncker algorithm, and the approximants are precisely those obtained by taking successive
convergents to the continued
fraction of 
.
The fact that if 
is an approximation to 
, then 
is a better one (the 
case) was known to Theon of Smyrna in the second century
AD (Wells 1986, p. 35).
Another general technique for deriving this sequence, known as Newton's iteration, is obtained by letting 
. Then 
, so the sequence
 |
(11)
|
converges quadratically to the root. The first few approximants to 
are therefore given by
 |
(12)
|
Wolfram's iteration provides a method for finding square roots of integers using the binary representation.
