2 / 1 = 2
(2 + 1) / 2 = 1.5
2 / 1.5 = 1.333333
(1.333333 + 1.5) / 2 = 1.4167
2 / 1.4167 = 1.4118
( 1.4167 / 1.4118 ) / 2 = 1.4142
etc.
The more times you perform the above procedure, the more accurate your answer.

##</code><code>##

#!/usr/bin/perl -w
use strict;

while (<STDIN>) {  # GET INPUT FROM COMMAND LINE
	chomp;
	$_ > 0 && ( /^\d*\.\d+$/ ) && last; # ASK AGAIN UNLESS VALID INPUT GIVEN
	print "Please enter a positive number only!\n";
}
sqroot(1, $_); # PASS INITIAL GUESS OF '1' AND RADICAND TO SQROOT SUB
sub sqroot {
	my ($guess, $x) = @_;
	while ( get_accuracy($guess, new_guess($guess, $x))  > .000000000000000001) { # CHECK DIFFERENCE BETWEEN OLD GUESS AND NEW GUESS
		$guess = new_guess($guess, $x); # MAKE OLD GUESS THE NEW GUESS AND RUN WHILE LOOP AGAIN
	}
	print $guess; # PRINT GUESS IF (OLD GUESS - NEW GUESS) IS VERY CLOSE
}

sub new_guess { # REFINE THE OLD GUESS TO MAKE MORE ACCURATE
	my ($guess, $x) = @_;
	my $x_div_guess = $x / $guess; # DIVIDE RADICAND BY OLD GUESS
	return ($guess + $x_div_guess) / 2; # RETURN, AS NEW GUESS, THE AVERAGE OF OLD GUESS AND RADICAND DIVIDED BY OLD GUESS
}

sub get_accuracy { 
	my ($guess, $newguess) = @_;
	return (abs ($guess - $newguess)); # COMPARES HOW CLOSE OLD GUESS AND NEW GUESS ARE
}