we know that values of f(a) and f(b) above
must be nonzero and each of different sign. That is, if f(a)
is negative, f(b) must be positive. Likewise, if f(a)
is positive, f(b) must be negative. In light of the Intermediate-Value
theorem, the ``intermediate-value'' that we seek is zero, which certainly
lies between the positive and negative numbers represented by f(a)
and f(b). The conclusion of the Intermediate-Value theorem
is that there is a point, 
, between a and b where 
. That is, the solution to f(x)=0 lies somewhere between
a and b.
The bisection method checks the midpoint, c, of the
interval [a,b]. If f(c) = 0, the task of finding
a root is complete. If 
, then for the same reason, there must be a root in either [a,c]
if f(a)f(c) < 0 or on [c,b]
if f(c)f(b) < 0.
This procedure defines the bisection method. At each
step the interval in which there is guaranteed to be a root of the equation
is halved (bisected), and the method terminates as soon as the width of
the interval containing the root is less than some error tolerance 
. Since the bisection method keeps a bounded interval where there is at
least one root at each step, it falls in the category of bracketing
methods.
A algorithmic definition of the bisection method is as follows:
