The Staircase Method

Given a graph of the iterator

we are able to give a geometric representation of the iterative scheme used to approximate the square root of 2. Starting with a (somewhat unlikely) guess of 4, the Babylonian technique can be written as

It is readily verified that this iterative scheme leads to

By referring to the graph of the iterator F(x), we obtain these numerical values as ordinates - i.e., 2.25 = F(4), 1.56... = F(2.25), 1.42... = F(1.56...), etc.

Another way of locating the points (4, 2.25), (2.25, 1.56...), (1.56..., 1.42...) , etc. is to include the line y = x in the above graph of y = F(x).

Now, instead of marking off values on the x-axis and then measuring the lengths of the corresponding ordinates, our "staircase diagram" represents a path from (4,0) to (4,2.25) to (2.25,2.25) to (2.25,1.56...) to (1.56...,1.56...) to (1.56...,1.42...) to ... .

Aside from making things more convenient, use of the line y = x leads to an important observation: The square root of 2 is a fixed point of the iterator F - i.e., a value of x for which F(x) = x.

.

For information about how a graphing calculator can be used to implement such iterations and draw the corresponding staircase diagram, click TI-82.

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