Functions as Iterators

Underlying the Babylonian technique for approximating the square root of k is a rulefor getting from x_i to x_i+1. This rule can be described verbally as:

"calculate the average of [the last guess] and [k divided by the last guess]."

However, a better way to describe this rule is to use functional notation. The function F defined by

embodies the same rule, but F applies this rule to x rather than [the last guess].

Using such functional notation, the Babylonian technique allows a very concise formulation:

.

Because it embodies an iterative process, we will refer to this particular function F as aniterator for approximating the square root of k.

With such an iterator at hand, we will be able to give the Babylonian technique a useful geometric representation. But before turning to this task, it is useful to note that a little algebra allows us to write this iterator as

.

Exercise. For k = 2, the iterator F(x) can be written as the sum of x/2 and 1/x. By sketching the graphs of y = x/2 and y = 1/x and then adding ordinates, obtain a graph of "the Babylonian square root iterator" when k = 2.

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