A Special Continued Fraction for Square Root

Note: more than any other idea or feature of Ted's World, it is the following discovery and a desire to share it that prompted me to set up a website in the first place.  Subsequently I have found other web pages featuring this method; but I believe that you will find this one easiest to read.

Simple Elegance

The square root of any positive number can be expressed as a continued fraction.  The well-known Babylonian equation is one of those:

The approximation of the root is e.  To calculate sqrt(27) with an estimate of 5:

For the second iteration, 5.2 would become the new e, and so on.  This procedure is described on a multitude of web pages; so I'll not elaborate further.  Equally many sites also discuss other continued fractions, detailing exotic constructs such as this approximation of sqrt(19):

This is indeed an interesting pattern, having a "period" of six iterations.  Much less frequently mentioned is this fact:

• Any square root can be expressed as a continued fraction with a period of one.

For reasons unclear to me, most demonstrations of continued fractions feature a numerator of 1.  Well, that restriction might be the natural order of things for the mathematical highbrows; but it is not the only valid setup.  Reducing our own CF to something useful involves just a simple adjustment to the Babylonian equation:

Using this arrangement, if e is the greatest integer which square is less than n, then the iterations approximate just the fractional, or decimal portion of the root:

Now that is elegant!

In the example, 4 happens to be the integer closest to the root, but in fact e can be any positive value.  The following continued fractions all represent sqrt(19):

Notice that e does not have to be less than the actual root.  If it is greater, then the continued fraction will have a negative numerator.

When e is less than sqrt(n), then successive approximations oscillate around the root, starting below it; otherwise, the series converges downward toward the root:

The iterations are quadratic; that is, the relative accuracy doubles with each loop.

When the digits in a continued fraction never change, calculating its value becomes a simple procedure.  This fact is put to good use on these other pages: