Ted's Math World - A Better Continued Fraction for Square Root

$Ted's Math World$

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:

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 next 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, virtually all references to 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:

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

Now that is elegant! It also can quite useful, as shall be seen.

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: