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A table of continued fractions terms for square roots

Everyone knows what a fraction is in arithmetic, it is one number divided by another. Some examples are Usually each number is whole number, also called an integer. In algebra we have fractions like

A continued fraction is a special form of fraction that looks like

The usual circumstances under which continued fractions arise require that all of the numerators be 1 and that all of the terms (3, 7, 15, 1, 292 in the first example) be integers greater than zero. (Continued fractions might better be called continued reciprocals since all of their numerators are 1.) There may or may not be an initial term; in the first example it is 3 while the second does not have one. Note that the fraction part of a continued fraction (all but the first term if it exists) will always have a value which is less than 1. (Since everything under the first fraction bar is greater than 1, its reciprocal will be less than one.)

The three examples given above are called finite continued fractions because they stop after a certain number of terms. While there is a certain symmetry and structural attractiveness to continued fractions, they are devilishly difficult to perform calculations with. The idea of trying to add or multiply two continued fractions to get a third continued fraction is hopeless.

While continued fractions are not good for arithmetic, they are a very informative way of representing numbers which are not integers, and also in addressing certain problems in the theory of numbers. Just as there are finite continued fractions, there are also infinite continued fractions. An infinite continued fraction has a non-ending sequence of terms. Most interesting continued fractions are the infinite kind. From this point on, we will use the name "continued fraction" to mean an infinite continued fraction. If we ever need to refer to a finite continued fraction, it will be specifically described as such.

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Perhaps you are wondering what what is happening here. First, continued fractions are hopelessly difficult to do arithmetic with. Now we say that the interesting ones have an infinite number of terms. Things seem to be going from bad to worse.
Trust me. If this weren't interesting, no one would bother with it.

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To address the awkwardness of writing continued fractions, we will adopt a short hand notation. In this new notation, the first example above would appear as the second as and the third as This last form using numbered subscripts will often be the most convenient.

Coming next: Depth and Transfiguration. Thrill to the elevation of a common decimal number to the highest social standing: a continued fraction. Stand back in awe as the unexpected becomes commonplace.